# Full text of "Kinetic Theory Of Gases With An Introduction To Statistical Mechanics"

## See other formats

INTERNATIONAL SERIES IN PHYSICS LEE A. DuBRIDGE, Consulting Editor KINETIC THEORY OF GASES The quality of the materials used in the manufacture of this hook is governed by continued postwar shortages . INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS G. P. Hartwell, Consulting Editor Bacher and Goudsmit— ATOMIC ENERGY STATES Bitter— INTRODUCTION TO FERROMAGNETISM Brillouin — WAVE PROPAGATION IN PERIODIC STRUCTURES Cady— PIEZOELECTRICITY Clark— APPLIED X-RAYS Curtis- ELECTRICAL MEASUREMENTS Davey— CRYSTAL STRUCTURE AND ITS APPLICATIONS Edwards— ANALYTIC AND VECTOR MECHANICS Hardy and Perrin— THE PRINCIPLES OF OPTICS Harnwell— ELECTRICITY AND ELECTROMAGNETISM Harnwell and Livingood — EXPERIMENTAL ATOMIC PHYSICS Houston— PRINCIPLES OF MATHEMATICAL PHYSICS Hughes and DuBridge— PHOTOELECTRIC PHENOMENA Hund— HIGH-FREQUENCY MEASUREMENTS PHENOMENA IN HIGH-FREQUENCY SYSTEMS Kemble— PRINCIPLES OF QUANTUM MECHANICS Kennard— KINETIC THEORY OF GASES Holler—’ THE PHYSICS OF ELECTRON TUBES Morse— VIBRATION AND SOUND Pauling and Goudsmit— THE STRUCTURE OF LINE SPECTRA Richtmykr and Kennard— INTRODUCTION TO MODERN PHYSICS ' Ruark and Urey— ATOMS, MOLECULES AND QUANTA Seitz— THE MODERN THEORY OF SOLIDS Slater — INTRODUCTION TO CHEMICAL PHYSICS MICROWAVE TRANSMISSION Slater and Frank — ELECTROMAGNETISM INTRODUCTION, TO THEORETICAL PHYSICS MECHANICS Smythb— STATIC AND DYNAMIC ELECTRICITY Stratton— ELECTROMAGNETIC THEORY White— INTRODUCTION TO ATOMIC SPECTRA Williams— MAGNETIC PHENOMENA Dr. Lee A. DuBridge was consulting editor of the series from 1939 to 1946. KINETIC THEORY OF GASES With an Introduction to Statistical Mechanics BY EARLE H. KENNARD Professor of Physics , Cornell University ■ ' ^ v * .• LIBRARY, First Edition Fifth Impression McGRAW-HILL BOOK COMPANY, Inc, NEW YORK AND LONDON 193 8 JI/V Lib Copyright, 1938, by the McGraw-Hill Book Company, Inc. PRINTED IN THE UNITED STATES OF AMERICA All rights reserved. This book , or parts thereof, may not be reproduced in any form without permission of the publishers. PREFACE The kinetic theory of gases is a small branch of physics which has passed from the stage of excitement and novelty into staid maturity. It retains a certain importance, however, and an adequate treatment of it in books will always be needed. Formerly it was hoped that the subject of gases would ultimately merge into a general kinetic theory of matter; but the theory of condensed phases, insofar as it exists at all today, involves an elaborate and technical use of wave mechanics, and for this reason it is best treated as a subject by itself. The scope of the present book is, therefore, the traditional kinetic theory of gases. A strictly modern standpoint has been maintained, however; an account has been included of the wave-mechanical theory, and especially of the degenerate Fermi-Dirac case, which has not been written up systematically in English. There is also a concise chapter on general statistical mechanics, which it is hoped may be of use as an introduction to that subject. On the other hand, the discussion of electrical phenomena has been abbreviated in the belief that the latter voluminous subject is best treated separately. The book is designed to serve both as a textbook for students and as a reference book for the experimental physicist; but it is not intended to be exhaustive. The more fundamental parts have been explained in such detail that they are believed to be within the reach of college juniors and seniors. The two chapters on wave mechanics and statis- tical mechanics, however, are of graduate grade. Few exercises for practice have been included, but a number of carefully worded theorems have been inserted in the guise of problems, without proof, partly to save space and partly to give the earnest student a chance to apply for himself the lines of attack that are exemplified in the text. To facilitate use as a reference book, definitions have been repeated freely, I hope not ad nauseam. The chief requisite for convenience of reference, I think, is a careful statement of all important results, with definitions and restrictions not too far away. Ideas have been drawn freely from existing books such as those of Jeans and Loeb, and from the literature; many references to the latter are given, but they are not intended to constitute a complete list. I am indebted also to Mr. R. D. Myers for valuable criticisms, and to my wife for suggestions in regard to style. Ithaca, New Yobk, Earle H, KenNARD. January , 1938. Preface . CONTENTS Page v CHAPTER I Elements of the Kinetic Theory of Gases 1 1. The Kinetic Theory of Matter 1 2. Atoms and Molecules 3 3. Statistical Nature of the Theory 5 4. Gaseous Pressure 5 5. Calculation of the Pressure • 7 6. Dalton’s Law k 9 7. Mass Motion 9 8. Reversible Expansion and Compression 11 9. Free Expansion 10. Isothermal Properties of the Ideal Gas 15 1 1 . Avogadro ’s Law g 12. The Temperature 13. The Thermodynamic Temperature Beale 20 14. The Perfect-gas Law 22 15. Molecular Magnitudes 24 16. Rapidity of the Molecular Motion 26 CHAPTER II Distribution Law for Molecular Velocities 28 17. The Distribution Function for Molecular Velocity 28 18. Distribution Function in Other Variables 30 19. Remarks on the Distribution Function 31 20. Proofs of the Distribution Law 32 21. Molecular Chaos 33 22. The Effect of Collisions upon / 34 23. Velocities after a Collision 35 24. The Inverse Collisions • 37 25. The Rate of Change of the Distribution Function 39 26. The Equilibrium State 40 27. Rigorous Treatment of the Equilibrium State 42 28. Maxwell’s Law 45 29. Use of a Distribution-function in Calculating Averages 48 30. Most Probable and Average Speeds 48 31. Mixed Gases. Equipartition 51 32. Uniqueness of the Maxwellian Distribution. The //-theorem .... 52 33. Reversibility and the //-theorem 55 34. Principle of Detailed Balancing 55 35. Doppler Line Breadth 58 vii CONTENTS Paoh V1U CHAPTER III Genbbal Motion and Spatial Distribution of the Molecules 60 Unilateral Flow of the Molecules 30 Effusive Molecular Flow 60 37 Formulas for Effusive Flow 01 38 Moleoulai Effusion 64 30 Thermal Transpiration , 00 40 Knuds en’s Absolute Manometer 67 41 Evapoiation 08 42 Observations on the Rate of Evaporation 70 43 Test of the Velocity Distribution in Effusivo Flow 71 The General Disti lbution-function 44 A Gas m a Force-field 74 46 Density m a Force-field 74 46, Maxwell's Law m a Force-field 76 47 The Tompoiature of Satin ated Vapoi 70 48 The Terrestrial Atmospheic 70 49, Cosmic Equilibnum of Planetaiy Atmospheres 81 60 The Gene ial Distribution-function 85 61 Differential Equation for the Distribution-function , 86 62 Applications of the Differential Equation , 80 The Boltzmann Disti lbution Formula B3 The Classical Boltzmann Disti lbu lion Formula 00 64 The Boltzmann Foimulft m Quantum Theoiy 03 66 Special Cases of the Bolt/mann Formula 06 Free Paths and Collisions 56 Molecules of Finite Sizo 07 67 The Mean Ficc Path and Collision Rate 08 68 Dependence of L and O upon Density and Tomporaluio t 100 69 Disti lbution of Ficc Paths, Absoiption of a Roam 101 00 The Mutual Collision Cioss Section 103 61 Tho Moan Fice Path in a Constant-speed Gas 105 62, A Molocular Beam in a Maxwellian Gas 107 03 Mean Free Path and Collision Rato at Constant Speed 100 64 Moan Free Path and Collision Rato m a Maxwolhan Gas 110 65 Magnitude of the Correction foi Maxwell's Law 113 00 Modo of Detei mining L and S or <r 113 07 Collisions m a Real Gas , 113 Molecular Scattering 08, The Scatteung Coefficient , , 116 00, Clasaioal Scattering Coefficient for Symmetrical Molocules with Fixed Soattoicr , ,110 70 Examples of tho Scattering Coefficient 118 71 Relative Scattoung 120 72 Classical Scattering Coefficient for Fico Symmetrical Molecules 122 73, Tho Experimental Determination of the Collision Cioss Sootion 124 74, ICnauor's Observations on Scattering , f , 126 76* The Wave Mechanics of a Particle , 127 70, Tho Indotcrmmalion Pimciplo 130 77, Wave Mechanics and Molecular Collisions 181 78. Wavo-moohamcal Scattering Coefficient , t , f , , t CONTENTS ix Paob CHAPTER IV Viscosity, Thermal Conduction, Diffusion t 135 A Viscosity 70 Viscosity , 136 80 Fluid Stresses m Gonoral , , 137 81 Simple Theory of Viscosity 138 82 Tho Menu Freo Path across a Fixed Plane 141 83 Correction for tho Velocity Spiead M2 84 Further Correction of tho Viscosity Foimula 14f) 86 Now View of tho Moloculai Procoss 140 86 Final Viscosity Formula Magnitudes of L and <r 147 87 Variation of Viscosity with Density 148 88, Variation of Viscosity with Tompoiatuio 160 89 Viscosity and Tompoiatuio with an Tnvoiso-powci Foico 162 90 Viscosity and Tompoiatuio on Sulhoi land's Ilypothosis 164 91 Viscosity and Tomporatuio Othoi Hypotheses, 167 92 Viscosity of Mixed Gasos 160 B Conduction of Heat 03 Tho Kinetic Theory of Heat Conduction 162 94, Simple Theory of tho Conductivity 1C3 96 Thermal Conductivity of Symmetrical SmaH-fiokl Molecules First Step 106 96, Thermal Conductivity on Moyer's Assumption 168 07 Thermal Conductivity Second Stop 109 08 Effect of Ono Collision upon Zv x v* 172 99 Average of tho Effect on 173 100 Total Effect of Collisions on XvxV* 176 101, Thermal Conductivity Final Appioximato Foimula 177 102 Final Collection of tho Conductivity Formula 179 103, Comparison with Observed Conductivities 180 104, Conduction of Heat by Complov Moloculos 181 106 Properties of tho Conductivity 182 O Diffusion : 106 Diffusion 184 107, Tho Coefficient of Diffusion 186 108 Simple Theory of Diffusion 188 109 Approximate Coefficient of Diffusion for Spherically Symmetrical Moloculos ♦ , , 190 110 Solf-diffuBion , , , 194 111 Tho Corrected Diffusion Coefficient , 196 112, Experiments on tho Variation with Composition 190 113 Diffusion at Various Pressures and Tompoiaturas , 197 114 Numerical Values of the Diffusion Coefficient 109 116, Forcod Diffusion , , , , 201 116 Thormal Diffusion 204 CHAPTER V Thh Equation of Si'atb 206 117, Tho Equation of Stale 206 118, Tho Equation of van dor Waals , 200 110 The van dor Wiuils Tsothormals , 208 X CONTENTS Paow 120 Quantitative Tests of van dor WaalB* Equation 210 121 More Exact Theory of the Pressuic in a Douse Gag 2U 122 Hard Attracting Spheres The Repulsive Picssuro 214 123 Equation of State for Haid Attracting Sphoies 217 124 The Value of & , 218 125 Ofchei Equations of State 218 12G Scries for pV, Vmal CoofTioionts 221 127* The Second Vmal CoofTicicnfc 222 128 The Second Vinal Coefficient and van dor Wftnb' Equation 223 129 Thcoiy of the Socond Vmal Coefficient, B 225 130 Nature of Molcculai Forces 228 131 B with an Inverse-power Force 220 132 Classical Calculations of B 230 133 Calculations of B by Wave Mechanics 232 134 B for Mixed Gases , 234 13&, The Vinal Theorem , 235 CHAPTER VI Enbugy, Enthopy, and Specific Heats , 238 Infoimation Obtainable from Thoi mo dynamics 136 Some Definitions and Basio Principles 239 137 Differential Equations foi tho Enoigy U « 240 138 Experimental Measurement of Enoigy and Enliopy, tho Specific Heats 242 139 Speoific-hoat Relations 244 140 Variation of tho Specific Heats 245 141 Thermodynamics of Perfect and van doi Wanls Gases 240 Specifio Heat of tho Perfect Gas 142 Molecular Energy 248 143 The Classical Theory of Spool Pic Heat 249 144 Comparison with Actual Specific Heats 251 145, The Spcoifio-heat Difference ■ 254 146 Tho Pioblcm of tho Internal Enoigy 254 147 Quantum Theory of tho Spoeific lloat 256 148 Variation of Specifio Heat with Tomporaturo , 267 149 Tho Case ofllai monic Oscillators 258 160 Hydrogen , 260 161 Para-, Ortho-, and Equilibrium Hydiogon 262 162 Specific Heat of Hydrogen , , , 204 163 Specific Heats of Mixed Gases , , , 4 205 CHAPTER VII Fluctuations , , , v 267 Phenomena of Dispersion* 154 Tho Simple Random Walk 208 155 Tho Vaiied Random Walk , 271 150, Dispersion of a Gioup of Molecules , 272 157 Molecular Scattering of Light 273 Fluctuations about an Avciago 158 Theory of Fluctuations about an Avoiage 275 159, Examples of Molcculai Fluctuations , , , , „ 270 CONTENTS xi Paqb Diffusion and the Brownian Motion: 100, Tho Brownian Motion 280 161. Theory of tho Brownian Motion 281 162, Observations of Brownian Motion 284 163, Diffusion as a Random Walk 280 164. Brownian Motion under External Force 287 CHAPTER VIII Properties op Gases at Low Densities 291 Motion in Rarefied Gases : 166. Viscous Slip. . 292 100, Steady Flow with Slip 293 167. Maxwell’s Theory of Slip 295 168. Discussion of the Slip Formula 296 169. Observations of Slip 208 170. Frco-molcculo Viscosity 300 '171, Freo-inoloculo Flow through Long TuboB 302 172. Tho Long-tube Formula 304 173. Flow through Short Tubes 300 174. Observations of Frce-molooulo Flow 308 175. Stokes’ Law for Spheres 300 Thermal Conduction in Rarefied Gases: 176. Tomperaturo Jump and tho Accommodation Coefficient 311 177. Theory of the Tomperaturo Jump 312 178. Frce-molooulo Heat Conduction between Platos 316 179. Frce-molooulo Conduction between Coaxial Cylinders 318 180. Observed Variation of tho Accommodation Coefficient 320 181. Magnitude of tho Accommodation Coefficient 322 182. Spectral Emission by an Unequally Heated Gas 324 183. Theoretical Calculations of tho Accommodation Coo (Tie lout 325 Thormal Croop and tho Radiometer: 184. Thormal Croop 327 185. Tho Creep Velocity . . i . 328 186. Thormal Pressure Gradients and Transpiration , 330 187. Thermal Gradients at Modomto Pressures 381 188. Tho Radiomotor and Photophoresis . ’, 333 189. Tho Quantitative Theory of Radiometer Action 336 CHAPTER IX Statistical Mechanics . , 833 190. Naturo of Statistical Mechanics 338 A. Classical Statistical Mechanics: 191. System Phase Space 830 192. Representative Ensembles 340 193. Tho Ergodio Surmise 341 194. Liouvillo’s Theorem 343 196, Tho Ergodio Layer and the Microoanonical Ensomblo 344 190. Tho Point-mass Per foot Gas. . . 3fl6 197, Tho Molecular Distribution, Molecular Chaos 848 198, Tho Loose Many -molecule System . .... 350 xh CONTENTS Pacib 199 The Most Pi obablo Distribution 352 200 The Most Probable ns a Normal Distribution 354 201 Some Generalizations of the Looso Many-moleculo System 355 202 Inti oduotion of the Toinperatuio 357 203 Entropy 360 204 Entiopy of tho Monatomic Gas 361 205 The General Boltzmann Distribution Law 362 206 Tho Equip aitition of Enoi gy , 364 207 Tho Canonical Distnbution and Ensemble 366 208 Entropy uncloi a Canonical Distribution 367 209 Tho Second Law of The l mo dynamics 367 210 Entiopy and Probability 368 211 Relations with Boltzmann's II 371 212 Entropy as a Measure of Range in Phase 371 213, Relativity and Statistical Thcoiy 372 B, Statistical Wave Mechanics 214 Tho Wave-mechanical Descnption 373 216 Tho Exclusion Principle 375 210 Tho State of Equilibrium 375 217 A Pi ion Probabilities ♦ 377 218 Tho Mnuy-moleoulo System without Intel action 370 219 Foimi-Dnno and Bose-Emstom Sots of Similar Molecules 380 220 Tlio Loosoly Coupled Many-moloculo System 382 221 Statistics of tho Loose Many-moleoule System 382 222, Inti oduotion of the Temper alui o 385 223 Case of Largo Enoigics Classical Theory aB a Limit Form 387 224 Entiopy of a Loose Many-moloculo System , 388 226 Statistics of Mixed Sys terns 380 220 Tho Canonical Distribution in Wave Mechanics 390 227 The Entiopy 301 CHAPTER X Wave Mechanics of Gases 303 228 The Porfoct Gas in Wave Mechanics 303 229 Tho Po ml-mass Poifect Gas , , 305 230 Tho Two Types of Pomt-masa Gas 397 231 The IIomogoncouB Point-mass Gas in Equilibuum 307 232 Tlio Appioach to Classical Bohavioi 401 233 Tho Numbci of States 402 234, Tho Zo 10 -point Entiopy 406 236 Chemical Constant and Vapor Piessuio 407 236 The Eormi-Diiao Gas of Pomt Masses 409 237 Tho Degenoiato Fonm-Dnn© Gas 412 238 Tho Boso-Einstem Gas of Point Masses 416 239 Complox Gases 417 240, Fieo Elections m Metals , 419 241, Dogonoi acy in Actual Gases 421 242 Dissociation « 422 243 Dissociation in the Classical Limit . 425 244 A Gas Not m Equtlibuum . 427 CONTENTS xiii Pao» CHAPTER XI Electric and Magnetic Properties op Gases 432 Tho Dielectric Constant: 24B. Polarization and the Dielectric Constant 432 246. The Local Electric Field . . . . ' 433 247. Tho Moan Molecular Moment g 436 248. The Molecular Polarizability a 430 249. Tho Clausius-Mossotti Law 437 260. Polarization Duo to a Pormanont Moment 438 261. Behavior in Intense Fields 441 262. Tho Variation with Temperature 441 263. Quantum Theory of Polarization 442 264. Initial Polarizability by Perturbation Theory 444 265. Wave Mechanics of the Dumbbell Molecule 448 266. Polarizability of a Dumbbell Molecule with a Structural Momont. . 460 Magnetic Susceptibility: 267. Magnetism and Molecular Magno inability 4B4 268. Langoviii’s Theory of Paramagnetism 466 259. Quantum Theory and Magnotizability 466 260. Sources of Molecular Magnetism 467 261. Wave Mechanics in a Magnetio Field 468 202. Magnotizability by Perturbation Theory 460 203. Diamagnetism 462 204. Paramagnetism 403 Motion of Electricity in Gases: 206. Tho Motion of Ions in Gases 404 200, Tho Mobility 400 207. Mobility of Heavy Ions 467 268, Tho Mobility of Free Electrons 409 209, Elementary Theory of Electronic Mobility , 470 270, Actual Electron Mobilities 473 Some Integrals 477 Important Constants 478 Index 479 KINETIC THEORY OF OASES CHAPTER I ELEMENTS OF THE KINETIC THEORY OF GASES Of all states of matter tho simplest is the gaseous state. The laws of this state were discovered long ago; and during the middle part of tho last century the essentials of an adequate theoretical interpretation of these laws were worked out, This theory, of course, constitutes only a part of a general kinetic theory of matter, which is the ultimate goal. It has become clear, however, that tho theory of liquids and solids must necessarily involve an extensive uso of quantum mechanics, and such a theory is as yet only in its beginning stage. Since tho methods and conceptions required to handle these condensed phases of matter are thus widely different from those that are appropriate to tho treat- ment of the gaseous phase, it is still convenient to treat tho theory of gases by itself as a distinct subdivision of physics. The purpose of the present volume is to give a concise account of the kinetic theory of gases as this theory exists today, and of its princi- pal applications to tho results of experiment, 1, The Kinetic Theory of Matter. The structure of matter is a very old problem, and for the beginnings of modern theory wo must go back to the early Greek philosophers. Driven by their basic urge to see in all phenomena tho operation of a few fundamental principles, these philosophers sought to reduce to simpler terms tho immense variety of natural phenomena by which they were surrounded. Hera- clitus, about 500 b.c., played a large part in initiating one of the princi- pal lines of Greek thought by advancing tho view that everything is composed in varying proportions of the four elements, earth, water, air, and fire; the elements and all mixtures of them ho assumed to bo capable of unlimited subdivision into finer and over finer particles without losing their essential properties. On tho other hand, Leucippus and especially Democritus (about 400 B.o.), holding that motion would be impossible unless there existed empty space into which a moving body could move, preferred tho hypothesis that all mattor consists of very small particles separated by void. These particles were supposed to be of many different sizes and shapes and to bo l 2 KINETIC THEORY OF GASES [Oiiap I engaged m continual lapicl motion, the vauous properties of material objects aiismg fiom diffei cnees m the lands of atoms 01 from differ- ences m thcii motion This view was rejected by Anstotle, but in spite of Ins gieat authonty it lcmaincd cunent to some extent, and wo find it developed at gieat length m the poem "De rcrum naluia” by the Roman wntei Lucictius (a d, 55) The kinetic tlieoiy of mattei thus arose m fairly definrlc form among the ancients, but they discovered no pioofs of its tiutli which weie of the convincing quantitative soit so characteristic of modern science. In ancient times and dining the Middle Ages the theory lemamcd nicicly one among scvcial alternative speculations, accepted by some thinkois on the basis of its geneial atti activeness, but emphatically 1 ejected by otheis Aflci the Revival of Learning the tlieoiy underwent a slow giowth as physical conceptions took on more precise fonns The phenomena of heat weie frequently attributed to the assumed motion of the atoms, and Gassendi (1658) lccogmzcd the relatively wido spacing of tho particles in a gas such as air On this basis Hooke (1078) attempted, although in a confused way, to give an explanation of Boyle's law, Tho fust cleai explanation of this law, howovei, including the assump- tion that at constant Lcmpciatiue tho mean velocity of tho particles loniams constant, seems to have been given by D Bernoulli in 1738. The contributions of Bernoulli to the thcoiy might be rcgaulcd as tho beginning of the liuly quantitative kinetic theory of gases Soon aflci 1800, a veiy impoitant development of tho atomic tlieoiy occurred, in the use that Davy and others made oi it to oxplain the law that chemical combination always occuis in simple piopoi Lions. Then, between 1840 and 1850, the woik of Mayei and Joule on the mechanical equivalent of heat, lcinforcing the qualitative but veiy convincing experiments of Rumfoid and Davy thirty years befoie, finally bi ought geneial conviction that heat is not a substance but a hum of energy, consisting, at least in laigo part, of the kinetic energy of motion of the molecules The time was now ripe for the development of a thoroughgoing quantitative kinetic thcoiy of gases, and wo find tho pnncipal frame- woik laid within the next twenty-five ycais (1857-1880) by Clausius, Maxwell, Boltzmann, and others, The lemammg years of the nine- teenth conluiy wero then a ponod of quiet during winch only a few fuithor refinements weie added The twentieth century has scon not only a still fuithor slow growth of tho thcoiy along the linos initiated by Clausius and Maxwell, but also successful applications to now experiments which served finally Sec, 2] ELEMENTS OF TJIE KINETIC THEORY OF GASES 3 to convince the last doubters of its truth. There came first the experiments of Perrin (1908) and others on the Brownian movement, which showed pretty definitely that here we are actually witnessing the eternal dance of the molecules that is postulated by kinetic theory, the suspended particles playing the role of giant molecules. Then followed a number of phenomena in which detectable effects are pro- duced by one atom or molecule at a time, such as the scintillations made by alpha particles upon a fluorescent screen, or Ijhe click in a Geiger counter recording the passage of a single electron; and finally, during the last twenty years, there has come the overwhelmingly successful quantum theory of the internal structure of the atom itself, At the present time the atomic theory of matter and, as a special case, the kinetic theory of gases, is perhaps second only to the Copcrni- can theory of the solar system in the completeness of its experimental verification, Today we know the mass and size — in so far as it possesses a size! — of an atom of any chemical element with the same certainty, although not yet the same degree of precision, as the astrono- mer knows the mass and size of the sun. 2. Atoms and Molecules. One of the most important advances in atomic theory was the distinction introduced a hundred years ago between atoms on the one hand and molecules on tho other. This distinction has lost a little of its sharpness of late, but wo can still say with substantial truth that an atom is the smallest portion of matter which 1ms the property of remaining essentially intact in every chemical reaction, whereas a molecule is the smallest portion which possesses the chemical properties of a definite chemical substance, Until some thirty years ago, atoms were commonly described as indivisible, and tho fact that nowadays wo are breaking them up rather freely is sometimes hold up as an example of the mutability and hence unreliability of the results of science. Such a view rests, however, upon a misinterpretation of scientific statements, for which the blame must sometimes bo laid at the door of scientific men themselves, espe- cially the writers of textbooks, Those old statements about indivisi- bility really meant only that the division ‘could not be accomplished by any moans known at that time, and this statement remains (so far as we know!) as true today as it was then; the modern methods of break- ing up an atom require experimental methods that have been dis- covered only recently. Of the internal structure of atoms* we possess today a fairly com- plete theory. All known facts agree excellently with the assumption * A good summary of tho theory can bo found in “Introduction to Modem Physics/ 1 by F. If. Richtmyor, 2d od., 1034, 4 KINETIC THEORY Of OASES tCiu* 1 that each atom consists of a positively chaiged nucleus siu rounded (in geneial) by a cloud of negatively charged elections Under ordinal y conditions the number of elections present is just sufficient to make the whole atom elcclncally neutial Chemical combination then consists in a union of two or moic atoms into a close gioup or molecule, one or moio of the outer elections of each atom changing position, m many oases, so as to belong xatlier to the molecule as a whole than to any pai ticulai atom m it In the gaseous state of matter the molecules aie scpai ated most of the time by distances that aie lathei laige in com- panson with their diametois (about nine times, in ail of normal density) In liquids and solids, on the othei hand, the spaces between them tend to be even smallei than the molecules themselves, and in many ciystal lattices (c g , potassium chloride) the anangement is such that the identity of the molecule seems to become entholy lost — * such a ciystal is xeally itself “one big molecule” (Biagg), In a gas we should expect, accoidingly, that the molecules would move about ficely dining most of the time; they should move almost in stiaight lines until two of them happen to come so close together that they act stiongly upon each othei and something like a “colli- sion” occius, after which they sepaxato and move off in new directions and probably with different speeds Just what happens in a collision must depend on tho laws of moleculai force-action As the gas is made raier, however, the collisions must become less frequent, and the intervening free paths longer; then the details of the process of collision will be less impoilant and only the resulting changes of molecular velocity will be significant Thus wc are led to idealize the gas foi the pin pose of making the fiist stops toward a thcoiy, as was done almost unconsciously by the earliest thoomers, For oui first deductions we shall assume an ideal 01 perfect gas in which the moleculos aid negligibly small, 3 o , cxoit appreciable forces upon each other only when their conleis of mass appioach within a distance that is very small compaiod with their avoiage scpaiation m the gas, In latoi chapters we shall then endeavor to soften, and if possible to remove, this restriction Wc assume, of course, that the molocules obey tho laws of mechanics For phenomena on tho moleculai scalo these dopaib so far from tho Nowtonian laws as to raise leal doubt whether tho conceptions of classical kinetic thcoiy are applicable to actual gases at all, Wo shall lctiun to this subject later (Chap III, end, and Chap X), it appeals that a theory based upon classical mechanics is novel theloss practically correct foi tho ideal, indefinitely laio gas at high temperatiu e Such a theoiy still possesses, thcicfoio, gieat usefulness as a first appi oximation to tho tiue quantum-mechanical theory, which is decidedly complicated, Sec. 3] ELEMENTS OF THE KINETIC THEORY OF OASES 5 Heat energy we shall, of course, interpret as mechanical energy of the molecules. At least part of it will be kinetic energy of translation of the molecules as wholes, but there may also be kinetic onorgy of rotation; if the atoms constituting the molecule are capable of vibratory motion relative to each other, there will furthermore be internal energy of vibration; ancl finally there may bo energy of motion of the electrons in the atom, or rather its quantum-mechanical equivalent. Finally, corresponding to the forces that act between the molecules when they approach closely, there will bo in greater or less degree a storo of inter- molecular potential energy; in liquids and solids, in so far as we can employ the classical picture at all, this part of the energy must be large. In our thinking wo must not forgot that all of those various forms are included in the “heat energy" of the substance and contribute more or less to its specific heat. Friction wo assume, of course, to be entirely absent in the interac- tion of the molecules themselves; these constitute conservative mechan- ical systems. The ordinary conversion of mechanical work into heat by friqtion we interpret morely as a conversion of mechanical energy into molecular forms whose oxact nature can no longer be recognized by the ordinary methods of experimental physics; in the case of gaseous friction the mechanism by which this conversion is effected is easily followed and will bo discussed later, The motion of a visible body represents an organized component in the motion of the molecules; after the body has been brought to rest by friction, thoro may be, especially in a rare gas, just as much motion as there was before it was stopped, only now the motion formerly visible 1ms become part of the completely disorganized heat motion and so is no longer perceptible as motion at all. , 3. Statistical Nature of the Theory. Since wo cannot possibly follow every molecule and calculate its exact path, wo must in kinetic theory be content almost entirely with statistical results, and those are ip all cases sufficient for practical purposes. By the density of a gas, for instance, what wo really mean is the ratio of mass to volume for a maeroscopically small volume, i.o., a volume just small enough so that for the experimental purpose in hand it can bo treated as indefinitely small. If the gas is dense enough so that such a volume contains many molecules, the density as thus defined will vary only to a negligible extent as individual molecules enter and leave the volume. If the density becomes too low for this condition to hold, wo may still bo able to secure a sufficiently steady macroscopic density by averaging the number of molecules in the volume over a maeroscopically small interval of time, during which individual molecules pass into and 6 KINETIC THEORY OF GASES [Chap, I out of the volume many times, or we may turn our attention to tho Jlticluaiions of the density caused by the megular molecular motion When all such devices fail, the common methods of kinetic theoiy simply become inapplicable and wo aio compelled to tieat the molecules as individuals, A similai discussion applies to all other magnitudes associated with a gas (01 with any other physical body, foi that mattei), such aspicssure 01 tcmpei atuie These conceptions aie all statistical m natuie, and the relations between thorn that aie expicssed by our formulas, while they may be mathematically exact, lepiesent the ical situation only with a ccitain degiee of appioximation 4 . Gaseous Pressure. The most characteiistic property of a gas, as contrasted with a liquid or solid, is its tendency to expand indefi- nitely or, if confined, to exert a positive piessuio upon the walls of tho containing vessel, This propeity is mtci preted in the kinetic theoiy as aiismg fiom the continual motion of the molecules, and on this basis a quantitative expression for the prcssinc is easily obtained In piacticc, however, it is also convenient to think of any poition of tho gas as exoilmg pressure upon contiguous poitions of tho gas itself To cover this case, a definition of the pre&siuo is convenient in which * the latter is loprcsonted as stream density of momentum. If an lmaginaiy plane surface is drawn thi ough a mass of matter, momentum is continually being transmitted acioss this suiface in both directions, cither by means of forces oi by being earned acioss it by molecules which themselves actually cioss tho suiface Let us choose a ccitain cliicction noimal to tho suiface as the positive one and take only tho component of the momentum m this dncction, Then tho pressure acting acioss tho surface can be defined as tho net i ate at which momen- tum noimal to it is being tiansmitted acioss it per unit area in tho positivo direction, momentum tiansmitted in tho opposite dncction being counted as negative, Wc shall show fust that in a gas in equilibnum tho piessure so defined is the same in the midst of tho gas as tho pressure on tho walls of the vessel Tho piessuio upon a rigid wall arises fiom forces cxoitecl upon it by those molecules which happen at any instant to be undergoing collision with it If &F n denotes tho sum of the components noimal to the wall of all such foiccs acting upon a macroscopically small element of area dS , then the piossuic is $F n / 8 S (or the time aver- age of tins expression ovci a macioscopically shoit time). These forces can be supposed to bo impaitmg normal momentum to the wall at a late equal to the piessure; if the wall nevertheless stands still, that is only because other foiccs acting simultaneously from without impail Sec. 6] ELEMENTS OF THE KINETIC THEORY OF OASES 7 to it momentum in the opposite direction at an equal rate. Corre- sponding to SF n there are also, by the law of action and reaction, equal and opposite forces exerted by the wall upon the gas molecules, and these impart equal and opposite momentum to the gas. Suppose now we draw an imaginary plane surface S through the gas parallel to the wall. Then, gas and wall being assumed at rest, tho amount of momentum possessed at each instant by those molecules that lie botwcon the surface S and the wall remains constant. The con- tinual inflow into this region of momentum directed away from tho wall, due to tho action of the wall upon tho gas, must therefore bn balanced by an equal inflow across S of momentum directed toward the wall. Thus the pressure of the gas lying beyond S upon the layer of gas inside of S, defined as stated above, must be oqual to tho pressure of the gas upon tho wall. Tho pressure can, therefore, be calculated as a transfer of momentum between contiguous portions of the gas. 6. Calculation of the Pressure. Consider a maeroseopically small plane of area SS drawn anywhore in the midst of a stationary mass of an ideal gas as defined above (cf, Fig. 1) . In such a gas the only appreciable mechanism for the transfer of momentum is that of molecular convection, the mo- mentum being carried across by molecules which them- selves cross the plane; for wo can neglect the very rare cases in which two molecules lie close enough together, one on one side of S and one on tho opposite, to exert forces upon each other. Now a molecule mpving with speed v in a certain direction will cross SS during a given interval of time dl, provided at the beginning of dl it lies within a cer- tain cylinder of slant height vdt drawn on SS as a base (cf. Fig. 1) ; and if it does cross, it will carry over a normal component of momentum mvx where m is its mass and rj. denotes tho component of its velocity in a direction perpendicular to SS. Tho volume of this cylinder being Vx dt 5 S, the number of such molecules lying within it at tho beginning of dt will bo n v Vx SS dt, n v being the number of molecules per unit volume that are moving in the manner assumed. The total normal component of momentum thus transferred will therefore be (fM'jL &S dl)mvx- (1) All of tho molecules can bo divided into such groups. Let vis take v L to be positive when it has that direction along tho normal to SS which we choose as the positive one. Then molecules with a positive valun of Vx can cross and transfer positive normal momentum mv x in the posi- tive direction across SS; those with negative Vx will carry negative 8 KINETIC THEORY OF OASES [Chap I momentum, but they also cross in the negative diiection, henco their effect upon the momentum in the gas lying on the positive side of SS is the same as if they had ciosscd positively canying an equal amount of positive momentum, and accoidingly the same expression (1) can be used for their conti ibution to the net momentum transfei Hence we get the total transfer of momentum by summing (1) over all groups of molecules in the gas Dividing the lesult thus obtained by dS dt } wc find foi the amount of momentum noimal to 8S transfer! ed acioss it by convection in the positive diiection per unit aiea pel second oi, by definition, tho pressiue, p ~ We can also wnto foi this simply p = Zmvi 2 , (2a) the sum extending over all molecules in unit volume (more exactly, since only molecules near 8S can contribute to the pressure, the sum is to be extended over all molecules in a inaci oscopically small volume and the result is then to bo divided by the volume) ♦ The same result is obtained, of couise, fiom the conventional calculation of the piossure on tho wall The momentum delivered to the wall by molecules falling on it at a certain angle is twice as large as expzession (1) because each molecule has its normal component of velocity reversed; but then the final sum extends only ovoi thoso molecules that aio moving towaicl the wall, which is half of thorn, and the pressiue thus comes out as given in (2a) Equation (2a) holds for any ideal gas at rest, In tho special case in which all molecules have the same mass } we can take out m as a con- stant factor m all toims of tho sum and write p ~ m'Zv i 2 = iwwj} = pvp (26) wheion - total number of molecules per unit volume, p » mn *=* ordi- nary density of the gas, and the bar over a symbol denotes the average of that quantity takon for all molecules m unit volume Let us assume, as wo should expect to be tho case in a gas in complete equilibrium and shall veiify later, that all dhections of motion aio equally probable (cf* the principle of molecular chaos in Chap II), Then, taking tho aj-axis of a set of cartesian coordinates in the direction of tho positivo normal to SS and denoting tho components of the velocity by v X) v VJ we have from symmetry v* = ** 5J, and, since v 2 ~ v 2 *» v* + v 2 + vl — 3ti| - 3^x 2 Ilonco (26) can be written in either of tho foi ms: p « £ MM 2 « i pv 2 ) pV = \ V 2 ) V » 1/p standing for the volume of a gram (3a, b) Sec. 7\ ELEMENTS OF THE KINETIC THEORY OF GASES 0 If tho molecules are of several different kinds with respective masses m< and densities n< in terms of molecules and p; in terms of grams, wo can treat in the preceding way the part of tho sum in (2a) that corre- sponds to each kind and obtain thereby for the total pressure, as an extension of (3a), ' p = = W X i summed over the different kinds of molecules, As an alternative expression, since the kinetic energy of translation of a group of mole- cules of mass m is 2^ mv 2 — \ nmv 2 } by (3a) or (4) we can write for either a simple or a mixed gas P-ftfj (5) where if stands for the total translatory kinetic energy of tho molecules per unit volume. Problem . Show that in a two-dimensional gas p ■= J pifi » K } p being here the number of molecules in unit area and It, as before, their mean translatory kinetic energy, 6, Dalton’s Law. From (4) we have at once the important result that, according to our theory, the pressure of a mixture of two or more perfect gases is simply the sum of tho pressures which they would exert if each occupied the same volume by itself. This is Dalton’s law and is known experimentally to be true at sufficiently low densities; the departures from it are at most of tho same order as the departures from the perfect-gas law for each of tho component gases, becoming noticeable in ordinary gases only when under considerables pressure. For example, mixtures of equal parts of argon and ethylene, of oxygen and ethylene, and of argon and oxygon, when actually exerting a total pressure of 100 atmospheres, according to Dalton’s law should bo exert- ing a pressure respectively 8 per cent, 7,2 per cent, and 1.45 per cont higher; at 30 atmospheres, however, the argon-ethylene mixture shows a deficit of only 0,85 per cent. In some other cases tho departure from the law of additivity of pressure is in tho opposite direction, * 4 7 . Mass Motion, We often speak of a gas as boing “at rest” or as “moving” with a certain velocity, From tho molecular standpoint those statements are obviously to bo understood as referring to tho mean velocity of the molecules, It will be useful to consider at this point the relation between this mean velocity and tho true rapid and irregular molecular motion, * Cf . ^Iabson and Dolley, Roy, Soc, Proc„ 103, 524 (1023). 10 KINETIC THEORY OF GASES [Chap I The symbol v will be used consistently to stand for the actual speed of a molecule lelative to whatevei basic frame of reference is being employed at the time; to denote its velocity both in dhection and in magnitude, 1 e , logarded as a vectoi, we shall use v punted in heavy type, and the tluee components of the velocity lef cried to a set of caiteslan axes wc shall denote by u X) v V) v z , Thus at all times * 2 - Vl + vl + v*. (6) Foi the mean 01 mass velocity we shall similaily wute Vo or Vo; it is defined by the equations Vo - V, 1C, Vox = Vzi Vq v « l V) Voz - v* t (7) the aveiagcs being taken ovei all molecules m a macroscopically small volume sunounding the point in question A gas at lest is then one in which Vo = 0, i o , hx - hu = Vo, - 0* (8) Foi the total velocity we can then write v - Vo + v' (9a) or v 9 — Vox + *4 Vjf = Voy + Vy } V z =» Voz + Vz, (96) the sum in (9a) being a vector sum, the new velocity v' thus defined is called the velocity of theimal agitation and obviously has the piopcrty that always v 7 ~ 0, ie, SJ ~ ~ 0* (10) It is then v' and not v for which all directions aie equally piobablo when the gas is in equilibnum, so that = 7* - W 1 * Conesponding to tins division of the molecular motion into mass motion and motion of thermal agitation, theie exists an important theorem concerning the kinetic eneigy. The tianslatoiy kinetic energy of the molecules m any macroscopically small clement of volume Sr can be wntten mv 2 ~ 2'} m(vl + + vf) - 2'i m(vl x + v\ u + t>&) + 2'Mvq x v' 9 + VoyVy + Vozv'z) + m(v f x 2 + V * v 2 + V **) * In a mixed gas whoso composition vanes from point to point, so that mtoi- cliffusion of its constituents is going on, the mass motion is diffeiont for clifieient kinds of molecules and tho theory requires modification to make it entirely satis- faotoiy, but such refinements seem to ho of no practical lmpoitanco at piesent Seo. 8] ELEMENTS OF TIIE KINETIC THEORY OF OASES 11 by (6) and (9b). Now suppose that the molecules are all alike, or, if different kinds are present, let the mass velocity y<j be the same for all kinds. Then in any sum like 2'wty»*v£ we can take out uo 3 as a common factor and write H'mv txvlt = voxh'mv’x — y 0 gS'wt/SX/, i 2 'y',- representing a sum over all molecules in or of kind no. j and the sum 2' extending over all the different kinds. But by (10), 2'y',- = 0, since the mean value S' for molecules of kind no. j is simply 2'y'^ divided by their number in Sr. Hence, after reasoning in the same way about the y~ and 2 -components, wo have 2'mvoxvl — 2'mvnyvh = = 0. Furthermore, by (6), S'! wi(y<i» + vl u + vl z ) = S'-) mv\ — | (S'm)y*. Thus the expression given above for the kinetic energy reduces to 2'f mv 2 = ! (2'?n)y§ + S'! mv ' 2 , in which each sum extends over all molecules in St. If we now divide this equation through by St and note that H'm/Sr = p, the density, we finally obtaiu for the translatory kinetic energy of the molecules per unit volume s! mv 2 - !pyg + 2 ! mw' 2 , (11) the sums now extending over unit volumo in the sense explained just under oq. (2), Thus the total kinetic energy of translation of the molecules is simply the sum of the kinotic energy due to the obsorvable macroscopic mass motion and the kinetic energy of thermal agitation. Another quantity that requires reconsideration in the presence of mass motion is the pressure. The pressure in a moving gas is best defined as the rato of transfer of normal momentum across a surface that is moving "with the gas," i.e., with a velocity equal to the mass velocity v 0 . Its value is obviously given by (2a), (3a, b) or (4), with v replaced by v'. 8. Reversible Expansion and Compression. The conceptions and the theorem of the last section find an interesting application in the molecular interpretation of those reversible expansions and comprcs- 12 KINETIC THEORY OF GASES [Chap I sions which aic so impoitant in theimoclynamics It may help in forming cleai conceptions of the molecular processes in a gas if we analyze to some extent a case of this soi t Considei, for example, a mass of gas that is being compressed veiy slowly by a moving piston, m a cylinder whose walls do not conduct heat The motion being slow, the gas will be eveiy whole close to equilibrium and moving with a mass velocity a 0 that grades downwaid, fiom a maximum value at the piston equal to its speed, to zeio at the other encl of the cylinder (cf Fig 2), Under these cncumstanccs the piston does woik upon the gas; the amount of this woik as the volume V of the gas deci eases by —dV is, as shown in elementary phys- ics, — p dV f p being the piessuie, and at the same time the gas docs negative work p dV on the piston, Since no eneigy is allowed to leave in the form Tig 2 Adiabatic compression Q f ] lca ^ ^llO W01 ] c clone upOll the gas must iomaui stoicd in it in the foim of an increase in its “internal” oi “mtiinsic” energy, which wo shall call simply the eneigy of tho gas Let us now view this pioccss fiom tho molecular standpoint, Accoiding to tho laws of mechanics the molecules that strike the moving piston rebound fiom it with an inciease in then lunotic energy icpresonting tho woik done on them by the piston. This does not imply an inciease of equal magnitude in tho energy of thei mal agtiafoon of these particular molecules, howovei Suppose, for example, a molecule moving at velocity v stukcs the piston Then just befoie tho impact its Ihcimal velocity, according to (9a), is v' cs v “ Vo whoie v 0) tho mass velocity of the gas at tho piston, is the same as tho velocity of the piston itself, Thus v' is also the veloc- ity of the molecule lolativc to tho piston, and since, accoichng to tho laws of clastic impact, relative velocity undergoes an alteiation only in clixcctiou but not in magnitude, wo see that the molecules that strike the piston do not themselves expencnco any gain m energy of thermal agitation at all l Of couise, it is an observed fact that the heating producod by tho compression under these circumstances is distributed equally through- out the gas. To see how this becomes about, lot us consider first tho flow of translator y kinetic energy of tho molecules acioss any cross section QQ' that is moving with the gas m its mass motion According to the analysis of the last section, tho motion of molecules acioss QQ f will bo determined by their ihcimal component of velocity alone, and tho number ciossing unit area pei second with a thcimal component 13 Sac. 8] ELEMENTS OF THE KINETIC THEORY OF OASES of velocity i>x' perpendicular to QQ' will be Swi/ summed over unit volume, as in the deduction of the pressure. Each molecule that crosses carries with it total kinetic energy mv i ; hence an amount of kinetic energy is carried across unit area of QQ' per second equal to ?2w»iV. Now we can obviously write v 2 = Vx s + an 2 , «n being the component of v parallel to QQ'; and by (9a) vx = »o + «jl', t>n = V> denoting tho mass velocity and won being zero and v 0 ± = in our case. Hence, for tho rate of transfer of energy wo have 4 Evivx'v 2 <=* -|2m»x'(«j. 2 .+ »n 2 ) = v [t>o2mt>j/ + 2a 0 Y,mvx n + Y,mvx'(vx rz + I'd'*)]. Hero Emvx' — 0 by (10), or by the argument used in order to dispose of EmvoxV' in arriving at eq. (11). We should also expect tlvo last sum in tho equation to vanish by symmetry, positive and negative values of vx' occurring equally often for the same value of tho quantity (vx 2 + t>n' 2 ) ; and wo shall find later that this is correct so long as there is no temperature gradient. In tho middle term, finally, Emvx ,s - 'P> the pressure [ef. (2a) and the end of the last section]. Tho expression for tho translatory kinetic energy carried across unit area of QQ' per second thus reduces to pv<> or to tho rate at which, according to ordinary mechanical analysis, tiro gas behind QQ' is doing work on that ahead. Now if wo consider tho mass of gas that lies between two such moving cross sections, as between QQ' and Q"Q'", tho flow of energy will bo greater across tho first than across the second because of the difference in the values of t»o- Kinetic energy, therefore, is accumulating between these two cross sections. Since, however, tho total energy is tho sum of the thermal energy and tho energy of the mass motion, as shown in the last section, and tho mass motion is constant, the increase must occur in tho thermal kinetic onorgy alono, except in so far as this may subsequently pass over into energy of vibration or tho lllco inside tho molecules. To seo in detail just how »', tho thermal part of tho velocity, comes to increase, is a bit tedious, but we can understand it qualitatively if we note that those molecules which mi agio at a given moment in a given region of tho contracting gas have come from neighboring regions whoso relative mass motion was one of mutual approach, and tho molecules thus mingle with higher relative velocities than they other- wise would. Because of simple relations such as those it suffices to develop a large part of tho kinetic theory for a gas at rest. Accordingly, here- after mass motion will bo understood to bo absent unless tho contrary is KINETIC THEORY OF GASES 14 [Chap I specified The extension of the results to moving gases can then be made easily when lequirod Pioblem Show that the total momentum of an element of the gas is that due to the mass motion, 01 , as a vectoi, it is pVo per unit volume, 9. Free Expansion. Quite a different case fiom the preceding is presented by flee expansion, in which a gas is allowed neithei to exchange heat with its aui roundings nor to do external woi ]t The ideal way to perfoi m such an expansion would be to put the gas into one compaitment A of a vessel with a vacuum in an adjoining compaitment B } and then suddenly to open m the partition holes so tiny that the molecules of gas could go through only one by one (of Fig 3) It is very difficult, however, to peifoim an expansion in this ideal mannei, and in practice loughor equivalents must bo substituted Such an equivalent was tiied by Joule in 1845; impiovmg upon an auangement used by Gay-Lussac in 1807, lie simply connected a vessel of air suddenly to an evacuated vessel With this auangement, when the stopcock in the connecting tube is opened and some of the an rushes into the vacuum, the air left behind is cooled greatly by an approximately reversible expansion undoi pressure. If, however, the gas could then be left to itself foi a time without exchanging heat with its suuoundmgs, it would soon come to rest, and eventually, by conduction of heat through the gas itself, it would come to the same uniform tompeiatuio as would have lesultcd from an ideal fieo expan- sion Joule evaded tho difficulty of so thoroughly insulating the gas by sunoundmg both vessels with a water bath and looking foi a change in the tcmpciatuie of the water, which would ccitamly have occunod if theie had been a final net change in the tempoiatuio of the air Being unable to detect any change, ho concluded that the heat of free expansion of ail, l o , the heat that must be added to a gram of it to keep its temperature constant when it is allowed to expand freely, is eithei zero or at least very small, In latei experiments by Kelvin and others* the gas was caused to expand slowly and continuously through a poious plug, such as a wad of glass wool closing a tube through which the gas was forced to flow, and the difference of temper atuie between the gas onteiing and the gas leaving tho plug was noted [tho Joule-Thomson effect, cf Fig, (4)] With this auangement, however, the tiue effect of free 4 Cf, T Pres row, "Heat,” pp 269, 771, Edsejr, “Heat," igv ech, p 376, Fra 3 — Free expansion Sho. 0] ELEMENTS OF THE KINETIC THEORY OF OASES 16 expansion is overlaid by another. As a volume V of the gas enters the plug under pressure p, the gas behind does work pV upon it; as it loaves the plug under a lower pressure p', it in turn does work p'V' upon the gas ahead; but usually p'V 1 is slightly different from pV, and the energy of the gas is thus altered by an amount equal to the negative difference of the two works or by -A (pV) = -{p'V' - pV). ' Fl0 ' 4, ~ Tho porou8 -I )lu e r>rrAngomont. The latter effect can be calculated from known values of pV as a function of the pressure, and so allowance can be made for it. If we may judge from the few cases that have been tried, tho heat of free expansion is always positive, but it is extremely small in the case of tho almost perfect gases, as is also the more complicated Joule-Thomson effect itself. In air at 0°C, for example, the Joule- Thomson cooling at moderate pressures amounts only to 0.26° per atmosphere drop in pressure; in carbon dioxide under tho same conditions it is 1.5°, but in hydrogen there is a heating of 0.03°. It may be of interest to see how from such data wo can calculate the heat of free expansion, and also the temperature drop in an adiabatic free expansion. Suppose a gram of gas enters the porous plug at 0°O and under a pressure of 2 atmospheres, and emerges at a temperature ( ST) jt and a pressure of 1 atmosphore; (8T)jt thus represents the Joule-Thomson temperature change per atmosphore. Now imagine this gas restored to 0°G, but still at a pressure of 1 atmosphere; to do this wo must give it heat -c v {ST) Jr , c„ being its specific heat at constant pressure measured in ergs. During each of these two processes tho gas does external work equal to its change in pV ; hence tho net external work that it has done since entering tho plug is the change in pV as p changes from 2 atmospheres to 1 atmos- phoro at 0°C, which will bo denoted by S (p F) . Accordingly, by conservation of energy the gas has on tho whole gained an amount of energy (measured algebraically) equal to tho boat absorbed loss tho work done or SU — —c p (8T)jt ~ Now the gain in energy depends only on tho initial and final states of the gas. Honce the samo gain would have occurred if wo had allowed tho gas to expand freely, without doing work, into tho samo final volume as it occupied in the first case after being brought back to 0°C. If at tho same timo wo supply enough heat to lcoop its tem- perature at 0°C, this heat will be, by definition, the heat of freo expansion, L,,; and it will also equal the gain in energy. Ilonco, equating tho two values thus found for SU, wo have for tho hoat of freo expansion 10 KINETIC THEORY OF GASES (Oiiap I L p = -c p (8T) jt - 8(pV) (12) ergs pei giara and pei atmospheie diop This equation connects the Joule-Thomson effect with the heat of fioe expansion On the othci hand, in the alternative piocess just described wo might have allowed the gas to expand without supplying any heat, the gas then changing in tempeiatuie by a ceitain amount dT, and then we could have brought it back to 0°O by supplying heat ~c v ST, cr being the specific heat at constant volume. In this case the gas, just after expanding, occupies the volume that it occupies at 0°C and 1 atmospheie, but at a temperatmo 87', Its pressure at that moment can differ only slightly fiom 1 atmospheie, howover; hence, we can lcplace ST appioximately by the drop that occurs m a fico expansion fiom 2 atmospheies to exactly 1 atmospheie, which wo shall denote by (ST),. Then, the change in cneigy being the samo as bofoie, we havo ~c v (ST), = L„ or (8T), = —L p /c V) and fiom (12) (ST), - y\ (STU + ~ S(pV) 1, (8T)„ - i (8T), - I 8(pV), t l p j y g p (13a, b ) where y — c v /cv Now for an ( ST) jt = —0.26 as stated above, 7 = 1 41, op - 0 24 X 4 186 X 10 7 , and 8(pV)/poVo — 0 00060 wheic p«, Vo lefor to standaid conditions, so that 8(pV) «= 0 00060 X 1,031 X 10«/0 001293, honce (ST), = 1.41 (—0.26 + 0.047) = -0 30° For hydrogen, (8T) jt - +0 03°, y = 1.41, c p = 3 4 X 4 186 X 10 T , S(pV)/p*V o = -0 00060, and 7 0 - 1/0 0000899, hence (ST), = 1 41(0 03 - 0 047) = -0 024°. Thus a fico expansion cools hydrogen just as it does air, only much loss, the positive Joulo-Thomson effect for hydrogen is due to tho decrease in pV upon expansion The heat of fico expansion in calories per gram per atmosphere L drop, or 4 X 8 0" X IQ 7 ' conics ou ^ ® 061 ca ^ ^ or au anc ^ &kout 0.06 cal for hydiogen 10. Isothermal Properties of the Ideal Gas. To mako further progress wo need now an understanding of the relation betweon molecular energy and the tiling that wo call the teitipe) atw e The Sbc. 10) ELEMENTS OF THE KINETIC THEORY OF CASES 17 common method of measuring the temperature of a gas is to place a thermometer in it and read the thermometer. Lot us see what this implies in regard to the molecular motion. A molecule of the gas impinging upon the wall of the thermometer must sometimes lose energy to it and sometimes gain energy from it, for the result of an impact deponds, according to classical conceptions, both upon the motion of the molecule and upon the motion at that instant of the particular wall molecule that is struck. In consequence of these impacts, a stato of statistical equilibrium soon comes into being in which on the whole the gains and the losses of energy balance each other, and when this state has been established, the mean trans- lator energy of the gas molecules and also their v 2 will have definite values which may be associated with the temperature shown by the thermometer. Now the reactions of the soparato molecules with the wall must bo independent processes, provided tho density of the gas is very low, the effect of an individual impact being in practically all cases unaffected by the positions or velocities of the other molecules. We should expect, therefore, that in the state of equilibrium the value of v 2 for each type of molecule would bo independent of the density and, therefore, a function of the temperature only. This surmise we shall find to be confirmed later by the elaborate method of analysis known as statistical mechanics (cf. the treatment of equipartition of energy and of temperature in Chap, IX). Accordingly, it will be assumed that in a rarefied gas in thermal equilibrium v 2 for a given kind of moleculo is a function of the tem- perature alone. It follows then at once by (3) or (4) that in our ideal gas, when the temperature remains constant, p is proportional to p, or pV is constant, V being tho volume of a given mass of the gas. This is Boyle’s law, obtained here as a deduction from kinetic theory. The law is found by experiment to hold very nearly for alL real gases when tho density is only a small fraction of tho critical density. Tho fact that v 2 depends only on tho temperature and not on the density suggests, as a further conclusion, that the energy, also, oughf to be independent of the density at a given temperature. To bo sure, the molecules will usually possess not only translational but also internal energy of various sorts, such as energy of rotation or of vibra- tion of the atoms. The distribution eff the energy between these forms, on tho one hand, and tho tmnslatory kinetic energy on tho other, comes about, however, through the agoncy of collisions, and there should bo, therefore, a definite average ratio for each kind of molecule between tho amounts of the different forms; increasing tho density 18 KINETIC THEORY OF OASES [Chap. I must increase the fiequency of collisions and so mciease the rapidity with which the equilibrium state is set up, but it ought not to alter the distubutional chai acteristics of the equilibiium state itself. This conclusion, again, we shall find to be confiimed by statistical mechanics. We conclude, theiefoie, that the energy of oui ideal gas will be, like P, a function of its tempeiatuio only and not of its density It follows then also that its heat of free expansion will be zero From the experimental standpoint it has been found convenient to define a “perfect” gas as one which both obeys Boyle’s law and has a zero heat of free expansion. Tho Joule-Thomson effect foi such a gas must then likewise be zeio The ideal gas of kinetic theoiy has thus the essential properties of the perfect gas of experi- mental physics. It appears fiom experiment that all gases become pci feet in this sense in the limit of zeio density 11. Avogadro’s Law. According to a famous theoiem of statistical mechanics the mean kinetic enei gy associated with each degree of freedom of a mechanical system in statistical equilibiium has tho same value (cf. Sec. 206 in Chap, IX). In the case befoie us, this means that m gases m equilibiium at a given temperatuie the averago translatory kinetic energy of all molocules has the same valuo; for two kinds of molecules with masses m i, m 2 and velocities Vi, V 2 , wo have thus \ m iti\ = § m 2 y| (14) According to the theorem of equipai tition this holds whcthci tho molecules are in different vessels 01 are mixed togotkci. Hence at a given temperatuie the root-mean-square speed, v, = (r 2 )*’* for different kinds of molecules is inversely proportional to the squaro root of the molecular weight On tho other hand, if we also make the picssuio tho same for separate masses of gas composed each of one kind of molecule, wo have by (3a) | nimjvl - $ ntfniol (15) Hence, dividing (14) into (15), we find that 4 ni = n 2 . (16) We reach thus tho very important conclusion that all porfoct gasci at the same piessure and temperatuie contain the same number oj molecules per unit volume. This statement, which is of considerable Sec. 12] ELEMENTS OF THE KINETIC THEORY OF GASES 19 utility in chemistry,* was proposed as a hypothesis by Avogaclro in 1811 to help in explaining the chemical fact that gases unite in simple proportions both by weight and by volume, and it is very often referred to as Avogadro’s hypothesis; we shall prefer to call it Avogadro's law. It appears here not as a separate hypothesis but as a deduction from kinetic theory. From Avogadro’s law it follows that the densities of different perfect gases at the same temperature and pressure are proportional to their molecular weights. The values of the product pV are thus inversely proportional to the molecular weights if V stands for the volume of a gram; but if V stands for the volume of a gram molecule or mole (i.e., a number of grams equal to the molecular weight), then at any given temperature and pressure V itself is the same for all gases. The volume occupied by a gram molecule of a perfcot gas under standard conditions is thus a universal constant, and it has been made the object of very careful experiment. The usual method is to observe at a given temperature the values of pY for a series of decreasing pressures and then to extrapolate to p — 0; from tho limiting value of pV as thus found, V can bo calculated subsequently for any pressure. The accepted experimental value of V for a gram molecule at 0° and a pressure of 1 atmosphere is Vo = 22,414 cc. (17) The volume Vi of a gram of a gas whoso molecules have moloculai weight M (in the chemical sense), and the density p of Buoh a gas, under standard conditions are then; 7i = cc, p = ~~ - 4.461 X 10-Wg/cc. (I7o) 12. The Temperature. In our references to temperature wo lmvo hitherto said nothing at all about any temperature scale. This was justified by the fact that wo have been employing only the equilibrium property of temperature, viz., the fact that sovoral bodies placed in contact oomo ultimately into a state of mutual equilibrium, whereupon we say that they possess tho same temperature. Wo must now intro- duce a scale for the quantitative comparison of different temperatures. Tho first temperature scale to bo widely adopted was that deter- mined by the expansion of mercury in a glass tube. This scale has important advantages but is limited to the range between tho freezing \ * See W. Nhiinst, “Theoretical Chemistry," or H. S, Taylok, “Troaliao on Physical Chemistry." 20 KINETIC THEORY OF GASES (Chap. I and boiling points of meicury; furthennore the selection of a paiticular substance such as meicuiy foi the theimometnc substance is a very arbitrary procedure Hence, when it was found moie than a century ago that at least the common gases expand neatly equally with liso of temperature and also almost uniformly as judged by the meicury thermometer (Chailes’s law), the pioposal was made to adopt the perfect gas as the basic thermometric substance; and during the last century the constant-volume hydrogen theimomoter was actually adopted for the ultimate standard as constituting the best piactical approximation to a perfect-gas thoimometer All perfect gases would necessarily lead to the same scale, since according to eq (14) thoir mean kinetic energies, and hence also, according to oq (3a), thoir pressures, vary at the same late with temperature We might, therefore, define the absolute temperature T as a quantity proportional to the pressure p of a perfect gas at constant volume The ratio T/p must then be proportional to the volume V, since by Boyle's law p « 1 /V when T is constant; thus we should have T « pV or pV = RT, where R is a constant for a given mass of gas. On the other hand, with the development of thermodynamics during the last century there aioso the possibility of setting up a temperature scalo that would not be dependent upon the special properties of any body whatevei ; and this scale has now come to bo regarded as the ultimate one. Fortunately it agices exactly with the perfect-gas scale, as we shall proceed to show. The argument is a somewhat abstiact one, however, and any student who prefois to bo satisfied with the peifect-gas definition of T can omit the proof and pass at once to Sec, 14. 13, The Thermodynamic Temperature Scale. The thermodynamic absolute tomperatuie T is most concretely defined as a quantity which, like the temperature on any scale, has the same value for any two bodies that are in thermal equihbnum with each other, but which at two different temperatures is proportional to the heats absorbed and rejected in a Carnot cycle woiking between those temperatures The theory of the Carnot cycle for a gas is not especially simple, how- ever, unless one adds the customary further assumption that tho specific heat is independent of temperature, which is by no means necessary for tho validity of the result that we here wish to establish. On the other hand, thermodynamic reasoning leads also to the equiva- lent but more abstract idea that, when a little heat dQ is imparted to a body in a reversible manner, we can write for it dQ = T dS where dS is the differential of another quantity, called the entropy, which has a single definite value corresponding to each possible state of the body. Smc. 13 ] ELEMENTS OF THE KINETIC THEORY OF OASES 21 This principle serves to define T equally well with a Carnot cycle and it is very easy to apply it to the perfect gas in the following way : Let U denote the energy of a mass of gas whose pressure and volume are p and V, and let a small amount of heat dQ be given to it in a reversible manner. Then, by the conservation of energy, dQ - dU + pdV, pdV representing, as is shown in elementary physics, the loss of energy from the gas due to the work it does on its surroundings. Now dU is the differential of a single-valued function of the state of the gas, or of any two independent variables such as temperature and volume that may be employed to define its state; for the energy U has always the same value when those variables take on given values. The reversible heat dQ , on the other hand, is not the differential of any such function. If, for example, we consider two different paths on the pV diagram by which the gas can bo carried from a stato A to a state B, whereas the change J^dU in U is the same along both paths, J'dQ or the total heat absorbed must be greater along that path along which the pressure is larger in order to provide for the larger amount of external work that is done (the excess being represented, of course, by tho area enclosed between the two paths on the diagram). For the same reason p dV is obviously not the differential of any single- valued function. But suppose now wo divido the above equation through by pV, thus: dU.dV pV + V' According to Boyle’s law, pV is a definite function of the temperature alone (as measured on any scale); and wo saw in Seo. 10 that for a perfect gas, U is likewise a function of tho temperature alone. Hence dU/pV must be tho differential of some function of tho temperature, which could bo found by evaluating J dU/pV. Also, dV = d(log V). Honce tho right-hand member of the equation is now the differential of a definite function of tho temperature and volume as independent variables. Tho same must, therefore, bo true of dQ/pV, 22 [Chur. KINETIC THEORY OF GASES & Accordingly, we can wiite dQ/pV ~ ity, wheie is some function o the temperature and volume, Suppose, now, we define the tempera ture T by the equation T — apV, wheie a is a constant, Thei dQ — Td(>p/a) , and this agiees with the thermodynamic equation dQ — TdS, if we define $ as S = \p/a The constant a can then bi chosen foi each body sepaiately so as to give the temperature thi propoi ty of being always the same for two bodies m theimal eqiulib num, for, if this is done at one temperatuie, it will remain tiuo at al temperatiues because by (36) pV == v 2 and hence, in conscquonci of equipartition as expressed in eq (14), pV must vaiy with change of temperaturo in the same latio for all gases, Since this requiremenl fixes only the latios of the values of o for different bodies, T stil remains arbitral y to the extent of a constant factor There are, to be suie, other quantities that might bo employed ii ordei to throw dQ into the general fonn T dS, fiomthe mathematical standpoint what we have shown is merely that 1/pV, and hence also lfapV -for any value of a, is an “mtegiatmg factor” foi dQ. It can be shown mathematically, howevei, that all integiating factors arc closely connected with each other, and that all whose leoipiocals can be given tho necessary comparative propeity foi different bodies arc simply proportional to each other and so must be included in the general form, 1/apV t 14. The Perfect-gas Taw. Whether we adopt tho thoi modynamic or the perfect-gas definition of tho absolute temperature T, wo arnvo at the usual equation foi a peifect gas, pV - RT. (18) If we then acid the requirement that the temperature interval between melting ice and saturated steam under 1 atmosphere shall be 100, the absolute temperature of the ice point is easily found to be given by the formula, piFi — p^Va — 100 poVa/T 0 or m _ 100 poVo 0 PiVi-PoVo poV a and piVi being observed values at the ice point To and at the saturated-steam point, T 0 + 100°, lespectively, This formula cor- responds to the actual method employed nowadays in the experi- mental measurement of r 0j the most recent determination by this method gave, * in excellent agreement with others, To - 273 14° * Kbbsom, van per Horst, and Jaconis, Physica , 1, 324 (1934) ( 19 ) Sho. 14] ELEMENTS OF TIIE KINETIC THEORY OF OASES 23 The perfect-gas law, eq. (18), can be applied to any desired quantity of gas, the proper value of R being calculated as pV/T> In physics the mass is usually understood to be a gram; in that case it is often more convenient to write the equation in terms of the density, 1 p v : p = pRT. (18a) In physical chemistry, on the other hand, a gram molecule or mole is almost always chosen; then, in consequence of Avogadro’s law, the gas constant R has a universal value for all gases, which we shall denote by Rm and whose value, found by dividing To from (19) into Vo as given in (17), is Rir = 82.06 cc atm^deg - 83,16 X 10“ cm dynes/deg. (20a) For a gas whose molecules arc all alike Rm = MR (206) in terms of the molecular weight M, In the case of a mixture of molecules of different masses, if 1 g contains yj g of each kind no. j in a volume V, the partial pressure due to each kind will be R,T vt = y i~y> Rj being the constant for a whole gram of kind no. j, and by Dalton's law the total pressure will then be p = ^Vi ~ RT/V where R - J-V/B/ (20c) } and represents the gas constant for a gram of the mixture. The perfect gas could bo defined as one which obeys the perfect- gas equation, (18), instead of defining it, as abovo, by tho two con- ditions that it obeys Boyle's law and also has a zero heat of free expansion, For it is obvious that any gas which obeys this equation also obeys Boyle’s law, and it can bo shown from the laws of thermo- dynamics that any gas obeying the porfoct-gas equation must also have a zero heat of free expansion (Sec, 137, problem), Problem. Show that for a perfect gas tho coefficient of expansion a (at constant pressure) and tho coefficient of pressure increase (at constant volume) are given, in terms of values Vi or pi which hold at any base temperature T it by 24 KINETIC THEORY OF OASES [Chap. I 1 V — Vi _ 1 fdV\ 1 1 “ “ Vi T - Tx ~ Vi \dTJp ' Ti , _ X p ~ Pi - 1 / i£\ _ 1 Pi T-Ti Pi \dTjy Ti The coefficient at 0°C, a G » e 0 ~ 1/To - 0 003661, is of particulai * interest 15 . Molecular Magnitudes, Since the product nm of the mass ni of a molecule and the number n of molecules in unit volume equals tho density p, this product can be calculated at once, but our equations do not enable us to calculate n and m separately As a mattei of fact, kinetic theoiy by itself does not furnish any very exact method of estimating these two molecular magnitudes The best values that we possess today are derived fiom the following indiiect evidence The electncal chaige carried by a giam atom of a monovalent element such a a silver is easily measured and has been found to bo 96,494 international or 96,489 absolute coulombs (the Faraday) *; and each atom carries tho same numerical charge as the election For the latter it now appears that Millikan's value (4 774 X 10~ ll> ) was too low because of an erroi in the viscosity of air If Kellstzom's recent value foi the viscosity t is combined with Millikan's oil-chop results, the value 4 816 ± 0 013 X 10~ 10 is obtained for the electronic charge, this agrees within the probable error with the value calculated from x-ray wave lengths as measuzecl by means of a grating, which is 4 8036 ± 0 0005 $ Since accurate repetitions of some of these export ments are under way and indicate that the giating wave length is at least not too low, we shall adopt the value e *= 4 805 X 10-*° electrostatic unit, Dividing this number by 2 9979 X 10°* to convoi t it into coulombs and then dividing the result into 96,489, we have then as the number of atoms in a gram atom, or of molecules in a gram molecule or mole, often called Avogadro's (or Loschmidt's) number, No - 6 021 X 10 23 , (21) and for the number of molecules in a cubic centimeter of peifccfc gas at 0° and 1 atmosphere pressuie, n 0 * No/Vo or, by (17), n 0 = 2 686 X 10 10 , or 2 69 X 10 10 (22) * Cf Birqb, Rev Mod Physios, 1, 1 (1929) t Nature, 136, 682 (1935) j JBirqe, Phys Rev, 48, 918 (1935) Sec. 16] ELEMENTS OE THE KINETIC THEORY OF GASES 25 to three places. The mass of an imaginary atom of molecular weight 1 is then the reciprocal of No or Wo = 1.661 X 10 -a4 (7; (23) and the mass of any atom or molecule is the product of this number into the atomic or molecular weight, A related number of great importance to theory is the Boltzmann constant , or gas constant for one molecule, k = Ip = mR = 1.381 X 10 -18 cm dyne/deg, (24a) m being the actual mass of a molecule and R the gas constant for 1 g. In terms of k we can write for the pressure, in place of pV ~ RT, p = nkT; (24 6) for V — 1/p = 1/nm. On the other hand, from (36), (18), (206), and (24a), a 5 - 2RT = 3 RmT M 3JcT m (25a) M being the molecular weight. It follows that the root-mean-squaro speed of the molecules, v„ — («t 2 ) M , varies directly as the square root of the absolute temperature, and for different gases inversely as the , square root of the molecular weight. The same thing is true of the mean speed, which wo shall find in the next chapter to bo v = 0.921y a . The mean translatory kinetic energy of a molecule is also of interest: i m i - kT. (256) This last equation lias .sometimes in kinetic theory been made the basis of the temperature scale. Equation (26a) is difficult to tost experimentally, but it can bo employed the other way round as a moans of calculating v, and 0. Values obtained in this way for a number of gases at 15°C* are given in the table on p. 26, along with values of R (for 1 g) in absolute units, as well as the molecular weight M and the actual mass of a inoloculo m calculated in the manner described abovo. With three exceptions the values of M were taken or calculated from the table of International Atomic Weights for 1931 1> R hr atmos- * 16°0 was chosen instead of 0°C bocauso in niOBt applications tho actual tompornturo Is room tomporaturo. ' t Sco Jour. Amr. Chum. Soo., 53, 1G27 (1031). [Chap, I 26 KINETIC THEORY OF GASES pheres wa& then calculated as R v /M, and those values wore multiplied bv 1 01325 X 10 6 to get R m absolute units The throo exceptions aie as follows The atomic weight of H 2 , the atom of deuterium or heavy hydrogen, was taken from a note by Bambndgo. I'oi air, It m atmospheres was found by dividing 1000 by the accepted mow of ft normal liter of an, 1 2929 g, and by To = 278.14, and then mul- tiplying by 1 0006f to extiapolate to zeio density Foi tho elect! cm, M was calculated as F standing foi the Faiaday m clectromiignetio units, and e/m having the value = 1 7576 X 10 7 ,t m was calculated directly from e/vi and e M m (unit, 10-* g) R (unit, 10° eig/dcgj ir.(16°C) (unit, JO 3 cm/aeo) U(16°C) (unit, UP om/w'o) H* 2 016 3 349 41 25 188 8 174 0 Hj 4 027 6 880 20 65 133 0 123 I Helium 4 002 6 648 20 78 134 0 | 123 r> h 2 o IS 016 29 93 4 615 03 18 58 19 Neon 20 18 33 52 4 120 59 08 | 54 08 N* 28 02 48 54 2 968 60 05 40 07 O, 32 00 53 16 2 598 47 39 43 66 HC1 36 46 60 56 2 280 44 40 40 90 Argon 39 94 66 34 2 082 42 42 30 08 CO* 44 00 73 09 1 890 40 42 37 24 Krypton , 82 9 137 7 1 0030 29 45 27 13 Xenon 130 2 216 3 6380 23 50 21 05 Hg 200 6 333 2 4145 18 03 17 44 Air (28 96) (48 11) 2 871 49 82 46 00 Electrons 6 49 X 10~ 4 m = 9 119 X 10" a g R = 1 514 X 10® otg/deg » 9 - 11 44 X 10 fl om/sco C « 10 6 1 X HPom/mm 16, Rapidity of the Molecular Motion. Tho first calculation of molecular speeds by this method was made by Joulo in 1848 Tho values found are high as compared with most speeds produced by human agency The slowness of gaseous diffusion in spito of those! high speeds, whiph was at one time advanced as an objection againnt the theory, arises, of course, from the continual mterfeumeo of tho molecules with each other's motions Foi example, if chlorine gaa m * Bainbridqe, Phys Rev , 44, 57 (1933) f Cf HoLBORNand Otto, Zeiis Physik, 33, 1 (1925), whole tho unit of prosmirq, however, is 1 M of* Hg \ Birge, Phys Rev , 49, 204 (1936), SBC. 10] ELEMENTS OF THE KINETIC THEORY OF GASES 27 released in one corner of a room, it may be minutes before the odor is perceptible in the opposite corner; a molecvile of chlorine goes nearly a quarter of a mile in every second, but this long path is converted into a complicated zigzag by collisions with other molecules and is thereby tangled up into a space less than an inch across. A comparison with the velocity of sound is also interesting. The familiar formula for the velocity of sound in a gas is (iff = ( T pV)Y‘ = (yl IT)V> where y <5 f . Comparing this quantity with v s or the square root of v 5 as given by (25a), we see that the velocity of sound is less than v» in the ratio (y/3) w or something under £, It could hardly exceed v B) since the sound waves are actually propagated by the motion of the molecules, so this result really constitutes a confirmation of the theory; but it may seem surprising at first sight that the two velocities should be so nearly equal. The high values of the molecular velocities and the enormous magnitude of N 0 or n 0 serve to explain why matter behaves in so many ways as if it were continuously distributed. Problems 1. Calculate values 'of from cq. (3a) for hydrogon, air and oarbon dioxide at 0°C, using actual values of p under standard conditions, and oomparo results with tho values given in the table. (Tho slight discrepancy is duo, of courso, to depar- tures from tho porfect-gas law.) 2* Compute tho temperature at which the roob-momi-squaro speed Is just oqual 1 to tho " speed of cscapo” from the surfaco of tho earth (i.o., tho minimum speed necessary to carry a molecule to infinity) for (a) hydrogen and (6) oxygon. Repeat for tho moon, assuming gravity on its surfaco to bo 0.104 as strong us on the earth. CHAPTER II DISTRIBUTION LAW FOR MOLECULAR VELOCITIES S. In the last chapter we found that the piessuie and tempoiaturc of perfect gases depend only upon the mean square of the molecular speeds and are independent of the manner in which the molecular velocities vary among themselves Theie aie other propci tics of gases, however, which clo depend to some extent upon the actual distribution of the velocities, and a knowledge of these is, theiofoie, needed. Accordingly, we shall take up next, in this chapter, the law according to which the moleeulai velocities aie distiibuted in a gas in equilibrium 17. The Distribution Function for Molecular Velocity. If we could follow an individual moleculo in its motion, we should obseivo it to undergo many and laigc changes in velocity as it moves about and collides with otheis, For example, one ,can easily invent collisions which, accoiding to the laws of mechanics, would leave one of tho colliding molecules aftei the collision momentarily at lest, and otliois which would give to one molecule, in consequence of a string of collisions m which it is struck repeatedly from the side, as laigo a speed as might be desired We should expect at any given moment, therefore, to find the individual molecules moving in all directions and with speeds vaiying all the way fiom zero up to values many times as great as the average All that we can hope to do as physicists undei sufch circumstances is to describe the situation in statistical terms To obtain a descrip* tion in mathematical form, let us as usual denote the vector velocity of a molecule by v and its cartesian components by and lot us fix our attention upon those molecules whose components at o given moment lie lespectively between a certain value v x and a slightly greater value v x + dv x > between v v and v y + dv V} and between v x and v, + dv gt We shall say for short that the velocity of such a molecule lies in the range dv X} dv V) dv z < When the number of molecules is very great, we should expect that even for small ranges the number of included molecules will be propoitional to the product dv x dv u dv g) hence we can write for this number Nf(v Z) v Vi v 9 ) dv x dv v dv t} where N stands fgr the total number of molecules and / for some function Sbc. 17] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 29 of v x , v y , v„ For brevity it will be convenient, however, to indicate the variables in / by writing just /(v) and to think of / as a function of the vector velocity v. The function / is called the distribution function or probability function for molecular velocity and obviously has the fundamental significance that f do x dv v dv, is the fraction of all the molecules that have velocities in the range dv x , dv V) dv„ Because of this significance, it obviously satisfies the equation ff'ff dv£dv v dv, = 1, (26) dvi v* the integrals extending over all values of v~, v„, v, from — co to + «> ; the total number of molecules thus comes out correctly as f f fty dv, dv v dv z = Nffff dvx dv v dv z - N. (27) The situation can be visualized if desired by imagining the veloci- ties plotted in a velocity space in which v Xl v V) v, servo as cartesian coordinates (cf. Fig. fi). A particular velocity v is then represented either by the vector drawn from the origin to that point in this space whoso coordinates are v x , v y , v z , or, if pre- ferred, just by this point itself; and the small velocity range dv x , dv„, dv, is obviously represented by a small, parallelepiped with edges having longths dv,, dv V) dv„ respectively. Frequently, however, we shall write *>.— 'Molooulnr potnta in volocity more briefly for an element of volume h| I C °' in volocity space simply die in place of dv, dv„ dv„ and the clement then need not be a parallelepiped but may have any shape. In general f will vary with the time. When it does not, the gas is said to bo in a steady state. The latter term is often restricted to refer to gases in complete equilibrium, in which not only is the dis- tribution of molecular velocities a steady one, but also the mass acceleration of the gas vanishes everywhere, and neither energy nor matter is flowing into or out of it at any point. These further restric- tions servo to eliminate, among other things, steady states of heat conduction or of viscous flow, whoso treatment requires special methods of attack. This chapter and the next will be concerned with gases in such a state of complete equilibrium. 30 KINETIC THEORY OF OASES [Chap ll The number of tho molecules which aie under consideration has so fai been left indefinite If desned, it can bo taken to be tho total numbei m a certain mass of gas Moic frequently, howcvei, f ref cm to the molecules in a macioscopicaliy small element of spatial volume; in this case / may vaiy with the position of the element, so that in geneial it is a function of the seven variables v *, v Vf v Zj x } y, z i l In tho state of complete eqiuhbiium, however, even / defined fox the mole- cules in an element of volume turns out to be independent of $, y f z, and l and is accoidmgly a function only of v*, v Vf v? 18, Distribution Function in Other Variables. Often it is moro convenient to employ polar coouhnates in velocity space When we do this, v plays the iole of r, the distance from the ongin; then the element of volume in velocity space is sin 6 dO d<p dv and v 2 f(v) sin 0 dO d<p dv (28) represents the fraction of the molecules that aro moving with speeds between v and v + dv and in a dii ection which makes an angle between 0 and Via e,—' Velocity pomis q -p WI th the polfti a\is and lies in a piano thiough the axis making an angle between <p and <p + d<? with the lefeience plane for <p For futuic reference several othci ways of grouping the velocities may also be noted at this point When we aie interested for tho moment piimanly only in the directions of motion of the molecules, but not m their speeds, it is often useful to imagine long lines drawn from an oiigm in the clnoctions of the various velocities and to talco as representing the velocities themselves tho points in which thoso lines cut a splioie of unit ladius drawn about the origin as center (cf Fig 6, m which all the points aie supposed to be on tho suifaco of the spheio), Consider, now, the impoitant case in which the velocities aro distributed equally as regards then directions The points will then be distnbuted umfoimly ovci the sphere, Now a small solid angle dw diawn at the center intercepts an area dw on the unit spheio, whose total area is 4?r Hence we have the useful result that, wlion the molecules aie moving equally in all dnections, those that aro moving in a dnection lying within an element da> of solid angle con- stitute a fi action Sec. 19]. DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 31 of the whole number. If polar coordinates are being used we usually write du = sin 0 dO dip. But in the ease under discussion we can also give to dw the form of a ring including all velocities whose directions make an angle between Hand 0 -f- dO with a given line, regardless of <p\ the area of this ring, which is a narrow strip on the unit sphere dO wide and 2r sin 0 in circumference, is 2 ir sin 0 dO, hence these velocities will constitute a fraction 2 ir sin 0 ~ = i sin 0 dO (30) of the total. This holds when all directions are equally probable. As a check, we note that sin OdO — 1.* In other connections, however, tho distribution function for a separate component of the velocity is needed, This is easily found from / by integration. Denoting this function for v x by f„(v x ), wo note that all velocity points for which y 3 has a value in a given range dv x , constituting the fraction f x dv x of tho whole, lie between two parallel planes drawn perpendicular to the v x axis ancl a distance dv x apart (cf. Fig. 5) ; the fraction of the whole numbor of points included is thus equal to J J Jf dv x dv u dv l - dv x J Jf dv„ dv„, to tho first order in dv x . Hence, equating this expression to f x dv x and canceling dv x , we have /* = J J f dv u dv,, (31) integrated over all values of v v and v, whilo tho value of v x in f(i) x> v V) v,) remains fixed. 19. Remarks on the Distribution Function. In our mathematical work we shall treat the ranges dv x , dv„, dv, as infinitesimals, as wo have already done in writing such integrals as those in eqs. (20) and (27). This might seem objectionable in view of tho fact that when these ranges are made very small the numbor of included molecules must be small, perhaps oven mostly zero, and tho numbor must in any case jump discontinuously by unity every timo the shrinking element of volume in velocity spaco happens to pass ovor a molecule. This objection can bo met in several different ways. We can say that dv Xl doy, do i are to bo made only macroscopically but not mathe- matically small, i.e., they are to be small as compared with tho scale of physical observation but large relatively to tiro spacing of tho * Tho uppor limit is ir and not 2ir, slnoo nil fuslmuthu around tho givon lino nro included within cnch do (of. Fig. 0). 32 KINETIC THEORY OF GASES (OlJAl*. II molecular velocities; a piocess of mathematical integration ft ( > is then employed merely for reasons of convenience mid yields i ca la differing slightly but not appieciably from the truth A second ant better way would be to regard N/cIk as- jopicsonting merely tho average number of molecules m the element of velocity space dtc duung a time that is macroscopically shoit but still long enough t allow many molecules to entei and leave tho clement, tho demon itself can then be made as small as we please, perhaps so small that it never contains moie than one moleculo and during most of the lime none at all The quantity / (h then i epi esents tho fraction of tho time during which the element does contain a moleculo, The best view, howevei — but also the most abstract ono is probably to treat f as being of the nature of a probability In this view, Nf (Ik represents the expectation, or / da itsolf roprosonts tho fractional expectation, of molecules in ch If Pq is the chance that there is no molecule in <A, Pi the chance that thoio is one, Pi that there are 2, and so on, then Nf (Ik = /\ + 2P 8 + * •+* jP/ "H ' ' ' ■ If we were to make a great many observations with conditions remain- ing the same, the aveiage of all the dilfeient numbers of molecules that we should find in dn would be Nf da Tho moaning hoi o is tho same as in the common term, “expectation of life,” We shall commonly speak of / m this book in terms of tho prob- ability interpretation and shall work freely with / rather than with Nf The logical argument is thereby made a little moio conoiso. Headers who dislike to attach meaning to the probability of a single event and piefer to interpret all piobabihties as representing averages of some sort should find no difficulty in modifying our treatmont to fit their preferences, it is only necessary to multiply our equations through by N so as to be dealing always with Nf, and to substitute for our language a description ih terms of ono of the alternative views 20. Proofs of the Distribution Law. The velocity distribution law for a gas in equilibrium, known as Maxwell’s law, was first guossod and partially established by J 0. Maxwell (1869)*; tho proof of it by direct methods was fiist earned to completion (in so far as this is possible) by L Boltzmann Pi oofs of this type havo beon given, howevei, only for ceitain simple cases The much moro general methods of statistical mechanics, on the othei hand, furnish a proof resting on a fiimer foundation and applicable to all cases (in so far as classical theory itself applies) *J C Maxwell, Phil Mag , 19, 31 (1860); Set Papers, I, p. 377 Sec. 21] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 33 From the standpoint of strict logic it might seem more natural simply to rely upon this proof by statistical mechanics, which is given in Chap. IX. The analysis involved in Boltzmann’s proof gives, however, such a lively idea of the processes at work in a gas that it seems worth while to take it up at this point, for the simplest case only. The discussion will be so conducted that the thread of the argument can be followed if desired without reading the rather voluminous mathematical details; any reader who prefers to do so can, as an alternative, pass over the proof entirely and proceed at once to the discussion of Maxwell’s law in Sec. 28 without encountering any difficulties with the notation. 21. Molecular Chaos, The Maxwell-Boltzmann proof proceeds by calculating the effect of collisions between the molecules upon the distribution of their velocities.* This effect depends, of course, upon the distribution already existing at the moment. Accordingly the procedure is to seek such a distribution that the effect of collisions upon it vanishes, and this distribution is then taken as that proper to a state of complete equilibrium. Now the frequency of collisions of any particular type depends upon the positions as well as the velocities of the molecules. At this point Boltzmann simplifies the analysis by making a famous basic assumption called that of “molecular chaos.” This assumption states that in a gas whose molecules interact only during collisions all possible states of motion occur with equal frequency. Thus each molecule is as likely to be found in one position as in another; further- more, except for the simple fact that the molecules cannot get inside each other, there is on the average no correlation whatever between the positions and velocities of different molecules. If, for instance, we know that a certain molecule is at a certain point and moving with a certain velocity, then at that moment another molecule is just as likely to be at any given point in the neighborhood of the first and to be moving with any given velocity as it would be if the first molecule were in any other position or moving with any other velocity. Perhaps the assumption of molecular chaos may scorn plausible enough, in view of the highly varied and tangled motion of tho mole- cules. There have been some who refused to accept it, however, and it certainly needs further support in order to be quite satisfactory as a basis for the theory, If the molecules wore to bo simply scattered around at random, the assumption would certainly be true, In reality, however, the distribution comes about as -tho result of mechani- cal motion, and it is quite thinkable that for this reason regularities * Cf. L. Boltzmann, “Vorlcsungon liber Gnsthoorlo, 11 vo!. I, 1890. 34 KINETIC THEORY OF OASES [CriAr II would occur m it; for example, molecules in a given neighborhood might tend, for all we can see, to have similai velocities Fuither- more, special states that arc not chaotic can, of course, easily be described; for example, at a given moment half of the molecules might be moving east while the othci half weie moving west, and the lafctoi half might be just on the point of sinking the first half in head-on collisions, Such a state of the gas Boltzmann calls "molccukuly ordeied,” He assumes that such states can be ignored, the gas being almost all of the time moleculaily unoideied (“molekular uiigeoidnot”) Fortunately it can be shown fiom statistical mechanics that in the state of complete cquilibiium the condition of molecular chaos does, in fact, exist during piactically the whole of the timo (of. Sec, 197) Part of Boltzmann's proof lequnes the existence of molecular chaos oven when the gas is not in eqiuhbuum, This, too, can bo justified to the following extent* It follows fiom statistical mechanics that states m which theie is an appieciablc depaituie fiom molecular chaos constitute only a small part of all possible states and so can bo expected to occur only very larely, This is about all that one could hope to prove. 22 The Effect of Collisions upon f. We shall assume fuither, with Boltzmann, that the distnbution function is the same at all points in the gas, The effect of dropping this assumption will be considered in Sec 50, Its importance lies m the fact that when / is unifoim, diffusion of tlie moleoules from one point to anothei does not tend to alter it, since as many molecules with given velocities arnve at any point as leave it, and vice versa, changes in / thus arise only as a result of collisions Finally, to avoid unnecessaiy repetition of details, let us assume for generality that two cliff eient kinds of molecules aie piescnt, with masses ni i and ?ra 2 and distnbution functions /i(vi) and / 2 (v 2 ), lespec- fcively. Both kinds will be assumed to be hard spheres fieo from mutual force-action, except m the collisions which aie assumed to be instantaneous. An extension of the aigument to other cases may be found in Boltzmann's “Gastheorie” and elsewhere We are now ready to analyze the effect of the collisions upon the distnbution function/ The method will be to select fiist a gioup of molecules having the same mass and almost the same velocity, and to study their collisions with anothei similarly selected group, calculating with the laws of mechanics the effect that these collisions have upon the distribution function / of the fiist group This effect depends upon the velocities of the colliding molecules, and also upon the position of their line of centers, which is a line joining the centeis of gnc, 22| DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 35 two colliding moleoules at the instant of collision. We then integrate over all possible velocities that the second group can have and also over all possible positions of the line of centers, and thereby arrive finally at expressions giving the rate of change of / with time. These expressions are stated in eqs. (42a), (426), and (43) below. As the two groups of molecules to be considered first, let us select a group of the first land having mass mi and velocity Vi lying within a certain range dni and a group of the second kind having mass ?n 2 and velocity v 2 lying in a range ck%. During a short time dt some of the molecules of the first group will collide with moleoules of the second group. Among these collisions let us select first those in which the lino of centers drawn from the first molecule to the second lies within a certain solid angle du of possible directions (Fig. 7). The number of these collisions can be written clown in sufficient detail for our purpose in the following way. Their num- ber will obviously be proportional to dt, to du, to the number of molecules in each group, and to the relative velocity between the two groups, whose magnitude we shall denote by v r - |vi - v*|. Now there are ni/i(vj) dx x molecules of the first group and nifnivt) dxt of the soconcl in unit volume, Wi and nt being the molecular densities for the two lands of molecules. Hence we can write for the number of these collisions in unit volume Fig, 7. — A molecular collision. nina<pv,fi(v\)Mvt) dm d« 2 du dt. (32) Tho factor of proportionality <p in this expression will probably depend upon tho angle between tho chosen line of centers and the direction of tho relative velocity v r (for example, glancing collisions may not have the same probability as central ones); but it cannot depend upon tho velocities themselves, for, with v r fixed, the velocities can vary only by tho vector addition of a common velocity to each group, and such an addition obviously cannot directly affect the number of collisions, nor can it affect their number indirectly by altering tho likelihood of the various possible positions of the mole- cules, since these positions, by tho principle of molecular chaos, show no correlation whatever with the molecular velocities. Nor can <p depend upon the position of the line of centers, for a rotation like a rigid body of the whole situation, velocities and all, while it may affect /i and ft, cannot affect because v r and all angles would bo unchanged and, by the principle of molecular chaos, the new positions 36 KINETIC THEORY OE GASES {Ciu* II of the molecules would be just as likely as the old It follows tlmt (p can depend only upon the angle between v r and the line of confers. The total number of collisions made by molecules of the fust gioup can now be found by integrating (32) ovei all values ol v 2X} v 2Uf vu and ovei all possible positions of the line of conteis, With an eyo to future developments, it will be convenient al^o to extend the first gioup somewhat by mtegiatmg vu, v\ V) Vn as well over a small but finite range Ai. We thus find foi the number of all collisions made in dt by molecules in the range Ai colliding with any molecules whatever of the second kind, per unit volume, TI 1 U 2 dtj* dwj* d/ci (S3) 23. Velocities after a Collision. To asceitam completely the effect of the collisions upon the distribution, we must now find out what velocities the molecules take on aftei colliding At each collision the laws of classical mechanics lequne the conservation of linear momen- tum and also, since no effect on any possible lotation of the molecules can occur, because of their assumed symmetiy, conservation of tianslatoiy kinetic eneigy Wave mechanics leads to the same conservation laws whenevci an experiment is arianged in such a way as to yield obscivations of momentum and kinetic eneigy. Accordingly, if Vi, V 2 be the lespective velocities of the molecules of the fiist and second groups after collision, 311 st as Vi, V 2 aie their velocities befoie, we have m\y lx + m 2 V2z ~ niV u 4* in 2 V 2Xf (34 a) niiv lv + m 2 V2 U = fniViy + m 2 V 2vy (34 b) tni Vu + mvu = rrhVu + m 2 Vu> f34c) I + i m&\ » | WiVf + £ m 2 Vl (3B) the first three equations expiessmg the conseivation of the three components of momentum. In addition to these four equations, however, two moie are needed in older to fix all of the six components after collision, Vu * Vi*, in terms of those before collision, Vu v 2z These additional equations arc furnished by the position of the line of centeis and can be obtained as follows. The increment of vector momentum impaitcd to each molecule by tho impact is in a dhection parallel to the lmo of centers, the momentum given one molecule being just opposite to that given to the other, hence the components of this inclement will be pioportional to the cliiection cosines \ p, v of this lme and, these Seo. 24] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 37 components. being mi(Fi* mi( Vu - »i,) _ X mi(Fi„ - vi„) n' etc., or vn), Wi(Fii/ — Vi„), etc,, we can write • X(Fi tf — »i v ) = /t(Fi# — m(Fl - vu) — v(Vi v — v iv ), v(Vu - v u ) = X(F a< - (36a, 6, c) Only two of these equations are independent, of course, for it is easily seen that one can always be deduced from the other two, in correspondence with the fact that the line of centers has only two degrees of freedom of position; nor do the corresponding equations in terms of v 2 and Vs add anything, beoause of (34a, b, c). 24. The Inverse Collisions. It is evidept from these equations, and also from elementary ideas, that the molecular velocities are usually altered profoundly by a collision and that molecules of the first kind which were in the velocity range A t to begin with, will very often be thrown entirely Fl0, lllvorao outside of it. For every typo of collision which removes a molecule of the first! kind from A*, however, another typo is possible that restores one to it. We can, in fact, take any two mole- cules that have just collided in the manner described above and, by shifting their positions without changing their velocities, cause them to collide again with the line of centers exactly reversed in direction and with the roles of initial and final velocities interchanged (cf. Figs. 7 and 8). This appears algebraically from our equations in the fact that they still remain true if the values of Vi and are interchanged with those of Vi and Fa; the necessity of reversing the lino of centers, however, i.e,, of replacing X, n, v by —X, — n, —v, becomes apparent only when we reflect that the molecules must be approaching each other just before they collide. Furthermore, the whole class of collisions formed by inverting the original ones in this manner actually includes all that can restore molecules of the first kind to the range Ai; for the result of inverting in its turn any invertod collision is to recover tho original collision, and accordingly, given any collision which restores a molecule to Ai, wo find an original collision of which it is an inverse merely by inverting the given restoring collision, Tho number of such inverse collisions corresponding to. the original ones will be given, obviously, by an expression similar to (33), via., Wh dlf daf fffff *VWt dV u dVi v dVu dVudV^dVu, (37) 38 KINETIC THEORY OF GASES (Chap II in which V r « |Vi - V 3 (, F i - /i(VO, F 2 = f 2 {V 2 )i ' 1S an element of direction for the inverted line of centeis, and the range of mtegiation for dVi* * dVu covers all values conesponding to values of v\*> Vi V) Vu in Ai Befoie making use of this expression, however, wo shall make a change in the variables of mtegiation in it To simplify the pro- cedure, let us, foi any given position of the line of centeis, imagine the axes of coordinates to be lotated so that the ^“axls is patallcl to this line, such a rotation having no effect, of course, upon the value of the definite integral containing dV\ x * dVu Then in (36a, b , c) X = 1, /i = v = 0, whence, using (346, c) as well, V ly 553 Vly f Viz - Viz) V 2 y — Vfy, V 2 z = V 2 g (38) (34a) and (36) then become, since v\ = v\ z + + vL etc , and since the y and 2-terms all cancel out, mi(v t z - V u ) « m 2 {Vu - v 2x ), rrh(vl - VI) = m »( 7 |, - 4 ,); and, dividing the fiist of these two equations mto the second and then solving, we find -it _ (wm — m t )vi x + 2imv 2x rr (m 2 — m,)u 2 # + 2wii»i» /ons yu mTm* (39) Accordingly, in (37) we can replace* dV j# d7 2 * by |/| dv Xx dv% x where dVudVu, dv if dv 3x = 1 »»i - nh, 2 m% Mj* dVjx {mi + m 2 y 2m h m 2 ~ mi dvi, dv 2(G so that |J| = 1, and, of course, by (38) dV dV \z dV 2 y d V 2 z == dt)iy dviz dv 2y dv 2 z The fact that in (37) we can thus replace dVi x * dV 2z by dv u • dv 2 z is an example of the famous Liouvillo theoiem that plays so impoitant a role in statistical mechanics Furthermoie, by (39) V TX - V 2 * - 7^ = -{v 2x - v lx ) - ~v r *> whereas by (38) V rv =a V 2y — Vu, = v 2y — v ly » v ru and similaily * This can be done, of course, only under the integral sign, as a nile, since dv\ K dt)u represents a rectangular area but the corresponding clement dn tho Vi? t Vt K plane, although of area |/| dv u dvu> will not usually bo a lectangle Sec. 25 ] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 39 V rz — v, e ; that is, wo have the familiar result that the component of the relative velocity in the direction of the line of centers is reversed by the collision, whereas its components perpendicular to that line remain unchanged. As a consequence, not only is V r = v r in magnitude, but also the angle between the relative volocity ancl the line of centers is the same after the collision as before, and the latter fact, according to what was said above about y, means that <I> = <p. Finally, do = do>, being just the vertically opposite element of solid angle. Hence (37) can be written Wi«2 dl J dwj' J' ipv r l'\I , \ (hi ch a. (40) 26. The Rate of Change of the Distribution Function. If we now subtract from this expression that given in (33), we have tho gain less the loss, or the net gain, of molecules of the first kind in the range Aj caused by all collisions with molecules of the second kind; the result can bo written nmi dlJ Ai dni J*j <pv r (F,l'\ — /i/ 2 ) doi, (41) the order of integration having been changed in preparation for the next step. There are, to be sure, some molecules which are merely transferred by collisions of the type considered to another point in Aj and so are not actually either lost from the group or restored to it; we could show that no error results from this circumstance, but wo may as well dodgo the issue by proceeding at once to make Ai infinitesimal, whereupon tho collisions in question become negligible in number and can be ignored. A similar expression with the subscript 2 changed to 1 throughout will then give the net gain in Ai due to nil collisions botween tho first group and all molecules of the first kind; and tho sum of this expression and (41) finally gives us the total not gain in unit volume during time dl of molecules of tho first, kind in Ai. On the other hand, this not gain can also be written Equating the two expressions thus found for tho net gain wo have: ’"‘“/J ai* — eki == dl Ja ^ Kt J J - /t/a) dxidw + n\ dt dsy j j <pv,(FiF{ - /,/{) dx[du. 40 KINETIC THEORY OF OASES [Chap II in which we have primed Vi or Vi and functions of them when they refei to a molecule of the fiist kind functioning as the second moleculo m a collision. Let us now suppose, as we have alieady done, that Ai is made indefinitely small, then in the limit we can replace each integral over Ai by the integrand multiplied by Ai (provided the integrand is assumed to he continuous) We thus find finally, after canceling ttiAi dt. In a similar mannei can be found for molecules of the second kind i " "•// ipVtiFiF'i — fiifi) chi + »ij* J* <?Vr(FtF\. — fifi) (hi dta (42b) These lesults aie easily extended to cases m winch more than two kinds of molecules are piesent On the othei hand, if only one land is present, we can diop the subscripts and wnte simply | = n J <pv r (FF’ - //') M dco (43) In this last equation, to repeat foi convenience in refei ence scattered statements already made or implied, do) is an element of solid angto within which the line of centois may lie at the instant of collision and p is a geometucal factoi expressing the likelihood of a collision between molecules apptoaching with velocities v and v' and i dative velocity tv; / = /(v) and lepresents the distribution function for molecular velocity, f — /(v'), and F;F f stand foi similar functions of the new velocities V and V' after collision; d is an dement of volume in the velocity space for The negative term lepre&ents the effect of collisions m lemov- mg molecules fiom a given legion m velocity space, while the positive term repiesents the effect of other collisions m throwing molecules into that region Analogous statements lipid for eqs (42a) and (426) n is the number of molecules in unit volume and the numbers of the two kinds taken separately 26. The Equilibrium State. The values of the time derivatives as given by (42a, b) oi (43) tell us how the velocity distribution changes with time m terms of its form at any given moment, whatever this form may be. The most important case, however, is that of a gas m Sbo. 20) DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 41 equilibrium, and we shall now confine our attention to this case. Equilibrium requires that the distribution of velocities shall be steady or independent of the time; and this means, when two kinds of molecules are present, that df 1 _ djf dt dt, - 0 (44) for all values of Vi and V 2 . If we insert in this double equation the values of the derivatives as given by (42a) and (426), we obtain two integral equations for the determination of /1 and/ 2 . Now, no simple general method of solving such equations exists, but in the present instance a solution is easily guessed; dfi/dt and dft/dt will certainly vanish, provided FiF[ =/,/!, FJ<\ = /,/, (45a, b, c) for all values of the independent variables. We shall first work out this solution in detail, and then later we shall take up Boltzmann’s proof that it is the only one. In (46a, 6, c), in turn, wo have functional equations to solve for fi, fi. They happen to become easier to handle if we introduce g — log /, so that (46c), for instance, becomes (after an interchange of the mombers) ffiOh*, »n/, Vu) + fi'aO'a*, %, t> 2 «) = ffi(Ni„ Fin, Vu) + 0i(V iai 7», V u ). (46) From this equation we see at onco that g must bo such a function of tho molecular velocity that the sum of its values for two molecules is unaltered by a collision. Now we are already acquainted with several quantities that have this property; tho kinetic energy is one and the three Components of the momentum aro three others. Obviously, also, any linear combination of those four quantities with arbitrary constant coefficients would enjoy tho samo property. This observa- tion suggests that g itself might be such a linear combination; certainly such a function does constitute one possible solution of eq. (46). A little reflection will serve, indeed, to develop a healthy doubt whether there can be any other form of solution ; but really to complete our proof we must actually show that there cannot bo another, and this we shall now do. It frequently happens in theoretical work that tho correct solution of a problem can bo guessed with ease, whereas tho proof that it really is the only solution' requires considerable labor; in such a case tho proof ought eventually to be sought, but evon before this has been 42 KINETIC THEORY OF GASES (Chaf II done, coneideiable importance should be attached to the result of the guess, since it is almost always light The reader who is satisfied with the aigument just given can turn at once to the results for ft and/ 2 or foi /, as expiessed in oqs. (63a) and (53 b) below together with the lestnctions expiessed in (64a, b), or in eq (53c) for a homogeneous gas, and proceed from that point without reading the ligorous treatment that is now to be given. 27 . Rigorous Treatment of the Equilibrium State. The usual method of solving ngoiously a functional equation such as (46) is to diflfeientiate and then tiy to eliminate the unknown functions one by one until a differential equation is obtained for one of them. Equation (46) must hold for all values of 0i» • * 0s*, and also foi all values of X, n, v, that make X 2 + p? + r 2 => 1, the quantities Via Vz, being determined by f34a, b, c), (35), and (36a, 6, c) It turns out, howcvei, that to reach our goal we need only consider such changes of the variables as leave Fi* • * V 2z unalteied, and the work is simplified by such a restuction because the lattei variables can then bo loft out of consideration altogether. With this fuithei lestuction the vanablcs »i* ’ • 0s* can vaiy only m such a way as to satisfy the following equations of condition, obtained by diffeientiating (34a, b, c) and (35) with Fi* • Fs* kept constant* dv ix 4- dv tl — 0, mi dv i„ + ms dv %v = 0, midviz -j- mz dt>t z = 0, (47) mi(vu doix + 0 i„ dv i„ + v u dv u ) 4* mz(vzx dv 4~ 02 u dvz v 4* 0 s* dvz z ) = 0. (48) Equation (46) will then also remain satisfied, provided dg i dvu do i, . , 1 ,. I 00*1 dv i* + ~ dvzx + ^; v dVtv + & dVlt = 0. (49) Now we can eliminate some of these chffeientials by solving (47) for dvsx, dvt Vl dv tz and substituting the values so found in (48), obtaining thus in place of (47) and (48) mi(vix — 0s*) dv u + mi(vi v — v iv ) dv i„ + mi(v u - v 2z ) dv u — 0, (48a) Any values can be assigned to dv i», dv iVl dv i, that satisfy this equation; the corresponding values of dvz x , dv a „, dv s, are then given by (47). Equations (36a, b, c) can always be preserved, with no change in Vu * * Fa,, by varying two of the quantities X, n, v, the thiid being then chosen so as to keep X 2 4- ju 2 + v 1 — 1, The aigument could S®a 27 ] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 43 be completed now by further elimination^ of differentials; but it is neater to make use at this point of Lagrange’s method of “undetermined multipliers.” To do this, we first eliminate dv 2zj dv a v , dvu in a similar way from (49) as well; then we add to the equation thus obtained eq. (48a) multiplied by a quantity Q, “undetermined as yet, obtaining + [St - ££ + Qm ‘(" u - *■■>] *- ” (M) The coefficients of the three differentials in (50) must now vanish just as if the differentials were completely arbitrary, Tor, suppose the coefficient of some one differential in (48a) is different from zero. Lot us choose both of the other differentials in (48a) arbitrarily, and then give to the first one such a value as will make (48a) true; lot uh cause the coefficient of the latter differential In (50) to vanish by means of a suitable choice of Q. Then the two remaining coefficients in (60) must also vanish because the other two differentials aro arbitrary. From the three equations obtained by equating to zero each bracket in (50), we can then in turn eliminate Q. For example, the firat two yield _ „ J tyi _ mi dgA _ , J dgi m t dg 3 \ /rn (vi y v 3v )\^~ — d —J ~ (vi* - - ~~ Q-). ( 51 ) This equation, like (40), must hold for all values of v ix ■ • • v 3 , as independent variables; hence wo can difforontiato both members partially with respect to any one of these variables without destroying the equation. Differentiating first by v lt , then by v iVl we thus find in succession, many terms dropping out: Ohi i lb,,) d 2 f/i dVudVix (Vu Vix) aVi 0vi»dv i„’ d Off i dri, Stiix 0. Choosing two other equations and proceeding similarly, wo find that ~ °* Henee §f^ x ‘ s independent of v lt and v lv and so Is a function of «i» alone. In a similar way one finds that ovory first derivative of gi or g% is a function solely of the variable named in tho derivative. Using this KINETIC THEORY OF OASES u (Chap II result and differentiating (51) by in* and tq„ in succession and then again by Via, we find successively (Vl U - Vly) d*g i dvl dfyi _ dfyi dv\ x 0v 2 J dg i dPu, m_i 5(72, m 2 dy ; 2V 2Si-o dvL u ’ (52a) (52b, c) In (62c) we have finally before us a differential equation from which we can find gi Integiating this equation three times we find gi = CiV\ x + CiViz + Gi{V\ v> Vu), where C l( C 2 , Gj are constants of integiation; Gi may, as indicated, be a function of V\ v and Vu, but Ci and C 2 cannot be because we found dgi/dvi x to be a function of v lf only, Working similarly with V\ v and then with Vu, one finds also that gi - Cs»l v + CiVi u + G»(vi X) Vii) and gi — Csfi* + CtVu + Gs(vi 0 , Vi v ) Substituting these various values of Qi in (52b), and in its analogue containing vu in place of Vt v , wo find, however, that Ci = Ci = C 6 Comparison of the three forms then shows that (J\ can be written finally in the form, lOg/l S3 01 = ~PKvjx + V\y + O + Ci'yVlz + <*\V it/ + ct'i'v u + Bi, (63a) —0\ being written in place of Ci for future convenience and a[, a”, a"', Bi being four other constants. In a similar way one finds log /a = Oi — ~0l(vl x + v% u + Vi,) + afrit + ot"vti i + a i' v a* H" (53b) in terms of five additional constants If wo then substitute these values of gi and p 2 in the three equations obtained by equating to zero the three brackets in (50) and then set all of the vaiiables Vi x • vs, equal to zero, we obtain such results as «i — = 0, and then, using this and making only vi» or vi* not zero liv 2 in the same equations, — 2j3f + Qm = 0 and — Qmi - 0, etc ; from these lesults we find that we must make m$\ - miPl, (64a) ffijal = Whaj, mia'l = mia'i, mia"' — mia 2 " (64b) (and incidentally Q — Aside from these restrictions the constants a, 0, and B are arbitiary , This assertion requires verification because of the way in which we restricted the variations of vu v it , but its truth becomes apparent Sbo. 28] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 45 when we reflect that we have carefully preserved the validity of (34a, b, c) and (35), and note that substitution of ffi and ff?. in (46), with any choice of the constants that satisfies (54a, 6), yields an equation that can also be deduced by adding (34a, b, c ) and (35) multiplied respectively by a! u a ", and — and canceling out in x . Finally, we may note that our work lias in no way depended upon the existence of a difference in the values of mi and ms and, accordingly, it will hold good also if only ono kind of molecule is present; the result in that case can be written log / = g — —fi 2 (v* + v* + v 2 ) + a'v x ■+■ ct"v v + a'"v, + B, (63c) where a', a", a'", ft, and B are all arbitrary, 28. Maxwell’s Law. The usual form of the velocity distribution law for a gas in equilibrium is now obtained if in (53c) wo put «' A a " „ « 0, viz., /(v) ~ f(v x , v„, v.) = Ae-fl 1 '’' (65) in terms of ft and a now constant A = e". Wo shall see in Sec. 30 below that (32 B K JL. = (50) p 2 kT 2 RT 2 R M T K \ m being the mass or M the molecular weight of a molecule and h, B and Em the gas constant for one molecule, 1 g and 1 gram molecule, respectively; hence wo can also write /(v*, v,) = = Ae- vViW . (57) When/ has this form, it is obvious from its symmetry that 1? 0, or the mass velocity vanishes. Accordingly, (56) or (67) has reference to a gas at rest. If wo then change to a frame of reference relative to which the gas has a mass volocity Vo, thoso equations will still hold for the velocity relative to the now frame or Clio volocity of thermal agita- tion v', so that the fraction of tho molocules with thermal velocities in the range dv x) dv' v , do 1 , is /'(v') dol (lv' u dv'„ where /'(v') « Ac-*”'*; (57a) but now for tho distribution of the now total velocities or values of v = Vo + v' wo shall have ( 076 ) 46 KINETIC THEORY OF OASES [Chap II since for the same group of molecules by definition of /' and / we have /(v 0 dv' £ d*>l d», = /(v) dv x dv v dv* and obviously dv* x ~ dv^ di>l = dv V) dv[ = dv Z) whcnco /(v) — /(V)* This last expression for /(v) can be obtained fiom the general form (53c) by choosing Vq and A so that v 0 * — <x'/2£ 2 , v 0u ~ a"/2p 2 , t>o* 858 a ft, /2ft 2 and = e* Thus we see that the more general result that was obtained in the last section had lefeience to a gas possessing umfoim mass motion with an arbitraiy velocity Vo. Equation (55) or (57) represents Maxwell’s famous law of velocity distribution in a gas at rest In view of what has been said it is obvious that a detailed discussion of this equation will suffice to covei the coso of a moving gas as well The constant A is fixed by the condition that « Jfda = Aje~P >c> dn — 1, the integral extending over all values of the velocity Now f 'er*v dK = dvyfjj-w dv, = ~ ( 68 ) since * e~ x ' dx — § VV, d$ = Vir, (69a, 6) as is shown in books on calculus Hence, by (56) Unfortunately a three-dimensional function like / cannot be ade- quately exhibited by a graph; we have tried, however, to suggest it by the distribution of the dots in Fig 6 Although only the speed v occurs as a vanable in /, it must not be forgotten that / has reference to a definite direction of the velocity as well, for the meaning is that out of, say, N molecules, Nf(v x , v„, v s ) dv x dv v dv x is the number having components respectively between v x and v x -j- dv mt v v and + dv u , v x and v x + dv t , and these limits effectively fix the direction of v itself within nanow limits Because of the symmetry, however, the distribution function for the moleeulai speeds irrespective of the direction of motion is also very useful, and this is easily found. The velocity points for molecules having speeds between v and v + dv he in a spherical shell in velocity Sec. 28] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 47 space of inner radius v and of thickness dv (of. Fig. 5) ; since dv is an infinitesimal, wo can write for the volume of this shell tho product of its thickness into the area of its inner boundary or dv X 47ri> 5 . Tho value of f is the same at all points of this shell, hence the fraction of tho molecules included in it is / X 4wV 2 dv, and if we also write for this /„ dv, where f v (v) is the distribution function for the speed v , wo find f v = 4t rr 2 / - ^rAv 2 e-^\ (Olo) A graph of %f v is shown in Fig. 9, drawn on the assumption that £ = 1. This graph really serves to give practically a complete conception of the distribution of velocities, for the only feature not exhibited in tho fact that all directions of motion occur with oqual frequency. » Problems. 1. Show that if f { d{' is tho fraotion of tho molecules whose kinetic energy, f *= £ mv", lies in a range then im 2. Show that the distribution function for one component, say fjvx), /» dv x being the fraction of the molecules with v x in dv x , is [cf. oq. (31)]. The curve for/, has thus tho shape of an error curve. It also is shown, on a reduced scale and with § — 1, in Fig, 0. 3. Show that the distribution of tiro y- and the 2-compononts of velocity among molecules selected with ^-component between v A and v x + dvx is independent of v s , (Oiut* II 48 KINETIC THEORY OF GASES 4 Show that / is the function that would result if each component of velocity were distnbuted independently of the others according to tlio law expressed by (62) (The assumption that the tluco components are independent constituted the basis of Maxwell's first attempt at ft dcduction of his law ) . 29. Use of a Distribution Function in Calculating Averages. Knowing/, we can easily calculate mean values of various functions or the velocity It may be useful to desciibc fust the gonoial method of calculating aveiages on the basis of a distribution function Suppose f(x) is the distribution function for some variable x, so that f(x) dx is the fraction of all cases in which x has a value lying between a: and x -f dx , and also J f(x) dx = 1, the integml extending ovor all possible values of x Then, if there are N cases m all, the average value of any function Q(x) is 2 Q(x)/N, 2 Q(x) denoting the sum of the values of Q for all cases In those cases in which a: lies in. any given range dx, however, which aie Nf dx in number, <3 has practically ft constant value; hence the contribution of thoso cases to 2Q will bo QNf dx Thus we can write 2Q = 2QNf dx = NEQf dx or N J Qf dx. The average of Q is then Q = N j Qf dx/N or Q ~ J Q{x) f(x) dx, (63«) the integral extending over all possible values of x. Frequently, however, the "normalizing factoi ” in / tlmt makes jfdx = 1 is not yet known, i.e , we have not / itsolf but a function of * proportional to it, say fi(x) = CJ(x) Then / = fi/C, Jfdx = Jfi dx/C <= I and 0~ J fi(x) dx In terms of the unnormalized distribution function fi(x), expression (63a) for the average value of Q can therefore be written 0 = J Q(i) fi(x) dx J fi(x) dx (631>) 30. Most Probable and Average Speeds. According to (55), Lho most probable single value of the velocity is zero, since /is a maximum for v = 0 On the other hand, the most probable value of the speed, or the value of v that gives a maximum value to /» in eq (6 la) , is found by equating dfjdv to zero and solving for v and is Sec. 30] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 49 (64) To find tho mean speed v t and the roo t-mean-squar e speed v 8) we substitute f v from (61a) in (63a) and set first Q = v and then Q « v 2 ; this gives us the two equations, V ~ 4?r dv t dv. Such integrals are readily reduced with tho help of an integration by parts, those containing an even power of v reducing finally to (59a, b); a table of the ones most used in kinetic theory is given at the end of the book, We shall work out only one example in detail: l v i e~ p,v> dv — df) by (59a), the change to x being made by writing f3v — x, and tho inte- grated part vanishing at tho upper limit because, if v — > <*> , v tt e~ av ' — > 0 for a > 0, s > 0 and any value of n. The equations previously obtained reduce in this way, after inserting A from (60), to (06a) ( 656 ) (06c) Wo note that the mean speed 0 exceeds v m slightly, obviously because tho curve for/* bulges a little toward larger values of V] the root-mean-square speed v, then exceeds i> in turn because the squares of large values of a contribute especially heavily to v*, Tho relations between v,„, v, and v, are indicated in Pig. 9. Somo physical phe- nomena, such as tho pressure, depend in a simplo way upon v„ but wo shall encounter others presently that aro more simply described in terms of S. Hardly any phenomena involve, v m directly. For many purposes of rough calculation, however, it does not matter much which average is employed. i • i Equations (666) and (26a) in Sec. 15 taken together yield the values of ft given in (60) above. Using theso values of ft, we obtain also the alternative expressions: ' 8 = | v m = 1.1284 v m ~ 0.9213 v„ ■yrP V7r 4,2 — JL, * 2 v, = {%V‘v m - 1.2248 v m = 1.0864 v. 50 KINETIC THEORY OF OASES ((’rw li _ J2BTY J2RhT\ h j2kTY , 8 = 2 \~j “ Tar j “ =W ' M being the molecular weight or m the actual mass of a molmjlp Another quantity derivable from / that is sometimes of \m* h %\* total fraction of the molecules that have speeds m velocity ponents within some given finite range Such fractions arc written down m terms of integrals. Fig 10 — Tho probability integral, $(x) • ~r fV*&. V* Jo For example, the fraction of the molecules that have speeds aim**' & certain value v is simply J* /« dv — 4rA v 2 e~ /3 ’’ 1 ' dv = ~= p(verP v> + tlvj, after integrating by parts and inserting A = /3 3 /V ?< Tho last integral here cannot be expressed in terms of ordinary functions but mu»f I® regarded as defining a new transcendental function Tho lntt«T, known as the probability integral, is commonly defined thus: *(*) = 4= P e “” dx, (Wj Vrjo so that $(0) = 0 and $(») =1 by (59a) Tables of <I> are given in many reference books (eg, in Peirce's "Table of Integrals” or (ft Jahnke-Emde's “Table of Functions”), a graph of it, along will* 1 — e’~ V**, whioh has the same area under it, is shown in Fig. Id In terms of $ we can write, since (S dv = *JT e~P v ‘dv — dv = dx — ** i f v dv 1 H — jz five-** 9 * — $((tv) \ir (Ml Sec. 31] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 51 Substituting values from the tables in this formula, in which wc put an « (3v = (2/ \/w) (v/v) by (65a), wo And for the percentage of the molecules that exceed various values of v/t> the following: v/0 0,6 1.0 1.6 2.6 3.0 % 88.80 46,70 12.66 1.70 0.12 0.01 ' Problem. Show that, if tq is any cartesian component of v, = (p/p) M or Newton’s value for the speed of sound in a gas. 31. Mixed Gases. Equipartition. When two different kinds of molecules are present, their separate distribution functions for the state of complete equilibrium are given by (63a, 6) above. Let us simplify these expressions by subtracting the common mass velocity Vo given by ®o« = «i/2j3f = <4/2/3!, v 0v — ot'//2p\ - ot'J/2p »« - m - i if. (Mo, 1)1; that is, we substitute v, = v 0 + v{, v 2 = v 0 + v$ with tho stated values of Vo in (53a, 6). We thus obtain: log/i = ~/3 Kofi + v[l + vZ) + U i + P14, log ft — —filivZ + v'/j + v'Z) + B 2 + whore, of course, vl stands for vl x + t>o„ + «or The fact that this simplification succeeds with the same value of Vo for both kinds of molecules shows that the mass velocity is the same for both. Wc shall drop the primes, however, since it suffices, as before, to discuss tho motion of thermal agitation only, and write as tho result: /i(vi) = Aitr?'** 1 , ft(v s ) = These equations represent a maxwellian distribution for each component gas. As an extension of (066) wo shall accordingly have Sf — | l//3f, ®| = 1 1/Pl and taken in conjunction with (54a), which can be written tthfil = ttijfil, this means that i miof = (69) LThus our analysis of tho effects of collisions loads to equality of tho mean translatory kinetic energy of gas molecules when mixed together find confirms eq. (14) in Sec. 11 of Chap. I. This is a special caso of the general theorem of tho equipartition of energy which is proved in statistical mechanics (Sec. 206). That collisions in a gas tend to set 52 KINETIC THEORY OP GASES [Chap II lip translatory equipartition of energy was shown in a less satisfactoiy but more direct way by Maxwell * All of the results that we have obtained for a homogeneous gas, which rested in part upon eq, (14), will accordingly hold with suitable modification of the constants for each component of a mixed gas This conclusion is easily extended to mixtuios of thice 01 moio com- ponents. Accoidingly, we can summanze by saying that m a gaseous mixture of molecules of diffeient sorts in completo cqiuhbiiuin onch kind of molecule has the same maxwelhan distnbution in velocity that it would have if the othei kinds were not present 32. Uniqueness of the Maxwelhan Distribution. The H-dheorem. In Secs 26 to 27 we showed that the maxwelhan distnbution was a possible equihbiium distribution foi the molecular velocities. Wo have not yet shown, however, that it is the only one. Any such distribution must make the right-hand membeis vanish m (42a, b) or (43), but our solution accomplished this in a diaslic fashion, for the integrand is made to vanish in each integral foi all values of tho variables of integration; it is conceivable that othei forms of tho function f might exist such that the integrals would vanish by mo 10 mutual cancellation of positive and negative conti lbutions from different paits of the legion of integration To this question we may also add the further one whether theie would be any tendency for an equilibrium distnbution to be set up if it did not alieady exist, tho possibility that the distribution of velocity might under some circum- stances never tend to become steady at all, while peiliaps not plausi- ble physically, is left open by the mathematics. Both questions were answered by Boltzmann. f He was able to show from his analysis of the effects of collisions that tho maxwollian distribution actually is the only steady one, and furthoimore that any other distribution would almost certainly be altered by the collisions in such a way as to approach the maxwelhan form. Since the ftigu* mept by which he did this, called the “H-theoi em,” possesses intorcst for thermodynamics as well as for kinetic theory, we shall give it boro, but only for the case of a homogeneous gas; the extension to mixed gases is easy to make Boltzmann studies the quantity H = fflog/dK (70 a) * Maxwell, PMl Mag , 20, 21 (1860) f L Boltzmann, Weiteie StucUon iiber das Warmegleichgewiohfc untor Gtx»- molekulen, K Akad TFm i}V%en) Siizb , II Abt 66, p 275 (1872) Sec, 32 ] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 53 in which as usual da stands for dv x dv v dv g) f « f(p m v Vi v g , t) or the velocity distribution function, and the integral extends over all values of v; wo have written t as one of the variables in/ to indicate explicitly the fact that / may be changing with the time. If / does not change, we have a steady distribution, and then II is constant. Other- wise II will be a function of the time with a derivative since the limits of the integral in the expression for II do not involve t. The 1 added to log / hero, however, adds nothing to the integral, for — 0 because ck must always remain equal to unity. Hence, if we insert the value of df/dt from (43) in Sec. 26, we have dll ell <pv r (FF ’ — ff') log f dx dx 1 du. Now this definite integral is distinctly asymmetric in the variables of integration. Such a circumstance should always arouse a suspicion that something interesting may turn up if the variables are- inter- changed. Wo can, for instance, interchange the variables v and v' (i.e., v x , »i/, v» with v' x , v' z ) without altering the value of the integral and obtain as another form of the equation the following: But wo can also change to tho velocities after collision as variables of integration, i.e., to V and V' as determined by (34a, b, c ), (36), and (3Ga, 6, c) in Sec. 23 in terms of v and v' ; tho method of making this change is just the reverse of that explained above in arriving at (40) of that section. The result is [cf, tho form of (37)] -f) log fdKdK'dQ. It is immaterial, however, what symbol ia written for a variable of integration in a definite integral, so nothing provonts us from simply replacing tho variables V, V' by v and v', respectively, whereupon / and F change places and likewise f and F\ and wo can also write du for dO; then <I> and V, become those functions of the variables which wo 54 KINETIC THEORY OF OASES (Ciup II previously denoted by <p and v,. This gives us anofclier possible form of the equation expressed in terms of v and v'; and then intorohmiRiiiH v and v' again gives us a fourth We thus obtain ^ « J J J - FP') log F dK da' do>, ^ “ */ J J tpVr( ' S ' ~ FF ’^ bg F> dK dK> d0> Now let us add these two values of dH/dt to the fust two ami div id* 5 by 4, This gives finally a form that is symmetric in the varmblr* v and v' : ~ J J J - FF 1 ) (log f - log FF>) dK dK ' do {im Here the integrand can never be negative, since log ff' ?! log FP* according as ff' | FF' Hence we have the important losull that in all cases Now for a steady distribution dll/dt must vanish. But in we now have an integral which can vanish only if the integrand vuni.'dir-* for all values of the variables of integration If this happens, wo uro led at once to eqs (45a, b , c) of Sec 26 and, from those equation*, ft* we have seen, to Maxwell’s law This shows that the mavwolhnu distiibution is the only steady one Any other form of /must undergo such changes by collisions Hint fl continually decreases It is easy to show,* however, that II line ft minimum value for the maxwelhan distribution as compared with nrty other that gives the same value to ifi, a quantity that does not cluing** with the time in an isolated ideal gas Hence we reach tho furlli*"r conclusion that the effect of collisions upon / is such as to makn If approach the value corresponding to a maxwelhan distiibution; and * In general, let fa, f be any two functions of v such that J/log/od* “ J7» log /o dK and ffdK = f/adK rn I, Then D ~ ff log fd. - J/„ log fo (Ik « // log (///.) <A * f f/log (f/fo) — / + fol dK (the terms added here canceling m*U other), and. [/log (///o) — / + /o] has an absolute minimum value of 0 for / ** /* Hence D ^ 0 As a special case, the first condition imposed on / is equivalent, in view of the second, to the conservation of if f Q =* Aer^ tv% Seo. 34] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 66 we can take this as an indication that the distribution must itself tend to take on maxwellian character with the passage of time. Light could be thrown upon the rapidity of tho approach in any given case by calculating the value of dH/dt from (706). In Sec. 100, however, we shall meet with a more convenient means of estimating the rapidity; it is shown there that equilibrium is approached exceed- ingly rapidly as judged by physical standards, 33. Reversibility and the H-theorem. Although tho proof just given of tho //-theorem is mathematically rigorous, an illuminating objection was formerly raised against it. Suppose we were to lot tho gas move, with H decreasing, for a short time and then instantaneously reverse all of the molecular velocities. Since every dynamical motion is reversible, the molecules would then retrace their paths, and // would increase, in spite of our proof that it never can do this I The explanation of this paradox is connected, of course, with tho fact that the formula for df/dt which we obtained above rests upon tho assumption of molecular chaos. Now, after a few collisions havo occurred, there is no reason to suppose the molecular motion to bo ordered toward the future, so that chaos should still exist so far ns subsequent collisions are concerned; but tho past motion may not havo been chaotic. Accordingly, when wo reverse tho velocities and thereby interchange future and past, it may happen that wo have con- verted the motion from a chaotic one to an orderod one (as regards the new future), and the //-theorem will then no longer bo applicable. Only in the case of equilibrium can it be assumed that tho revorso of every molecularly chaotic motion is again a chaotic one. Some further remarks upon tho //-theorem and in particular upon an interesting connection between tho quantity H and tho entropy will be found in tho chapter on statistical mechanics. 34. Principle of Detailed Balancing. As we have soon, tho equilib- rium of the velocity distribution is maintained by the molooulcs in an exceedingly simple fashion, for in each of eqs. (42a, h) or (43) of Sec. 26 the two terms of the integrand cancel each other for each individual type of collision, This moans, as is evident from tho origin of those terms, that for every typo of collision the inverse typo occurs with equal frequency and exactly undoes tho effect of the first. Many cases can be cited in which a physical equilibrium is main- tained in a similar simplo fashion. Tho characteristic feature is that for every process which alters the distribution thoro exists an inverse process, and in the state of equilibrium each process and its inverse occur with equal frequency and just offset each other's effects tending to disturb the equilibrium. This has been called the phenomenon or KINETIC THEORY OF GASES [Chap II 66 principle of detailed balancing* ancl it has sometimes been set up as a universal law of nature The pnnciple seems, howevci, to bo rather difficult to state in teims that aie broad enough to cover all cases of mteiesfc ancl yet so as to admit of no exceptions, nor can it leacbly be deduced ill general fcoims fiom the laws cither of mechanics or of theimodynamics It seems lathei to constitute a goneiahzation which is subject to a few exceptions but nevertheless possesses gieafc utility when we aie endeavoring to form conceptions of the piocessos that occur m natuie or to make a fhst-tnal analysis of an unfamiliar phenomenon In some instances the motion involved m the inverse process is just the reverse of that in the direct pioccss, but m many othei cases this is not so, and there does not seem to exist any fundamental connection between detailed balancing and the fact that mechanical motions can always bo “reversed” or made to occur m the levoise dncction (m classical theory at least) In the paiticular case of molecular collisions which we have been studying, for example, the mveisc of a direct collision which offsets the effect of the lattei is not the same as the reveise collision that results if we simply leveise the velocities of the two molecules after the first collision and allow them to letiaco tlioir paths, nor is it even (in general) this reverse collision lotated through some angle On the other hand, in phenomena such as the following it is con- venient to apply detailed balancing in such a way that the mveise process is the direct one reversed When liquid in a tube is m equi- librium with its saturated vapoi, just as many molecules condense ill every second at any point of the suiface as evaporate there, so that no mass streaming is set up, such as might conceivably occur m the liquid from the meniscus at the edge back toward the centci, oi in tho opposite direction, owing to possible differences m the late of evapo- ration over a flat and over a curved suiface Examples of detailed balancing of this soit are common in physical chemistry. As a lathor similar example in another bianch of physics, we may cite the fact that the emitting and absoibing powers of a body for radiation aro always equal, not only on the whole but also m every individual direc- tion and for every separate wave length Another case that is of particulai interest in connection with tho question of mechanical reveisibility is furnished by the motion of ions * Detailed balancing appears to be a bettor term than Tolman's li microscopic reversibility" (Nat Acad Pioc , 11, 436 (1926)) because the lattei suggests that the motions in the balancing process are just those in the dnect pioooss reversed* which is often not the case Sec. 34 ] DISTRIBUTION LAW FOR MOLECULAR VELOCITIES 67 or electrons in a fixed magnetic field. This motion is completely non- reversible; if an ion moves along a right-handed spiral in following a line of force in one direction it will follow a left-handed spiral on the return trip. To reverse the motion, we should have to reverse the field as well. Nevertheless, even in the fixed field, under conditions of equilibrium, there exists the usual maxwcllian distribution of velocity, so that at every point there are equal numbers of ions moving equally in all directions, and hence, for example, as many cross any area in one direction as in the other, as is required by detailed balancing. Perhaps the simplest example of a failure of detailed balancing is furnished by the effect of the walls of the containing vessel upon the velocity distribution in a gas. Suppose, for simplicity, that the density is so low that intennolecular collisions are rare and suppose that the walls reflect specularly (in other words, like the reflection of fight by a mirror). Impacts upon the walls will now serve to sot up and maintain an equal distribution of molecular velocities in direction, although, of course, without affecting their magnitudes, provided only that the vessel has not some special shape such as that of a rectangular parallelepiped. The equilibrium as to directions of motion is main- tained in a complicated circular fashion. For concreteness, lot the vessel be hemispherical in form. Then each shower of molecules moving in some definite direction downward toward tho plane base is continually being transformed by impact on it into a shower moving in a definite direction upward. Very few of the molecules of this second shower, however, are restored to the original downward-moving one upon striking the curved surface; they are scattered equally in all directions throughout a certain solid angle, which rises to a maximum of 2w in tho case of the particular shower that is moving vertically upward. Tho process inverse to reflection from the base is thus almost entirely absent, The first shower is continually being recruited some- how by reflection from tho curved Avail; but tho molecules that are thrown into it are drawn from a wide range of different upward-moving shoAvers, Perhaps we should say that detailed balancing fails in this case because the equilibrium is not a true one, and avg have, therefore, not really employed the fundamental form of the principle; a mass of gas enclosed in a vessel of any sort represents, in fact, a inctastable state of the system. To have true equilibrium, there must exist some path by which molecules can pass from one side of the Avail of the vessel to the other, since they arc capable of existing on both sides; and if tho wall itself is composed of molecules, then those must occasionally rearrange themselves so as to take on, in tho course of time, all possible 58 KINETIC THEORY OF GASES [Gimp. II arrangements If such changes were possible, it is probably true that in the long run detailed balancing would hold In other words, detailed balancing probably holds for all interactions between particles that are statistically free to get eventually into all positions that nro mechanically possible, and also for the emission and absoi plion of ituhtl- tion by such particles 35 Doppler Line Breadth. Aside fiom many phenomena in which the distribution of molecular velocities plays a lole, in the sense that an average must be taken ovei all velocities, there are a few which exhibit details coirespondmg to the distnbution itself The most straightforward one of these phenomena involves an additional bit of kinetic theory and is described in the next chaptoi as a cloliboiale lest of Maxwell’s law On the other hand, the effect of molecular motion on the shape of spectial lines, while it involves an additional process optical in nature, depends so directly upon the velocity distribution itself that it will be described briefly here The spectrum emitted by gaseous atoms and molecules consists under most conditions principally of lines which have a natural width that is too small to be observed with ease As seen by the spectres- copist, however, these lines are usually widened into nariow bunds in consequence of molecular motion When a souico emitting light of wave length Xo is moving in vacuo away from tho obsoivor with n component of velocity w, the wave length as measured by the observer is, by the usual theory of the Doppler effect, X = X 0 ^l + c being the speed of light Accoidmgly, if there are du molcculcn in unit volume, which are emitting radiation of natural wave long ill X# and are moving away from the observing appaiatus with a component velocity lying in the range du, the light from these will bo received in fl du spectral range dX = \ 0 — If we write J d\ foi this Xftdmtion, J representing, therefore, the intensity of the lino at wavo length X, we have the equation, J d\ = J\ 0 ~ = y nf u du, where y represents the amount of radiation received from one molecule If wo thon cancel out du and insert the value of /„ for a maxwelhan distnbution from (62), changing v , in that formula to u = -ft ~ x °), wo obtain Xo J = J oe ~ 2 J o standing for a new constant Sbc. 86] DISTRIBUTION LAW NOR MOLECULAR VELOCITIES 59 A plot of J against X thus has the form of an error curve with its maximum &tJ = Jo', and if log J is plotted against (X — An) 2 for either half of the spectral line, a straight line is obtained from whoso slope the temperature of the gas can be calculated (cf. Fig. 11). An experiment that confirms this form of the curve was performed, although for another purpose, by Ornstein and van Wylc in 1932.* In this experiment they studied one of tho linos emitted from an electric discharge passing through helium at very low pressure. Tho observed shape of the line corresponded closely to that duo to a maxwellian distribution but at a temperature some 50° above tho tubo containing tho helium; this difference was ascribed to heating of tho gas by the discharge. Some classical experiments made much earlier by Fabry and Buissonf had .shown only that the widths of certain lines emitted by tho gases helium, neon, and krypton varied with temper- ature very nearly in the right way to correspond with tho Doppler explanation, being direotly proportional to the square root of the absolute temperature. * Ounsteint and van Wyk, Zeila, Phyaik , 78, 73d (1932), t Fabry and Buisbon, Acad, Sci., Compt. Rend,, 164, 1224 (1012). CHAPTER III GENERAL MOTION AND SPATIAL DISTRIBUTION OF THE MOLECULES Up to this point we have discussed effects of the velocities of tho molecules without paying much attention to then positions, and wo have found that ceitam piopeitios of gases can bo adequately handled in this way There are other phenomena, howevei, which depend also m Laige part upon the distribution of the molecules in space It will be convenient to piepare foi the tieatment of such phenomena by taking up at this point ceitam aspects of the molecular motion which involve their distribution m space, The tedium of the abstract argument will be relieved at mtei vals by the discussion of concrotO applications* UNILATERAL FLOW OF THE MOLECULES 36, Effusive Molecular Flow* As our fust topic, we may take up for consideration the general flow of the molecules fiorn place to place in a gas A concrete example suggesting such a study is pie&entod by the effusion of a gas through a hole into a legion of lowei piessuro, The chaiactei of this lattei phenomenon varies greatly according as the hole is large 01 small, If the hole is sufficiently laige, tho motion can be handled theoretically by the methods of ordinal y hydiody ninn- ies, the treatment of this case accordingly lies outside of oui field, but its geneial conclusions may be cited for purposes of oompaiison. It turns out that, as the pressure difference between the gas in tho 1 vessel and that in the legion outside is increased, the outflow of gas varies at first in proportion to this diffeionce If, howover, * the outer pressuie is lowei ed so as to he below a certain cntical value, which is a little more than half of the inner pressuie, then the magnitude of the outer pressure no longer makes any difference at all m the outflow. The maximum velocity of tho gas in the issuing jet is then equal to tho * speed of sound thi ough it at the tempei at ure and density existing m tho 1 . jet, which is cooled by expansion The fact that undei these circum- stances the outside pressure has no influence upon the rate of outflow 1 can be explained by saying that it is no longer possible for a signal * to work back through the jet into the vessel and, as it were, notify tho 1 60 ! 61 Sec. 37 ] GENERAL MOTION AND SPATIAL DISTRIBUTION gas inside of the very low pressure existing outside. Under such conditions, and even before such speeds are reached, the issuing jet has a contracted form, narrowing to a certain minimum cross section at a short distance from the hole, beyond which it expands again, but often with oscillating cross section; and at speeds of outflow above a certain low minimum the jet is also surrounded by eddies and the whole motion is more or less turbulent. « Suppose, however, for contrast, we now go to the other extreme and make the hole small even as compared with the molecular "mean free path" or average distance traversed by the molecules between collisions. Then, according to classical theory, the molecules must issue independently of each other in the form of a molecular stream, each ono moving with the velocity it had as it came up to the hole. The loss of a single molecule now and then through the hole should disturb only slightly their general distribution inside 6f it; a trace of mass motion toward the hole must develop because of the absenco of those collisions that the lost moleeulo would have made with others on its return from the wall, but this effect will bo wiped out promptly by the molecular interplay, which is always tending to set up and preservo the equilibrium state. If there is gas in equilibrium on both sides of the hole, a process of effusion of this sort will occur in eaoh direction just as if the gas on the other side were absent. If the hole is now widened until its diameter is comparable to the mean free path, an intermediate typo of flow occurs. As the hole is widened, mass motion toward it develops more and more in the vessel as a result of intormolecular collisions (or their absence), and this mass motion, carried along in the issuing gas, tends to result in the formation of a jot of forward-moving gas outside. In this way a continuous transition occurs from molecular to what might bo called hydrodynam- ical streaming. In the case of the very small hole the outflow of molecules through it should be the same as their flow across any small plane area of equal size drawn in the body of the gas, Accordingly, wo shall now consider the latter more general case in detail, 37. Formulas for Effusive Flow. Consider a small plane of area S drawn anywhere in a mass of gas that is in oomplote equilibrium, Molecules will be crossing S continually in both directions; lot us fix our attention upon those that cross toward ono side only. These molecules, might be said to constitute a maxwellian effusive stream. Out of n molecules per unit volume, 4jr nAv 2 6~ ,>, ' ,, dv are moving with speeds in a given range dv, by (61a), and, since they are moving equally in all directions, the fraction dto/d?r of them are moving in 82 KINETIC THEORY OF OASES (Chap, III directions lying in a given solid angle du whose axis makes an anglo 6 with the noimal to S [cf (29) in Sec 181 Of these molecules, as many will cross S during a given intei val dt as lie at the beginning of dt withm a cylinder standing on S as base and having slant height v dt and hence a volume Sv dt cos 6 (cf Fig, 12) Hence the number so crossing is the number in the cylinder at the beginning of dt or 4tt ■nAv i e-P">' dv(~^Sv dt cos 0. Dividing this expression by S and by dt, we have, accordingly, as the number of molecules crossing S per unit area per second, with velocities lying in the range dv in magnitude and in the solid angle rfco of directions, 7iAv 3 e~P tv \ cos 0 dv do> (71a) where A = /3 3 /V* by eq (60) and /3 is given by (66) in Sec 28. Thus the molecules that cross a plane in a given time are distributed in direction according to the law that is familiar m optics as Lamberts or the cosine law As in the analogous optical case, it is often useful to sum further over all azimuths about the normal Taking the normal to S as the axis for polar coordinates 8 , ip, we can write do) = sin 6 dO d(p and then integrate (71a) over all values of <p (cf Fig 13) Since the integrand is independent of <p and f** d<p — 2?r, the result is 2mAv *e-P ivi sin 8 cos 0 dv dQ , (716) representing the total number of molecules that cross S per unit area per second with speed in the range dv and with a direction of motion that makes an angle lying in the range dO with the normal to 5; the values of A and P are as stated above, just below eq. (71a), Of course, we could also have obtained this result directly by writing sin B dO from (30) m the place of du/in The presence of v 3 in (71a) and (716), as contrasted with v 2 in the distribution formula, means that high speeds are relatively commoner among the molecules that cross a plane than they are m the gas in general, obviously because the faster molecules stand a larger chance of crossing the plane in a given time Sec. 37) 'GENERAL MOTION AND SPATIAL DISTRIBUTION 03 These formulas characterize adequately the distribution of veloci- ties in a maxwellian effusive stream at temperature T. In some con- nections, however, much less detail suffices; we may need to know only the total number of molecules that cross, regardless of their speed or direction. This number can be obtained by integrating {716) over all values of ■j r 0 from 0 to tt/2 and of v from 0 to °o . Since sin 6 coa 9 dO ~ and an integration by parts gives j' o ‘°v 3 e~9 ,u 'dv = l/(2/3 4 ), wo thus obtain \ mA/f}*, or, after substituting A — 3 3 /V^ from (00) in See, 28 and 1/(3 = -\Arv/2 from (65a), r n = J ms (72a) as the total number of molecules crossing unit area of S per second toward one side. Here v stands as usual for the moan molecular speed. For practical applications, however, it is more convenient to replace nv by immediately observable quantities. Writing r„, for tho mass and IV for the volume of the gas that crosses unit area per second toward one side, or IV for the pV value of this gas, wo have r M = toIV with nm — p = v/RT, Tv — T„/n and IV — pIV, and henoo, after inserting b = 2(2RT/v) H from (60a) in Sec. 30, T ~~ i f> ~ f {%r) ~ (&•«?')“' I ' 12b> Here p is the density and p tho pressure of tho gas, T is its absolute temperature, and R refers to a gram, or R = Ru/M, where Rm is tho gas constant for a mole and M is the molecular weight. Equation (725) shows that the rate of molecular flow (in either direction) across S is tho same as if tho gas wore moving bodily across it with a speed equal to one quarter of tho mean speed of tho molecules. According to what has been said above, those formulas should apply not only in tho interior of tho gas but also to effusion through a hole in the wall of the containing vessel, provided its diameter is small as compared with tho molecular moan free path. Tho symbol p in (72 b, d) then stands for tho pressure on tho side from which tho gas comes. If there is gas on both sides of the hole, effusion will obviously occur in both directions independently, for under these conditions a molecule runs very little risk of being stopped by a collision as it 64 [Chap IH KINETIC THEORY OF GASES passes thiough, the net flow thiough the hole is then the difference of the two opposing umlateial flows For convenience of futuie leierenco a number of olhor useful foimulas aie appended below, the proofs being left as oxoicisos foi tins student Problems 1 Show that the total numbci of molecules croHHiim unit aiea of S pel second in a dnection lying within an element dto of solid angle is i nv cos 0 clo 3 — ~ cos 0 cloi, (73 <0 Air 7 r \27r??z/ [E g , integrate (71a) and use (65a), (56) ] 2 Show [eg, fiom (62)] that the number of molecules ciossmS unit area of S per second toward one side with a component of velocity normal to S lying between and i>i + dvx is v x e-W dv± = dv x . (73 M 3 Show that, if the ct-axis is taken peipendiculai to S } of those* molecules that cioss S toward one side m a given time tho fraction — v x e~ fiivi dv x dv v dv z ( 74 ) 7T has velocity components in the langcs dv X) dv U) dv Z) where as UBUftl v 2 — v* + v 2 + v\ fcf Sec 28, Piob, 3) 4 Show that the mean tianslational enoigy of tho molecules Umfc cioss S in a given tune cxceedvS that of those piosont in a given volume* by the factor % and so amounts to 2kT cigs pci molecule or 2RT por giam, 38. Molecular Effusion, The effusion of gases through small holes* is sometimes applied m technical operations as a convenient roufgti means of determining gaseous densities * In such applications, how- evet, the holes employed are usually so laigc that mass motion occur** and the phenomena are consequently of littlo interest for kinetic theory Tho first case of true molcculai sti earning to bo subjected to expei imental study was the motion of gases through plates made of some poious material, such as gypsum oi meeischaum, the phenomenon being usually called transpiration If the canals tlnough such a plate are small enough, oi m any case when the gaseous density is low * Of Buckingham and Epwahds, Bull Bui Standi 16, 573 (1920) Sec. 38] GENERAL MOTION AND SPATIAL DISTRIBUTION 66 enough, the molecules should wander through them as individuals, little affected by collisions with other gas molecules, We cannot expect our formulas to apply quantitatively to such motion, but it seems clear at least that the mean rate of passage of the gas through the plate should be proportional to the mean molecular velocity, as in eqs. (72a, b, c, d), and so, according to (G6a) in Sec, 30, to V T/M, where T = absolute temperature and M = the molecular weight. The presence of another gas on the opposite side of the plate should make little difference, since collisions with a molecule of the other gas are much rarer than collisions with the walls of the canal; the two bodies of gas should transpire through the plate independently of each other or nearly so. The predicted variation with molecular weight was tested by Graham in the course of a series of experiments (1829 to 1863) and was found to hold reasonably well. Rates of transpiration have, accordingly, sometimes been employed in the determination of molecular weights, but it is difficult to make this method very exact. In his pioneer separation of helium from nitrogen Ramsey employed differential transpiration through unglazed clay pipe stems, the helium passing through the clay at a rate ( 2 %) M or 2.6 times as fast as the nitrogen so that by repeated fractionation .in this manner it could be got out fairly pure. True molecular effusion through a visible hole 1ms, however, been observed and studied by Knudsen.* Ho employed in one series of observations a hole roughly 0.026 mm in diameter in a platinum strip 0.0026 mm thick and worked with hydrogen, oxygon, and carbon dioxide at pressures ranging from 10 cm down to 0.001 cm of Hg, and at a temperature of either 22°C or 100°G. The gas was passed steadily through the hole under a pressure p' on one side and p" on the other, and the quantity of gas that passed through in a given time was noted. Effusion being bilateral in this case, eq. (72d) gives us for the not rate of flow measured in terms of pV (Knudsen's Q) (¥ )V - *">• The general result of Knudsen's work was to obtain a good check of the theory when the pressure was low enough so that the mean free path in the gas was at least ten times tho diameter of the hole. At higher pressures tho rate of flow was found to be somewhat greater than tho theoretical value as given above; and at a pressure sufficiently * Knudbion, Ann, Phydk , 28, 75 (1909), 60 KINETIC THEORY OF GASES Kjuap in great to make the mean fiee path less than a tontli of tho diameter of the hole he found agreement with the foimula furnished by hydro- dynamics for the case of slow isotheimal flow through an opening iKnudsen's results thus check the kmctic thcoiy veiy satisfactory. 39. Thermal Transpiration. A pecuhai case of comidoiuble theoretical interest is that called theimal tianspnation. Suppose two vessels containing the same kind of gas aie connected through a poi oua plug of the sort described above, and let them bo maintained at different temperatuies Then, accoidmg to (72b), tho laics at which molecules enter the poies of the plug pel unit aiea on tho two sides are pioportional to p/s/T If the poies were meioly small holes in an L indefinitely thin sheet, wo should, accord- * "V? m & ] y> expect a diffeiential flow to continue // until the gaseous densities became so adjusted 1 1 that the quantity p/y/T had the aaino value iiBH I on th « two sides, i e , until ■ (r 2 ^ 1 ^ 18 comm °nIy assumed that the samo ccmdi- 4% tion of eqmhbnum ought to oecui in tlio cnao __|f ?! an actual P lll g, and wo shall support this I i® Wwfa fflri theoretical conclusion in a later chap lor by I J \ \ analyzing the effect of a temperature giadimfc JJ \\ along a veiy narrow tube (of. Sec 180), <£/ k > \V Experiments of this typo weio performed To manometer ong ag0 Osborne Reynolds. * A plate of Fm 14 -Reynold.’ expen- ^thenware, stucco, oi moio usually rncor- ment on transpiration Schaum, ^6 to in. thick, Was mounted exposed on each side a cfrcX aieaTv^ “T *° ™ to ‘ CaV ° the rings were then !i!T f , ^ 111 m dl£ wnoloi (1% Id)} shaped spaces containing the tr S T? ° f tin S ° a8 to form difiIc " still other spaces were forme l anspmng gas > ancl outside of those ? y t ; nough ^ ^ on the other Conl l t? ° + ° lng watci known tompemlmo give the gas between the tin sheeSand the no IT Mhod upon to a TT7 °', the sW » ZSS#* opiMO ’“ m " Wy low, the*law ClTb*” mZZZt'l T ul.; held as closely as ho could judge * Reynolds, Phil Trans , 170, H, 727 (1870) Sec. 40] GENERAL MOTION AND SPATIAL DISTRIBUTION 67 with his rather crude way of observing tho temperature, but at higher pressures departures occurred; since the mean free path is, as we shall see presently, inversely proportional to the pressure, this is just what we should expect. He was not able, however, to observe tho actual size of the passages through the porous plate and so to make sure by comparison of these sizes with known values of the moan free path that the transition to eq, (75) occurred at a pressure of the right order. In an arrangement of this sort it is interesting to imagine the two vessels to be connected, not only through the plug, but also through a tube large in diameter relative to the mean free path. Then any tendency for a difference of pressure to be set up by transpiration through the plug will give rise to mass flow through the tube in the opposite direction. Thus there will occur a steady circulation of the gas as long as the temperature difference lasts. It might, perhaps, be thought that this steady circulation, or even the pressure difference in the original arrangement, involves a violation of the second law of thermodynamics. One might, for instance, connect an engine so as to take gas from tho high-pressure vessel and discharge it into the one at low pressure and thereby obtain an indefi- nitely largo amount of work out of the aystom. Thermodynamic compensation for this process is furnished, however, by the conduction of heat through the gas which is in the act of transpir- ing through the plug; it can easily bo shown* that the degradation of energy resulting from this conduction is ample to protect tho second law from violation. 40* Knudsen’s Absolute Manometer. Knudscn proposed to utilize the phenomenon of transpiration in a the construction of a gauge for the measurement of very low pressures.! He suspended a copper disk on T \ a torsion fiber (shown from above at a in Pig. 15); this disk hung with half of its surface very close and parallel to the plane face of a copper block which Kmuiaoka ’ n b- was heated electrically and surrounded by another soluto manomo- unhe&ted block 7 , designed to serve as a guard ring. 0 ‘ Temperatures of disk and block were read on mercury thermom- eters inserted into them. The narrow space between tho disk a and the block should then act more or less like a vessel in which tho gas has a temperature halfway between the temperatures of tho block and T\ of the disk, and if the pressure is so low that the mean free * Kennard, Nat, Acad., 18, 237 (1932). fKNUDSEN, Ann. Phynh, 32, 809 (1010). 68 kinetic theory of oases tciu* ni path m the gas is long relative to the distance between disk and block, the formula for thermal transpnation should apply Knudsen assumes that the disk has sensibly the same temperature as the suirounding gas, whose pressure p is what we dosno to measure, and this assumption seemed to be justified by his obscivations The piessure in the layer of gas between disk and block will accordingly bo, by (75), So that if T 2 — Ti is small, there is a net outward force per unit area on the disk equal to This force can be determined by observing the lesulting small dis- placement of the disk with the help of a minor mounted on it, and since Ti and T 2 aio known, p can then be calculated Knudsen showed by obseivations that the foimula held when the pressure was low enough and T 2 — T\ small enough; foi laiger T% ~ T\ he voiified a more accuiate foimula that he obtained by making a detailed analysis of the moleculai motion between disk and block and so intro- ducing a small collection on the mean tempeiaturo of the gas there, his result being that p f = p[(T*/Ti)H — 1] Because of the absence of doubtful constants in these formulas Knudsen called his instniment an "absolute” manometei In the moie convenient direct-reading foim devised later by Rieggcr, how- ever, this featme is sacrificed and calibration at one piessure with a MacLeod gauge is necessaiy * 41. Evaporation The foimula foi effusive molecular flow obtained m Sec 37 finds an inteiesting application to the late of evapoiation of a substance fiom the solid 01 liquid state. The eqinlibiium between a condensed phase and its saturated vapor is a kinetic one, molecules aio rapidly evapoiatxng all the time and passing off into the vapor, but at the same time a stream of other molecules is condensing out of the vapor, and when equilibrium exists, these two processes just balance each other If we upset the equilibrium by lowering the density of tho Rjeggeh, Zeits tech Physil, p 16, 1920 Shc. 41] GENERAL MOTION AND SPATIAL DISTRIBUTION 69 vapor a little, the rate of condensation is lowered in proportion; then evaporation exceeds condensation and there is a net passago of mole- cules from the solid or liquid into the vapor. What is commonly- called evaporation is this net flow of molecules. Similarly, if the density of the vapor is raised a little above the saturation value, the continual precipitation of molecules is augmented and not condensation occurs. In the case of volatile substances the unilateral evaporation and condensation go on at such enormous rates that only an imper- ceptible change in the density of tho vapor is required to produce the relatively slow net evaporation or condensation that ordinarily occurs under the limitation imposed by the necessity of supplying or removing the heat of vaporization. It follows that in such cases tho density of the vapor in contact with the liquid can be assumed not to depart appreciably from its Saturation value. The maximum possible rato of not evaporation would occur if arrangements could be made to remove the vapor as fast as formed, so as to eliminate condensation entirely; observations of this maximum rate would then give us also an experimental moasuro of tho unilateral rate of evaporation or condensation, which is not itself observable. On the theoretical side, a direct calculation of this maximum rate of evaporation is hard to make, but we can got at it indirectly by utilizing tho fact that when tho vapor is saturated, unilateral evapor- ation and condensation are going on at equal rates, and tho rate of condensation can bo calculated from kinetic theory. The mass of vapor molecules striking the surface of tho solid or liquid per second is easily found from oq. (72 b ) . A difficulty arises, however, from tho fact that some of these molecules upon striking the surface of the solid or liquid rebound from it without condensing, Tho best that wc can do, therefore, is to introduce an unknown factor « to represent tho fraction of the impinging molecules that do condense ; a has been called the coefficient of evaporation. The amount of a substance continually being evaporated from a liquid or solid phase, and also being returned to it from the vapor when this is saturated, measured in grams per unit area per second, is then G = eT,,, or, by (726), G m «p(~) = (2 viw*' (76) Hero 12 is the gas constant for a gram, T tho absolute temperature, and p the density or p the pressure of tho saturated vapor, so that p = pRT. G then represents likewise the "maximum rato of evaporation," which is observed when the vapor is removed as fast as formed. 70 KINETIC THEORY OF OASES [Chav III Unless a can be determined somehow independently, thoro is no hope of testing this result experimentally, beyond the fact that, since a cannot exceed unity, the equation obviously sets a theoretical upper limit which the rate of evaporation cannot exceed Wo can, however, turn the argument around and use obsoived evaporation latcs in combination with eq (76) to determine a This will bo done in tho next section It should be remarked that throughout the discussion up to this point we have been assuming foreign gases to be absent In a caso such as the evaporation of water into the atmosphoio, dcpaituic of tho vapor from the neighborhood of the watei suiface is impeded by tho presence of the air, so that if there is no wind to keep sweeping the saturated air away from the water, the vapor is compelled to diffuse away through the air, In such cases evaporation may bo a voiy slow process indeed, 42. Observations on the Rate of Evaporation. Intel eating observations of the maximum evaporation rate G foi moicury have been made by Knudsen, by Bi ousted and Hevesy, and by Volmcr and Estoimann.* The principle of the methods employed was to introduce the mercury into a high vacuum between the walls of a vessel shaped some- what like a Dewai flask, and then to contiol its tempeiaturo by means of a suiroimding bath while cooling the inner wall of the vessol Fig 16 —Arrangement with liquid air or other refrigerant so ns to for measuring tho evapora- , , , , . . , , , ... tion rate of mercury condense and hold upon it tho mercury that evaporated. The arrangement is shown in principle in Fig 16 The pressure was kept low enough so that the evaporating molecules on their way to the cold sui face would stand little chance of colliding with another moleeulo, and thoro is good reason to believe that reflection of mercury molecules from a mercury surface at liquid-air tempeiatures is negligible; tho ralo of deposition on the upper surface should therefore equal tho rate of unilateral evaporation from the mercury below In tho experiments of Bronsted and Hevesy the deposit on the cold surface was moasuiod by melting it and then running it out and weighing it * Knudsen, Ann Physik, 47, 697 (1915), also a book, "Tho Kinotlo Theory of Gases Some Modern Aspects” (Methuen) Bronsted and IIbvhsy, Nalurc , W7, 619 (1921), and Zeits Phys Chem , 99, 189 (1921) VoLMERand Ebtermann. Znls Phys , 7, 1 (1921) Seo. 43] GENERAL MOTION AND SPATIAL DISTRIBUTION 71 The accuracy obtained in such experiments does not go beyond possible errors of several per cent, but the results obtained by all workers point consistently to the conclusion that a is at least very close to 1 for liquid mercury, and also for solid mercury below --140 o C, while for solid mercury above — 100°C it is certainly less than 1, perhaps by 8 or 10 per cent. The fact that no definite evidence was found for a value of a exceeding unity constitutes a confirmation of kinetic theory. The rate of evaporation of mercury is diminished very greatly by any contamination of the surface, as was shown by Knuclscn; for the surface of a certain brownish-looking drop he found a to be only J-^ooo* This is presumed to be the reason that oven very small drops of mercury may lie around in the laboratory for days before they finally disappear. On the other hand, if we assume the truth of eq. (76) and somehow know (or assume) a } we can employ the formula the other way round for the calculation of vapor pressures from observations on the maxi- mum rate of evaporation. This was actually done for mercury itself by Knudsenj* and later a similar use of the formula was made by Langmuir, f The latter investigator had made an extensive study of the rate of evaporation of tungsten to servo as a guide in the prac- tical handling of tungsten filaments in lamp bulbs and vacuum tube»s, and as a by-product he calculated from his results the vapor pressure of tungsten at various temperatures, which would bo very difficult to measure directly in the case of such a nonvolatile substance. The rate of evaporation was found by keeping a tungsten filament at a known temperature for a given length of time and then weighing it to find the loss of material from its surface. Several not too definite considerations convinced him that a wtfa probably unity, so that tho equivalent of eq. (76) witli a omitted would give correct values of p in terms of the observed values of G, Some of the results thus found were: at 2000°K, 0 « 1.14 X 10” 13 g/cmVsec, and p ~ 6.46 X 10^ 12 mm I-Ig; at 2400°K, which is not much above tho operating tempera- ture of a common tungsten lamp, 0 — 8 X 10~ lo and p = 4.9 X 10~ 8 ; at the melting point of tungsten, 3640 o K, 0 = 0,00107 and p « 0,08 mm, 43. Test of the Velocity Distribution in. Effusive Flow. Several experiments have been performed which permit a more or less direct test of tho distribution of velocities in an effusing gas, and those are of considerable theoretical interest becauso they constitute a fairly * Knudse^, Ann , Physik^ 29, 179 (1000), f Langmuir, Phya, Rev> t 11, 320 (i013), 72 KINETIC THEORY OF GASES [Chap 111 clnect test of Maxwell's law In these experiments use is made of the method of pioducmg moleculai lays that was woiked out by Stern and his coworkeis at Hamburg * Fiom an oven heated unifoimly so as to become filled with vapor of some substance in theunal equihb- lium, vapor is allowed to stieam out thiough a veiy nanow slit in one sklo, and a nanow beam of molecules is selected out of this stream of vapor by means of a second slit placed opposite the first and at some distance from it Beyond the second slit the beam thus fanned is split up 01 otheiwise expenmented upon and is then measiued m some soit of lecoiving device The entue path of the beam outside of the oven lies in a high vacuum, all issuing molecules being condensed as they strike vanous parts of the appaiatus, which aio cooled if noecs- saiy by liquid air Special interest foi kinetic theory attaches to the arrangement invented by IS K Hall, constructed by Zaitmanf and impiovcd by Ko $ In this apparatus thorc is mounted above the second collimating slit (£2 m Fig 17) a rapidly levolvmg dium having a nar- row slit in one side and cairying on its innoi BUi face, opposite this slit, a glass plate P The molecular beam is thus received foi Iho most part on the extcnoi of the dium, but Fig 17 ball's velooity-spec- onc0 }n each evolution, when the slit m the drum comes opposite the exit slit ot ol the beam, a short spurt of molecules enteis the dium, crosses its interim, and is deposited upon the glass plate Since, however, the drum has time to turn through a certain angle while the molecules aie ciossmg its mtenor, the point of deposition on the glass plate vanes with the molecular speed, and m this way a velocity spectium is formed According to eq (71a), the number of molecules issuing through the slits with speeds between v and v + dv can be wntten dv, C being a proportionality factor The point at which molecules with speed v strike the glass plate is displaced a distance s from the point at which they are directly aimed as they entei the drum, given by s =* (D/v)(jrDn) =* 7 mD*fv> T> being the diameter of the dium and n its frequency of revolution The molecules in the range dv will thus bo spread ovei a distance ds ~ dv } and, if I is the number depos- * Stern, Zeiis f Physik> 39, 751 (1926) f Zartman, Phys Rev , 37, 333 (1931) \ Ko, Jour Franlhn Inal, 217, 173 '1934) d He found that the Sec. 43] GENERAL MOTION AND SPATIAL DISTRIBUTION 73 ited per unit length, wo have I da — —Cl J 3 <r' s '“ 1 dv and ri rit ttWDW iriiD 2 s 6 in terms of a new proportionality factor O'. The substance studied by both Zartman and Ko was bismuth, which deposits readily at room temperature, provided the surface is coated with a preliminary layer of bismuth. Unfortunately for the point we are interested in, however, bismuth evaporates partly as Bi and partly as Bi 2 , and probably even slightly as Bi B , so that several kinds of molecules are present in the beam; the determination of the molecular composition of bismuth vapor was, in fact, Ko's main object, rather than a check of Maxwell's law. observed deposits as measured photo- metrically after a run with the oven at 827°C were in fair agreement with the assumption that in the beam 44 per cent of the molecules were Bi, 64 per cent Bi s , and 2 per cent Bi«. In Fig. 18 is reproduced Ko’s Fig. 10, showing the observed points for the mass deposited, in comparison with the theo- retical curve; in calculating the latter the relative magnitude of tlie three Pw< 18 ._ Ko>s volooity Bpootrum component curves was adjusted to of n bismuth beam. Abscissa, « tho best fit with the observa- tSLA&T*** tions, but the form of each was deter- mined theoretically in terms of the observed oven temperature. From his data Ko deduced in the following way a value of the heat of dissociation of the bismuth molecule; and this he regarded as his most significant result. According to the last member in (726), for the separate components of a beam p a Y m s/Ti a since the molecular weight M « 1/22; but the number of atoms passing is proportional to r ,,,/M ; hence the pressure is proportional to the num- ber of atoms multiplied by ■s/M. Now the ratio of M for Bi a to M for Bi is 2. Hence in the example cited above the afcomio ratio 64/44 or 1.23 in tho beam means a partial-pressure ratio 64 44 X V2 = 1.74 for Bi s as compared with Bi. In addition, a second relation involving tho partial pressures was obtained experimentally by observing tho 74 KINETIC THEORY OF OASES [Chap. Ill total rate of loss of bismuth through the slit, the bismuth in tho oven being weighed before and after a run; the aiea of tho slit being known, these observations give the sum of the values for Bi and Bij of the quantity denoted by r m in (726), and from this sum another expression containing the partial pressures in the saturated vapor could bo found The two relations thus obtained were solved for tho actual partial pressures, and from values of these corresponding to two different temperatures the heat of dissociation was calculated with tho help of the theory of dissociation THE GENERAL DISTRIBUTION FUNCTION 44. A Gas in a Force-field. Up to this point we have dealt only with gases that are free from the action of external forces other than those exerted by the containing vessel Theio are many cases, how- ever, in which forces such as giavity are piosent, and it is iinpoilant to investigate the effect of such foices upon the molecular motion In almost all cases of this sort the external forces may bo regarded as due to a force-field, by which is meant that tho forco on oach mole- cule depends in some definite way upon tho position and other charac- teristics of that molecule alone In by fai the most important case, furthermore, the forces are derivable fiom a scalar potential function, so that a molecule possesses a potential energy depending only upon its spatial position. An example of a force-field of a moro gcnornl type is presented by ions in a magnetic field When a potential function exists, the force that acts on tho in ole- cule is the negative gradient of the potential-onei gy function; its direc- tion is that in which this function decreases most rapidly and i magnitude is equal to this maximum rate of deciease of tho function. In vector or in cartesian notation the force, donotod by F oi F x> b\, F„ can be written thus in terms of the potential-energy function «: or F = — F, - du> ~dx F u = - du dy Ft = dz (77) 45. Density in a Force-field. The effect of a simple potential- energy orce-field upon the spatial distribution of the molecules can be obtained very simply without any use of kinetic theory. Consider the gas inside an element of space having the form of a short right Znn °l c ™ ss ' sectl0nal area ss and height Sh (Fig 19). Its mass will be p SS Sh, where p is the density of the gas. Let Q donolo tho Sac. 461 GENERAL MOTION AND SPATIAL DISTRIBUTION 75 potential energy of unit mass of the gas due to the field; then the com- ponent of the force on unit mass in a direction normal to the faces of the cylinder can be written —SQ/dh, where A as a variable denotes distance in the specified direction. (We must choose one of the two directions along the normal as positive and then represent all dis- placements dh and components of force by positive numbers when they have this direction and by negative numbers when they have the opposite ono.) The field thus exerts on tho gas inside the cylinder a component of force normal to its faces whose magnitude can be written -» m fs|2 P SS dSl after replacing Sh dQ/dh by dQ, the difference in fi at perpendicularly opposite points on the two faces. This is allowable at least in the limit when tho cylinder is made indefinitely small, tho difference between the values of dQ for different pairs of opposite points becoming then negligible in comparison with dQ itself. To preserve equilibrium, this force must ^ now be balanced by an equal and opposite force arising from the pressure of the surrounding gas on the surfaces of the cylinder. Pressure on tho curved sides, however, causes no force in the direction normal to the faces. Lot p Ek.io . — Eauillbrjum in b denote tho pressure at any point on tho lower “ HLl " 10 face and p + dp tho pressure at the perpendicularly opposite point on the upper face ("upper” meaning, situated toward larger values of A); then the difference of tho pressure forces on the two faces can bo written V SS - (p + dp) SS - -dp SS, this expression again being accurate in the limit. The sum of this force and the force duo to tho field must now bo zero. Hence —pSSdQ — dpSS-0 and dp rn -p da. (78) By properly locating the cylinder, whose axis need not be vortical but may have any direction, dp and dQ can obviously be made to repre- sent differences between values of p and 0 at any two neighboring points. Hence eq. (78) holds throughout tho force-field. If a general relation between p and p is known, the integral of (78) can bo written down at once. Suppose, for example, the temperature 76 KINETIC THEORY OF OASES [Chap iU T is uniform thioughout Then, the gas being assumod pollock p = pRT, R being the gas constant foi a giam, and by (78) Mli? , „ da . P Integiating this equation, we find RT log p ~ + const , which can be written ft ft p = p 0 e ***, p = poc (70«) wheie po or p 0 — po/RT leplaces the constant of integration and has the significance of the piessuie or density at points (if there aio any) wheie Q = 0, Equation (79a) expiessos the law of isothermal dtstrt* buhon of a gas m a foice-field. Sometimes, howovor, it ih moio con- venient to wnte ft = co/m, m being the mass of a molecule and w, ins before, its potential eneigy in the field, if at the same timo wo intro- duce k = mR } (79a) becomes V = Poe hT , P = pa e hr (70ft) In a uniform gravitational field we can also write tt ~ gh, w = mgh, in terms of g, the gravitational acceleration and h, the olovation tvbovt* some chosen datum level, then (79ft) takes the special foun , _mh mgh V = w, p = Po c w (70c) In Fig 20 the isothermal distribution is compaied with the adia- batic distnbution described below, foi an, which has y = 1.4. It is assumed in both cases that T = T 0 where n — 0. 08 06 ■ ■ ■ ■ ■ I ■ 1 H 1 ■ ■ 1 1 1 I ■ ■ ■ ■ 1 |§ 1 I ■ 1 ■ B a i 1 ■ Mr wmm 20 — Donsity p m a Bold of potontial Cl Sbow that lf the distribution is adiabatic, so that pY r constant from point to point, (79a) is replaced by Sue. 4C] GENERAL MOTION AND SPATIAL DISTRIBUTION 77 p - f(l - — J#;)*' p - »(l - 3 k)^' (8® where 7 = ratio of specifie heats and po, pa and T a stand for values at any point where 0 = 0 (or may bo regarded as merely representing integration constants). According to these formulas p — 0 (and also p = 0 , T = 0 ) at any point where Q <= yR r l\/(y — 1 ), and there can be no gas at all in regions of larger 0 . 46. Maxwell’s Law in a Force-field. The next question that naturally presents itself is whether Maxwell’s law can hold for the molecular velocities in a force-field or whether, perhaps, it requires modification. As molecules move into regions of higher potential they must lose kinetic energy, and vice versa; it might bo thought, therefore, that there would be a tendency for a difference of temper- ature to be set up through the agency of thermal agitation, conceivably like the temperature gradient that is actually observed in the atmos- phere. The equilibrium distribution would then be one in which the temperature is a function of fl. < > To throw light on this question, let us consider first the concrete case of a homogeneous gas in the earth’s gravitational field. Take two horizontal planes Pi (and Pt so close together, that a moleculo can cross from ono to the other without appreciable] chance of a collision, and take the 2 -axis vertically upward (of. Fig. 21). Then if Maxwell’s law holds in the neighborhood of the ' 1 lowor plane, the Fia ‘ 31 ~ , 1,1 a forc0 * mean density of the gas being ni and its temperature 3T, according to (736) with /3 s = m/2k r P inserted from (5(3) in Sec. 28, there will ho , dVt (81) ' molecules that leave unit area of tho lower plane per second moving in an upward direction with ^-components of velocity in the range dv f . Each of these molecules upon roaching the upper piano will have lost kinetic energy equal to its increase Aw in potential energy. Only the ^component of the velocity is affected, however; hence, if we let v' denote tho velocity at tho upper piano of a molecule that left the lower with a ^component v, } wo have $mv f * ~ i mvl - A<o, \ rm\ « £ mv'* + Aw. ( 82 ) 78 KINETIC THEORY OF OASES (Chap HI From this equation we have v, dv, = v' dv', Hence, remcmbeiing that v i — a| + v* -f tij, v 12 == y ' 2 + y ' 2 + K* and noting that vZ = v*> v l = we can write in place of (81) for the number of the molecules under consideration n 1 e-^ kT (~^J i v , ,e dv',. Upon arriving at the upper plane P h these molecules form part of whatever distribution exists there, then they pass on upward as an actual part of the stream of molecules that is continually leaving m an upward direction from P\ The last expression found foi their number is, however, the same function of v, and dv f t that (81) is of t>« Aoj and dv„ except for the added factor e kT . Thus the expression is just what it would be if the gas at the upper piano had likewise a maxwelhan distribution at the same temperature T but a density decreased in the ratio e kT , and this lattei is just what the ratio of the densities must be in order to agree with the law of isothermal distribution in a force-field, as expiessed by (796) We may conclude, therefore, that a maxwelhan distribution at the umfoim temperature T will be left undisturbed by the flow of the molecules and, accordingly, that such will be the actual distribution when the gas is in a stato of equilibrium There are, however, two points that require further examination. In the first place, those molecules that start upward from the lower plane with y, so small that ■§■ mvl < Aw never reach the upper piano at all However, they aie not missed there, foi those molecules that have exactly % mv\ ~ Aw just barely arrive with t 1 ' = 0, and obviously all greater values of v’, aie adequately represented in the stream that crosses Pi. On the other hand, we can obviously deal m the same way with downward motion, locating the second plane below the first; then molecules leaving with v, < 0 will arrive with v' t < — (-2mAw)W (Aw being now negative), and very slow molecules might thus scorn here to be missing at the second plahe But now the molecules that should cross the second plane with a negative v’ t of smaller numerical magnitude than (— 2mAu)M will be supplied by exactly those mole- cules noted above which leave the lower plane in an upward direction and, after failing entirely to reach the upper plane, fall back and Sec. 48] GENERAL MOTION AND SPATIAL DISTRIBUTION 79 recross the lower one in a downward direction with v x just reversed (cf. Fig. 21); the maxwellian flow downward is thereby made complete. The second point concerns the effect of collisions, which we have completely ignored in our discussion. If, however, a maxwellian distribution exists at each point, collisions will throw as many mole- cules into a group moving in any particular direction as are removed ,from it, so that their net effect will be nil and we are justified in ignoring them. Since all of this reasoning is obviously applicable to a gas in any force-field that has a scalar potential, wo may conclude that when a gas is in complete equilibrium in any such field its density varies in accord with the cq. (79a) or (7%) that we obtained above, and also that the temperature is uniform throughout and a corresponding maxwellian distribution of velocities holds at every point. 47 . The Temperature of Saturated Vapor. The conclusion that Maxwell's law holds in a force-field with no inequalities of temperature can be drawn also from the general differential equation for the molec- ular distribution which we shall obtain presently [eq. (87) in Sec. 51], or, still more satisfactorily, from the Boltzmann distribution law [cf. (92) in Sec. 55]. On this latter basis the conclusion holds uni- versally for systems in equilibrium. It throws an interesting light upon the question as to the temperature of freshly formed vapor. One might suppose that tho vapor would be cooler than the liquid or solid from which it comes because tho evaporating molecules do a large amount of work in escaping from tho attraction of othors that stay behind, this work forming in fact almost the whole of tho ordinary heat of vaporization. Our results on the motion of molecules in a force-field indicate, however, that it should bo only tho fastest moving molecules which escape at all and that, by tho time they have escaped from the attractive field of the liquid or solid, they will have become slowed down exactly into a maxwellian distribution corresponding to the temperature of the region from which they came. Freshly formed vapor, therefore, ought to have tho temperature of tho surface of tho evaporating liquid or solid. Tho experimental facts bearing on this point seem to be somewhat uncertain but at least it may be said that they do not definitely contradict the theo- retical prediction. 48 . The Terrestrial Atmosphere. The most famous case of a gas in a force-field is, of course, the earth’s atmosphero. Modern work, however, has shown that many different influences arc at work here and, in consequence, the state of the atmosphere does not exemplify 80 KINETIC THEORY OF GASES (Oiiai* HI any one simple theoiy * We have space heio only to discuss briefly a few aspects of the subject that aie of paiticulai mteiest fiom tho standpoint of kinetic theory. Observations made from balloons show that with mcieasmg height the tempeiatuie of the atmosphcie, as a iule, diops at fiist appioxi- mately at the adiabatic late, ie, it vaiies with the pressure in the same way as it would in an adiabatic expansion (r °c p v ), Tho deciease ceases, howevei, when a certain minimum tempeiaUiro is reached, this temperatiuo varying fiom about —54° at a height of 12 km {1% miles) in latitude 45° to ~S4°C at a height of 17 km (m nulesj ovor the cquatoi The atmosphere below this height is called the tiopospheie, In the legion above, called the stratosphere, the temperature either is constant oi actually lisos with mci easing height The accepted explanation of thebe obseived feaiuies ascubes thorn to the cuciunstance that the ttopobphere xoeeives heat primauly by conduction at its base fiom the eaith and loses it thiough infra-rod ladiation to the stratosphere, the absorption of the sun’s lays by tho troposphere being only a mmoi facfcoi The continual warming of the tiopospheic at its base then seta up the tamihai processes of convection by which the air is continually earned up and down in storm movements and in the geneial terrestnal eu culation, and is thereby subjected to icpeated adiabatic expansions and compressions, as a lesult of which the approximately adiabatic distnbution of tem- peratuie is bi ought about. The continual mixing also causes tho composition of the air to be closely the same eveiy where, except, of couise, for the vauable content of water vapoi Thus it is only m the stiatospheie that considciations based upon kinetic theoiy are likely to be of impoitance Hcie clouds are rate and theie is probably little lapxd veitical movement Obseivations of am oi a! heights indicate that the stiatospheie extends at least to 300 km (or 200 miles) and even slightly to 1000 km (oi 600 miles) The density is extiemely slight at f»uch gieat heights, of course, and it seems to vaiy gieatly from day to night, and from winter to summoix ior example, at a height of 100 km (62 miles) the density, lelativo to that undei standard conditions, has been estimated to be some- thing like G X 10'” 6 on a summer day, 6 X 10~ 7 on a sujnmci mglit, 10^ on a winter day, and 3 X 10^ 7 on a wmtei night , At 300 km (nearly 200 miles) the estimated figtues are neaily 10 million times * Of w J HmiPHRBYS, "Physios of the Air," 2d ed , 1920 Sec, 40J ' GENERAL MOTION AND SPATIAL DISTRIBUTION 81 smaller, but even at the latter height there are still around a million molecules in each cubic centimeter. The most interesting feature for kinetic theory is to be found in tho very long free paths of the molecules at such altitudes, At 100 km the free paths are only a few cm long, but at 300 1cm the mean free path ranges from 200 km (125 miles) on a summer clay up to a maxi- mum of perhaps 15,000 km (over 9000 miles). At still greater heights the molecules can be thought of as .moving like tiny satellites in ellipti- cal orbits with tho earth at the focus. At such heights as these tho atmosphere must be very far from a state of thermal equilibrium. Many ions must be formed through ionization by the sun’s ultra- violet rays, and these ions will then spiral for long distances about the magnetic lines of the earth’s field; a spray of such ions produced in equatorial regions and spiraling off to descend into lower altitudes in the region of tho earth’s magnetic poles has been suggested recently as a possible cause of the aurora. Interest attaches also to the question of the distribution of tho various constituents of the atmosphere. If the latter were in iso- thermal equilibrium, we could apply eq. (79c) to each of the constituent gases separately, each one being distributed according to this law just as if the others were not present. Tho coefficient of h in the exponent in (79c) increases with tho molecular mass to; honce it would follow that the heavier gases are much more concentrated near the surface of the earth than tho lighter ones. On the basis of this result from kinetic theory, tho view has frequently been expressed that at great heights helium must form a much larger fraction of tho atmosphere than it does lower down; at the earth’s surface helium forms only 0.04 per cent of the total, but from 100 km, or 60 miles, up it should predominate over nitrogen and oxygon. At still greater heights the atmosphere should be nearly all hydrogen. Unfortunately, satisfactory observations to test these conclusions do not yet exist. Furthermore, it is by no means certain that there must bo any appreciable amount of hydrogen at great heights, even if there is a trace of it at tho earth's surface, which is in itself not certain; for. any hydrogen that wanders up into the upper atmosphere may bo promptly oxidized to water vapor by the ozone which is known to occur there in considerable quantities. 49. Cosmic Equilibrium of Planetary Atmospheres. The problem of tho upper boundary of a planetary atmosphere presents features of interest not only for tho astrophysicist but also for the student of kinetic theory, If such an atmosphere were isothermal, it would extend to indefinite distances from the planet, for in (79a) the potential g2 KINETIC THEORY OF OASES [Cuai\ Ill a would be finite even at infinity An adiabatic atmosphere, on the other hand, distributed in accord with eq (80), would lmvo ft slmip upper boundaiy at the level at which ^ — yR1'o/(y ) Y*!' of other properties of gases it is obvious, howcvei , that such a boundary could not persist foi any length of time, foi the lapuhty with wliteli inequalities of temperature aie ironed out in a ga» hi consequence of the conduction of heat becomes infinite at vanishing density, nnrt consequently the gas immediately beneath a bounding qut fa go of zero density would be brought quickly to ft condition approximating uniformity of temperature and would thereupon proceed to spread out toward infinity, Only two possibilities are open, therefore, in legard to tho upper- most part of a planetary atmosphere Either it passes continuously into a general distribution of matter m thermal equilibrium filling tlio surrounding space, or it is not m equilibnum and is continually stream- ing off into space, or being built up, although perhaps at a very slow rate Now, it is a fact that ceitain absorption linos in stollai speotia point toward the existence of diffuse matter scattoied throughout space, consisting largely of atoms of sodium and calcium, to tho extent of something like 10“* 24 g or 20,000 molecules pci cubic motor. Tlio density of an atmosphere that would be in equilibnum with such an interstellar gas can be estimated and turns out to be consistent With the observed densities On the other hand, a serious difficulty ia presented for such a hypothesis by the onoimous variation m tlio observed composition of planetary atmosplicios, foi on tho outer planets prominent atmospheric constituents are ammonia and mo th- ane,* which are not found in measurable amounts on tho caitiu In view of this latter fact it seems most likely that tho atmospheres of the planets are only in pseudo-equihbnum, it is usually supposed that they are continually leaking away into space but that tho l'ato of this leakage is so slow that the loss even during cosmological ages is not large An exact calculation of the rate of escape from an atmosphere mto empty space would icquile a knowlodgo of conditions in its uppermost layers, and these conditions are hard to do term i no theoretically because of the extiemely long paths that occur thoio. We can probably obtain a sufficiently accmate estimate to reveal the various possibilities, however, if we imagine simply that (1) tlio upper part of the atmosphere extends in isothermal equilibrium nfc least up to a certain great height ho, and (2) above that height the density is so low that collisions may be neglected altogether, Tho *Cf Science, 81, 1 (1035) Seo. 40] GENERAL MOTION AND SPATIAL DISTRIBUTION 83 reasonableness of assumption (1) is supported by the fact that, as we saw in Sec. 46, a maxwellian distribution of velocities in a force-field is preserved automatically without any help from collisions as tho molecules move about in the field. As a matter of fact, a slight extension of the analysis of that section loads to the conclusion that the stream of molecules leaving any level in an upward direction is the same in number as if the upward stream entering the bottom of the isothermal layer simply rose unhindered to higher levels, collisions merely substituting other molecules for the initial ones without pro- ducing any other change. Accordingly, to find the rate of loss to infinity we need only find at what rate molecules start upward from the bottom with speeds exceeding tho "speed of escape” from the planet. This speed, which we shall denote by v c , is so defined that a molecule leaving with speed v t has barely enough energy to carry it to infinity and leave it at rest there, provided it makes no collision on the way out; its initial kinetic energy therefore, equals the total work that it must do against gravity. Now the gravitational force on a molecule of mass m can be written mgrft/r 2 , where r denotes distance from the center of the planet, ro tho radius of the planet, and g the acceleration duo to gravity at its surface, the force thus reducing at tho surface to mg. Tho work done in escaping is, accordingly, f (mgr§/r a )dr = mgr„; henco = mgro and v? - 2gr o. On the other hand, by (71b) tho number of molecules leaving unit area with an upward component of velocity and with speeds above v c is f “v'e-P" dv r sin 0 cos 0 dO = — (0M + l)e^***\ Vi X Jo • 2/3 Vir It is convenient to divide this number by n and thereby obtain tho rate of loss expressed in terms of centimeters of thickness of tho gas, a form of statement that is independent of the density. Introducing tij “ 2gr$ as just found, p 2 = M/2RmT from (66) in Seo, 28 in terms of the molecular weight M, Rm = 83.16 X 10° and the absolute temperature T } we find finally for tho rate of loss, in centimeters of the gas at the bottom of the isothermal layer lost per second. RmT) groM Kx'i\ 84 KINETIC THEORY OF GASES [Chap III la a mixed atmosphere this formula will obviously apply to each kind of molecule sepaiately As a final result it is pci haps most illuminating to calculate the temperatuie at which a given depth of gas would be lost fiom vaiious bodies of the solar system 01 fiom the sun itself during a period of time that is cosmically long The follow- ing table shows a few absolute tcmpeiatuies, calculated in this way, at which a kilometei of vaiious kinds of gas would be lost m 10 billion (10 l0 ) years, which is several times the age of the oldest known rocks on the eaitli's surface The rate of loss is natuially veiy sensitive to the temperature, a change of only 10 to 20 pei cent in the latter would suffice to change the time of loss either to 10 7 oz to 10 12 years, or to make the removed layer, say, 10 km m thickness Foi different gases the tempciatures are propottional to the molecular weight Ha He HiO N a 1 Electrons Earth 350° 3130° 4860° Moon 16° 32° 145° 225° Mars Sun 70° 1 140® 630° In using this table it must be remcmbeied that the temperatuies refer to the stiatosphere or to an equivalent isothermal layer in fcho upper part of the atmospheie, So undeistood, it is clear fiom the table that accoiding to our estimate the eaith ought now to be holding all gases, in the past when it was molten it should piobably have lost hydrogen and helium Perhaps the free helium now in tho atmosphere has been produced subsequently as a consequence of radioactivity* The case of the moon is less clear, howevei Since its illuminated side is observed to reach a tempeiatuie of well over 300°K, it can probably hold nothing, and this conclusion agrees well with the entire absence of any detectable atmosphoie on the moon; but whether it might be able to hold a layer of nitiogen oi heavier gases of sufficient thickness to develop a stratosphere at a temperature considerably below that of the surface is a question not capable of offhand decision* Perhaps such gases were all lost long ago when the moon was hot Mais should now be retaining water vapor, in agreement with the fact that, according to its temperatuie as calculated from thei mo- pile observations of its radiation, the polar caps can scarcely consist of anything other than ordinary snow Perhaps nitiogen and heavier gases were lost when the planet was molten, but in that case the water vapor must have been evolved from the interior during the latei stages Sec. 51] GENERAL MOTION AND SPATIAL DISTRIBUTION 86 of cooling, Finally, not even the sun at 5500*0 or higher can retain electrons by means of gravitational attraction; they may, of course, be retained by the attraction of & positive charge, but there is some reason to think that electrons are actually being emitted freely from the sun. All of the other known facts concerning planetary atmospheres are in similarly good agreement with the view here described. 50, The General Distribution Function. In discussing above the distribution of a gas in a force-field we considered only the state of equilibrium. More general cases were dealt with in the last chapter, but the distribution function was assumed to be the same throughout the gas. At this point it will bo convenient to take iqi Boltzmann's treatment of the still more general case of a gas that may not be in equilibrium, and in which both tho density and the distribution of velocities may vary from point to point. We shall assume, however, that this variation is slow enough so that in any macroseopically small element of volume the molecular distribution can bo treated as practically uniform and as possessing tho property called molecular chaos, Usually the number of molecules in any such element is assumed to be large, but this condition can be dropped provided we interpret the distribution as referring to averages taken over a molecularly long but macroseopically short interval of time, an interpretation which will not invalidate any of tho conclusions that we shall reach. Such a distribution can be expressed by writing for the number of molecules per unit volume n(x, y t z f t) and for the (fractional) dis- tribution function /(a?, y ) z ) v Z) v z , t ); the product nf has then the significance that 7if dx dy dz dv x dv v dv, is tho number of molecules lying in the spatial element dx dy dz and also having velocities in the range dv * dv v dv K) so that nf can be regarded as a distribution function for position and velocity taken together. 61. Differential Equation for the Distribution Function. As a general foundation for statistical calculations, let us now seek an expression for the variation of nf with time, as represented by ^ to replace the expression obtained in the last chapter for the variation of / alone. We shall do this only for a homogeneous gas, but the resulting equation will then hold also in a gas composed of different kinds of molecules for each constituent separately, For generality we shall also allow external forces to be acting; the external force F fC»Ai' III 86 KINETIC THEORY OF OASES on each molecule will be allowed lo bo a function of both its position and its velocity, but we shall assume that this function is the same for all molecules of a given land In actual cases tho foico upon a mole- cule moving with given velocity usually is also sensibly constant over any region that is macioscopically small, as when olootiic 01 magnetic fields are applied to a gas, and such legions can usually be taken large enough to include many molecules; but even if the density is too low for this condition to be satisfied, the theory developed below can bo shown to apply piovided we mtei pi ct nf as representing an average over a macioscopically small inteival of time To find the rate of vaiiation of nf with time, wo select for study those molecules that simultaneously have their centers in. an olemont A^ A y As diawn about any point Or, y, z) in bpaee and thou velocities in an element Av x Av u Av t diawn about any point (v x , v Vl vf) in velocity space. Theio will obviously bo, to tho flint oidei of small quantities, nf Ax Ay As Av x Av y Av t of these molecules, and then number will change at tho rate of Ftq 22 — (nf) Ax Ay Az Av x Av v Av * (83) molecules per second, both nf and its derivatives having hero their values at the six-dimensional point (x } y i z } v x , v V} v z ) and at time L Such a change m the numbei can occur, howovor, only through passage of molecules across the boundaiy of one of tho elements. Let us calculate the change pioduced m this way, and let xm fust consider the effect of molecules that cioss those two faces of tho element Ax Ay Az which are peipendicular to the onixis (of, Fig 22), The space density of molecules with velocities in tho range Av* Av v &v t is nf Av x Av y Av z Hence, by an aigument such as wo have used sovoial times before, the numbei of molecules with theso velocities that cnlor the space element per second by ciossmg the left-hand faco m tho direction toward +x is Av x Av y Av z J Jnf v x dy dz intogiatod over the face, and a similar expiession foi the right-hand faco gives tho rate at which molecules leave the element by ciossmg that faco, The diffeience between these two expiessions then gives the net gain of molecules m the element due to passage ovci these two faces; it can be written in the following way as a single integral, v x having tho same nearly constant value ovei both faces • Sec. 51] GENERAL MOTION AND SPATIAL DISTRIBUTION 87 hv x Lv v Avi v x f f - (»/)>•] dy dz, (84a) the integral extending over Ay and A z, Hero [(nf)i — stands for the difference between values of nf at two points having the same values of y and z and located on the left ancl right faces, respectively; by the “mean-value theorem” we can replace it by — Ax where is the value of some intermediate point on L ox Ji ox the line joining the two points on the faces. This value of the deriva- tive, however, becomes indistinguishable from its value at the fixed point {x, y, z, v ff v V) v t ) in the limit as we make both of our elements indefinitely small; so we may as well insert the latter value in place of it at once. The integrand is then independent of y ancl z } and if we put all constant factors in front of the integral sign we have as an integral simply // dy dz, which equals Ay A z. Expression (84a) for the net gain of molecules thus becomes — Ax Ay A z Av x Av u Av £ . (846) The other two pairs of faces yield similarly - Ax Ay Az Av* Av v Av z . (84c) _ dy vz J Now the crossing of a face of the element Ax Ay Az by a molecule involves no change in its position in velocity -space. Accordingly, the molecules just considered remain in that element in velocity space in which they lay to begin with. The expressions just obtained represent, therefore, contributions to the net gain in the number of those molecules that lie simultaneously in both elements, On the other hand, molecules lying in the space element Ax Ay Az may, without leaving this element, cross the faces of the velocity element Av a Av y Av e through experiencing a change in their velocities, The force F causes the vector velocity of each molecule to change continually at a rate equal to the acceleration F/m and so causes the representative point to move through velocity space at a velocity equal to F/m, The resulting net inflow of molecules lying in Ax Ay Az, into the element Av x Av v Av g , across those two of its faces which are perpendicular to v x can, therefore, be written, in close analogy with (84a) and (846), 88 KINETIC THEORY OF GASES [Chap III Ax Ay A z j jm i(Fx nf)l ~ {Fz nf)r] dVy dt>1 ’ the integral extending ovei Av v and Av t) or — 1 & Vx (86a) m aVx Fx must follow the sign of differentiation heie unless we know that it is independent of v x The other two pans of faces of the element Av x Av y Av x yield similarly + ^ L y Az Av * Av > (m) In addition to the effect of the external force F, thcie lemains then finally the effect ot collisions As molecules in Ax Ay A z and Av z Av v Av e collide with otheis, they aie thrown entnely out of the velocity element Av x Av y Av t) and at the same time other molecules m Ax Ay A z undergo collisions of the inverse type and aie thereby thiown into the given velocity element This is the same effect of collisions that was studied in detail foi a paiticulai type of molecule in the last chaptci Instead of attempting a similai analysis here, we shall simply wnte down a symbolic expicssion to denote the con- tribution of collisions to the rate of change of the number of those molecules that lie in both elements, writing for it Ax Ay A z Av x Av y Av J • (86) L w Jfioii If no association or dissociation of the molecules occurs, so that n is unaffected by collisions, we can also write this in the form n Ax Ay A z Avx Av v A vj ) \d£/coii* and if the molecules aie hard spheies (df/dt ) 00 il has then the valuo given foi df/dt m eq (43) or by an expression similar to (42a) and (426) in the case of a mixed gas We shall employ here the more general form in order that our final equation may hold also m the case of a dissociating gas We may now add up all of the expressions (846), (84c), (85a), (866), and (86), theieby obtaining the total change in the number of molecules due to all causes, and then equate their sum to (83) above, After canceling out A;r Ay Az Av# Av v Av z on both sides of Sac. 62 ] GENERAL MOTION AND SPATIAL DISTRIBUTION 89 the resulting equation ancl moving all terms but one into the left member, we thus obtain d_ dl ( n f ) + + V v <W) | .. d(nf) (*>/) + ± (F t nf) (87) 62. Applications of the Differential Equation. In eq. (87) we have a differential equation for the general distribution function iif in a homogeneous gas or for the distribution function of each constituent of a mixed one; as remarked above, it holds even for each constituent of a dissociated gas. The equation lias a number of uses. In all practical cases, however, it can be simplified somewhat because F x is independent of v x , F v of v Vl and F, of v,; for this reason the force term is usually written Furthermore, the external force is usually either derivable from a potential or gyroscopic in character, or a combination of theso two types; as a general expression for such forces wo can write 77T I F, = + 7«tv - y v v, n <?<») | k + **■ - F* = + y v v* - y*v» where u(x } y } z f t) is the potential energy of a molecule in the external field, which may perhaps vary with the time, and y x> y Vi y t are com- ponents of a vector y which may likewise bo a function of position in the field, and perhaps also of the time. Examples of the potential type of force have already boon encount- ered in considering a stationary gas in a gravitational field, An example of forces of gyroscopic character is furnished by a group of ions in a magnetic field; if e is the charge on each ion in electro- static units and H the vector magnetic intensity, then y «* eH/c ,* Another example is encountered if we employ a rotating frame of reference, such as a frame rotating with the earth; the effect of such a * Cf. L. Paqb and Adams, “Principles of Electricity,” p. 244. 90 KINETIC THEORY OF GASES [Chap. Iir rotation with umfonn vectoi angulai velocity w, when without it y = 0, can be allowed foi simply by adding a tcim — £ mw 2 s 2 m 0 and setting y = 2ww, m being the mass of the molecule and s its porpendiculai distance fiom the axis of lotation * In the case of the earth the term — §mw 2 s 2 simulates a slight change m the potential energy due to giavity, and its effect is automatically included in tho ordmaiy “acceleration due to giavity” g P)oblems 1 Show that when the forces aie donvable from a potential or else aie partly 01 wholly gyioscopic in chaiaciei (1 e , of the foim wutten for F x> F V) F z just above) tho steady distnbution function w my* nf - Be & ( 88 ) is a solution of (87), the collision teim vanishing by the argument given m the last chapter and the teims that contain dcuvatives with lespect to % } y, z t v X} v V) v % canceling out; B is a constant such that § $ f j $ dxdy dzdi)xdv v dv z = N } the total numboi of mole- cules, and kT is just anothei aibitrary constant so faz as tho diflezential equation is concerned but can be shown to have its usual physical significance, l being the Boltzmann constant and T tho absolute tempeiatme. Equation (88) contains as a special case eq (796) and leads also at once to the conclusion z cached 111 Sec 46 that Maxwell's law holds even in a foi co-field along with uniformity of the tcmpeiaturc [cf eq (65) in Sec. 28], heie we have established the result foi the moio general type of foice-field described above As a paiticular example, we may draw the impoitant conclusion that the presence of a magnetic field does not affect the oqiulibiium 111 space of ions 01 electrons, nor their maxwellian distribution of velocity, in spite of tho lesulting cur Vat uie of their paths 2 Show that a gas can rotate as a rigid body with Maxwell's law holding at eveiy point (velocities being defined in teims of sta- tionary axes), provided the density is piopoitional to e mwWmT , where w = angulai velocity and s — distance fiom tho axis of rotation. THE BOLTZMANN DISTRIBUTION FORMULA 63. The Classical Boltzmann Distribution Formula. Our treatment of the distribution function m the last section was icstiictecl in scope in two ways For one thing, we tieated the molecules as if they were * Qf, Dams, "Dynamics,” p 88^ or othoi books on analytical median^ Sec. 63] GENERAL MOTION AND SPATIAL DISTRIBUTION 01 mass points, ignoring their internal motions. Then in the second place we also restricted the forces to be of the nature of a force-field. The mutual interaction between molecules, however, cannot bo regarded as a force-field, for the force exerted by one molecule on another varies not only with the position and internal condition of the molecule acted on but also with the position of the other molecule. Accordingly, we have been ablo hitherto to introduce such interaction only in the special form of occasional collisions between extremely small molecules, a restriction which does not correspond to the true situation in gases of appreciable density, A treatment in which both of these restrictions are removed can be profitably developed only by moans of the methods of statistical mechanics, and the fundamental basis for such a treatment will duly appear in Chap. IX below. The principal results there obtained are so simple and so useful, however, that it is convenient to oito them here and to proceed hereafter to make freo use of them. The reader who prefers a strictly logical order can readily secure it by reading alternately in Chap. IX and in the present chapter. Tho rest of the book can be understood, however, without reading Chaps. IX and X at all. In order to state conveniently tho statistical results just mentioned we need to have in mind the language used in general dynamical theory, which is explained in books on analytical mechanics. Each molecule can bo described by moans of a certain number of variables called coordinates, which we shall denote by (ji, qs, gz • • • q»\ their number is often called the number of degrees of freedom of the mole- cule. Corresponding to those coordinates there are then s other variables called generalized momenta, which wo shall denote by Pi, Pt> " ' • ft- The three coordinates of the conter of mass can bo taken as three of tho 5’s, the corresponding p's being tho components of the ordinary momentum; then, in general, the throe Euiorian angles representing the orientation of tho moloculo constitute three more; and there may bo any number of others representing different possible modes of Internal vibration. It is often useful to think of tho q‘s and p's as cartesian coordinates in a space of 2s dimensions. Now consider, first, the case of a homogeneous rarefied gas in which, as hitherto assumed, molecular interaction occurs for each molecule only during a very small part of tho time. In such a gas, when it is in thermal equilibrium at an absolute temperature T, statistical mechanics tells us that at any given moment tho fraction of tho molecules that have the coordinate qi lying between a given 92 KINETIC THEORY OF GASES [Chap III value qi and qi + dq i, sinnlaily in a lange dq^ and so on tlnough the p’ s, is P dqi dqi dq* dpi dp 2 • dp 8 , where P = CVT*^ Ci « [ Je^dqi • dq 9 dpi • #,] \ (89a) Here € is the eneigy of the molecule when its vanables have the values stated, h is the Boltzmann constant, and the mtcgial in the expression given foi the constant C 1 is to be extended ovoi all possible values of all of the vanables so as to make J P dq x dq * dq a dpi dp 2 • dp* « 1, [Cf eq (249c, e) in Sec 199 below] Or, fixing oui attention upon a particular molecule, we can mterpiot P dq\ dp a as the piobability that any given molecule is, at a given moment, in the condition speci- fied, or the fiaction of tho time dining which it is The cncigy e may include a teim to representing potential eneigy of the molecule as a whole in an external fixed foice-field, and this field may include gyroscopic teims of the sort desciibed in the last section, these being without influence on the piobability When the gas is not homogeneous , a foimula like (89a) exists for each kind of molecule separately, containing a common T but, in general, different values of Ci. When, on the other hand, mtw action between the molecules extends beyond the occunencc of almost mstantanoous collisions, this foimula no longer holds, at least not -accurately Then, however, we can fall back on a still more general conclusion from statistical mechanics. Let us number off m a single senes all of the Ns cooiclinates of all the N molecules in tho gas, denoting them by gu, • • * qN B) and tho momenta similarly: pi, • px a Then we can suppose that, while it is in thermal equihbiium at absolute tcmpeiatiue T ) the wholo gas spends a fiaction Pdqidq 2 * * dq^dpidpi * ♦ f dpx* of its time, or has a probability of that magnitude of being found with its variables lying in tho ranges specified, where P « C r *e~®, Ci = • dq$ B dp x dpx^ (89b) IS being now the energy of the whole gas [Cf eq (254) in Sec, 207 ] If the gas is in contact with other much laiger bodies so that its energy can fluctuate a little, it actually does what we here supposo it to do; if, on tho othei hand, the gas is isolated and its energy is there- fore constant, it does not really behave \n this manner, bqt c^lcula- Sec. 64] GENERAL MOTION AND SPATIAL DISTRIBUTION 93 tions based upon tho assumption that it does will nevertheless lead to correct physical results; for the physical behavior of a gas in equilib- rium does not depend, either in theory or in observation, upon the nature of its surroundings. Equation (89a) is coming to be known as the Boltzmann ( dis- tribution ) formula and is of extremely wide usefulness. Equation (89b) expresses what Gibbs called a canonical distribution in phase, but it can obviously bo regarded simply as an extension of tho Boltzmann formula to the whole gas (the physical reason for the validity of the formula is of the same sort in either case). The canonical distribution can be shown to lead to the ordinary Boltzmann formula as a corollary in any case to which the latter formula is applicable. 64. The Boltzmann Formula in Quantum Theory, Wo must next note the modifications that are required in theso principles when quantum theory is substituted for classical mechanics. According to quantum mechanics the description of a system in terms of q’s and p’s is only an approximate method whose usefulness is limited to cases of sufficiently high energy; the fundamental mode of description is quite different. The general quantum theory of gases will bo taken up in Chap. X, but only a few simple details are needed here; they can easily be understood without ever reading that chapter. In dealing with a system in thermal equilibrium wo can speak as if it wore always in some one of a definite series of possible quantum states (disregarding the fine question whether it really is in a single state) ; to each quantum state there corresponds a certain value of tho energy. Each of these quantum states then takes the place, for sta- tistical purposes, of a cortain region in tho classical q, p space of the system. By a “quantum stato” without further qualification wc shall always moan, as here, one of the complete fundamental series of stationary states for the moleculo. When several of those states have the same energy, however, they are often grouped into a single multiple “state”; tho number of fundamental states composing tho multiple ono is then called its multiplicity or statistical weight. The Canonical Distribution. The principle of the canonical dis- tribution now takes tho following form. The probability that tho whole system is in quantum stato i with energy Et can be assumed to be P< » Ce C = [2/ w ] , < the summation in the expression for C extending over all quantum states that are possible for the system. [Cf. ccp (272) in See. 226,] g4 KINETIC THEORY OF OASES [Chap III The resolution of a gas into molecules, oil the other hand, is A more ticklish matter m quantum than m classical theory. It turns out, however, that in neaily all piactical cases the use of quantum theory is essential only as legal ds the internal condition of Iho mole- cules, including then motions of lotation, and l>lml a hylmd tlicoiy ill which classical methods aie employed for the tianslatoiy motion is quite accuiate enough (cf Sec 241) Using tliis foirn of theory, vvo suppose each molecule to be in a ceitain mteinn,I quantum state while moving as a v hole with a cei tain momentum , and, under the conditions under which this theoiy is appioximatcly valid, it appears that Iho Boltzmann distribution holds in the limiting caso of a pcifoot gas ill the following form General Distribution Formula The fraction of tho molecules that at a given moment aie in moleculai quantum state j and also have the caitesian cooidinates and momenta, of thoir contois of mass in certain langes d%, dy, dz, dp x , dp v , dp s (or he in the element dx dy dzdp x dp v dp, of moleculai phase space) is Pjm dx dy dz dp x dp„ dp , where P,« = Ge k ?) (GO) here e is the total eneigy of the molecule and tho constant C has tho value C = kT dxdy dzdp x dp v dp,] i integrated over the volume oi the vessel and over nil possible values of P*> Vv> V‘ and summed over all of the internal quantum states Wo can also write for the energy « - f + « + (91) where f represents the kinetic eneigy of the center of mass, so that £ = 1 mv i — pI db $ d £• . f 2 mv 2 m whereas w is the potential energy of the molecule as a whole in whatever fixed foice-field may be present and 17 , is the internal eneigy cor- responding to the jth quantum state The foice-fiold may include gyroscopic terms of the type described m Sec. 52 This principle has been shown to hold well whenevei the scale of variation of « is laige relative to the molecular wave length \ — h/p Sag. 65] GENERAL MOTION AND SPATIAL DISTRIBUTION 95 In a mixed gas there will be a separate formula like (90) for each kind of molecule, T being the same in all but usually not C. These formulas are valid for actual gases only at low density, of course; even the division of the energy into parts as in (91) tends to fail at higher densities. 56, Special Cases of the Boltzmann Formula, All of the dis- tribution functions obtained previously and many others are included in the Boltzmann formula as special cases, and this fact gives them a basis independent of the special analyses of molecular processes by which we originally obtained them. It will be convenient to collect here some formulas for the principal cases that can arise. Distribution of Centers of Mass , Suppose that we are interested only in the translatory motion of the molecules. Then, disregarding all internal features and summing as given by (90) over all of the internal quantum states with the energy split up as in (91), we obtain as the fraction of the molecules with their centers of mass in the range dx dy dz clp x dp v dp * the value P m dx dy dz dp * clp v dp t) where P,n = %p im - C m e~ L & (92) i C m being a new constant" of magnitude C'2_ f c kT Since p x ~ mv X) J pu = mv v , pi = mvi, we see that dx dy dz dp x dp y dp, == m* dx dy dz dv x dvy dv, in the notation of Sec. 50, so that f — m a P m . Accordingly, C being also equal to mv 2 /2, the value of P m just found leads at onco to eq. (88) and so also, among other things, to Maxwell’s law. Distribution in Position Alone. If wo arc interested, not in the velocities or momenta, but only in the positions of the molecules, we can also integrate P m over momentum space and so obtain as the total number of molecules in the element dx dy dz of space the number Pmq dx dy dz where P mq , the total distribution function for spatial position, is P mq = (92a) G q being a now constant standing for C m J f f o hT dp x dp v dp,. Molecules in a Particular Internal State. Sometimes, viewing the situation in greater detail, we wish to select for consideration only those molecules that happen to be in a particular internal quantum state. According to (00) and (91), of all the molecules in state j, KINETIC THEORY OF OASES 90 [Chap III the fraction Pj” dz dy dz dp x dp v dp, will have their centers of mass in the range here specified wheie pm = Ce kT f Ce kT dx dy dz dp x dp v dp t the denominator lepiesenting the total fraction which molecules in state j form of the whole numbci and serving to make /Pj^ dx • • dp, = 1, or, aftei canceling the r\> lactoi, P$ « C'e~^, C = [ f e~ L & dx dy dz dp , dp v dp ,]" 1 Molecules with Definite Position and Velocity Reversing our choice we might select those molecules in a given range dx dy dz dp x dp v dp, of translational phase space and ask for their internal distnbution, The fi action of them that aie in state j is p m) m dp, = e-V* r dx • dp, ^e-V* 5 " ) ) Since P £ 5 18 independent of the quantum state and Pj m) of the spatial motion, we see that the distubutions of the molecules in space, in velocity, and m internal condition aie quite independent of each other. General Intel nal Dish ibution It follows also that the fraction of all the molecules in the gas that are in state j is the same as P$ m) and so can be wntten, for future reference, P, . g-ij/tr ' e -V*r i (036) If multiple quantum states aie employed, with multiplicities u>„ and energies we have from (93a) or (936) for their probability P K = pern = (93c) the multiple states being numbered here in order and the indicated summation extending over all of them Molecules mth Intel action For further use we may mention a special result that can leaddy be obtained in the classical case from Sec. 56] GENERAL MOTION AND SPATIAL DISTRIBUTION 97 the canonical distribution and constitutes a sort of extension of the BolUmann distribution law to the relative positions of the molecules. Suppose each molecule is surrounded by a force-field so that when another comes near it the two possess a mutual potential energy depending upon the relative positions of their centers of mass. Then if we fix our attention on two particular molecules and assume the first to be in some definite position, the chance that the second is at the same time in a given clement of volume dx dy d& or dr% near the first is proportional to wn 0 kT dn; ' (94a) or, under the same conditions, the chance that of two other molecules one is in an element dr 2 and the other in a second element dr 3 is pro- portional to e dn dr a, (946) and so on. Here stands in general for tho mutual potential energy of molecules i and j. In all such cases the distribution in velocity is independent of tho distribution in position. These latter results are obtained readily from (896) by fixing the variables of all the other molecules than those under consideration, and also the velocities of the latter, and noting that then o > 12 or W 12 + <ou + toss is the only variable part of the total energy E and dr or dr* dr 3 are tho only variable differential ranges. Problem. The ordinary states of the sodium atom are multiple; for the normal or lowest (a state) w = 2 , whereas for the next higher or first excited state, jumps put of which into the normal state result in emission of the familiar D lines, w =■ 6 ( 2 P ^ plus 4 ). The two states lie hv D = 3.37 X 10~ 12 erg apart. If a little sodium is introduced into a Bunsen flame at 1800°C and if thermal equilibrium may be assumed to occur, what fraction of the sodium atoms are excited? (All higher excited states may be neglected. Ans.: One atom in 4.3 X 10 4 .) FREE PATHS AND COLLISIONS 66 . Molecules of Finite Size. Up to this point in the present ohapter we have been considering aspects of the molecular motion in which tho finite size of the molecules is merely a disturbing feature, •one which has to be made negligible by assuming their size to be extremely small; these phenomena would not, therefore, be altered if we imagined the molecular diameters to decrease further toward 98 KINETIC THEORY OF GASES (Ciiai 1 III Fiq 23 — Path of a moloQulo zero There are other featuies of the motion, howevei, which depend directly upon molecular size, and it will be useful to turn now to some of these, particularly in piepaiation for the treatment picsentiy to be given of tianspoifc phenomena The molecules will still be assumed small in companion witli their average distance apaifc, but no longei vanishingly small The Jesuits that wc shall obtain will thus he accurate only m the limit of vanishing density, but, of couise, they will apply appioximately also to cases of sufficiently low but finite density, and consequently a theoiy strictly valid for zeio density constitutes a valuable fust appioxnnafcion to the couect theoiy foi an actual gas. Except as otheiwise specified, we shall in the rest of this ehaptei make the fui ther simplifying assumption that the mole- cules are entnely fieo fiom mutual force- action, except when they aio veiy close togethei We shall call molecules of this soit small-field molecules. The existence of a finite limited legion in which the moleculai field is effective is moro or less equivalent to the molecules having a ceitam size oi diamctei; and, of course, the requnement would be met if the molecules wero in reality small elastic solid bodies free from force except when in contact 57* The Mean Free Path and Collision Rate. The path of the center of mass of a small-field molecule must be an nregular zigzag having at each corner a collision with another molecule and con- sisting of straight free paths between these (Fig 23) The individual lengths of these free paths will vaiy widely; if, howevei, we follow the molecule until it has traversed a great many fiee paths, the avezage of their lengths will have a definite value, which is called the mean free path and will be denoted by L The collisions will likewise bo distributed m time in a veiy niegulai mannei, but ovei a pciiod long enough to include a great many them will be on the aveiagc in each second a definite numbei of them, this numbei is called the collision i ate and we shall denote it by 0 The mean fiee path and the collision rate necessanly stand m a simple and important i elation to each other. Foi in a longish time i the molecule moves a total dis- tance tit, D being its average speed, and this distance is biolcen up by Qt collisions into free paths of average length L ; hence vt = QtL and v - QL (95) Sbc. 57 ] GENERAL MOTION AND SPATIAL DISTRIBUTION 99 Both L and 0 may also be regarded as having reference to a group of molecules instead of to a single one, and this {other viewpoint is often useful, For it is obvious that if we take the mean value of all the free paths that are executed in a given time by all the molecules in a given volume, we again obtain L, provided the time and the volume are not too small. Similarly, if wo select at random a group of mole- cules containing a huge number N } this group will make collisions to the number of iV0 per second; for, when N is very large, this rate is sensibly steady, and the total number of collisions maclo by the group in any time t must obviously be N times the number made by one molecule or NOt. The total number of molecular impacts made by n molecules in unit volume is, therefore, nO; but the corresponding number of . comploto collisions, each involving two impacts, is, of course, only n0/2. Up to this point we have ignored the variation in molecular speed. For a given molecule, however, tho chance of a collision must vary with its speed, being certainly greater when the molecule is itself moving rapidly than when it is merely standing still and waiting to be struck, In some connections it is necessary to make allowance for this variation, and for this purpose wo need to consider .separately those molecules that arc moving at each particular speech Suppose wo select just those free paths that are executed by a molecule while it is moving with a speed between v and v + dv ; lot tho average of these paths be L v . Then if t v denotes tho total inte- grated time during which tho molecule so moves, not counting time during which it moves at other speeds, the total distance covered during this time will be vt v , This distance being assumed to be broken up into paths of average length L Vf tho number of collisions that terminate these paths is vt v /L v ; and if we write 0 v t v for this number, wo have v « 0 V L V (96) in exact analogy to (95), Tho quantity 0„, the collision rate for a molecule while moving at speed v } can also be thought of in terms of probability, and this view of it is perhaps tho simplest and most useful one, As a particular mole- cule moves along, during each short interval of time dt there is a certain chance that it collides with another one, This chanoo will bo pro- portional to the length of the interval dt; but if molecular chaos exists (cf. Sec, 21 above), the chance cannot vary otherwise so long as v remains the same, since in molecular chaos the position of one molecule has no correlation with the positions or velocities of others. Accord- 100 KINETIC TREOllY OF GASES [OlIAlP 111 mgly, if P v di denotes the chance of a collision dtuing dt, out of a laige numbct N molecules moving similarly with speed v the number NP V dt will collide during dt Let us suppose that, as each molecule of qul chosen group collides, we select anothei moving with the same speed to take its place, theieby keeping the total number constant Then we can integrate with lespect to t } legal ding N and P v as con- stants, and obtain fNP » dt = NP v t foi the number that collide in a finite time t But this is also lepiesented by NO v t, accoi cling to the definition of Q v Hence 0 V = P v Thus the collision rate O v at speed v repiesents the chance pel second that a paiticulai molecule collides; it might appiopnately be called the collision probability per unit time Even in a maxwellian gas we shall find that O v and L v vary some- what with v and are therefoie equal, respectively, to 0 and to L only at one definite speech 68* Dependence of L and 0 upon Density and Temperature. The actual values of the mean fiee path L and the collision late 0 foi small-field molecules will obviously depend m pait upon their shape and size, ie,, upon the chaiactci of their foice-ficlcls Without knowing anything moie about these fields, howevei, than is implied in oiu assumption that they arc confined within small limited legions, wo can discover the mode of dependence of L and 0 upon density and tcmpeiature Foi, m the first place, increasing the temperature is equivalent merely to multiplying the velocity of each molecule by a certain uniform factoi, the lolativo chstiibution of velocities being the same at all temperatures, The collision piobability per unit time 0„ of each molecule theieby becomes multiplied by the same factor as the velocities, and so does the general collision l ate G Thus 0 v « Vf. But then, according to (96), the mean free path L is independent of the tempeiature, The situation is quite different, however, when we keep tho tompeiatuxe constant and increase the density. Then, as a molecule moves along, its chance of meeting anothei one is increased, if tho other molecules were distubuted at random, this chance would bo exactly propoitional to the mean density of the other molcoules and the collision rate would accordingly be duectly, and the mean freo path indiiectly, propoitional to the density Now, according to the pimcxple of molecular chaos, the molecules actually are dis- tributed at random, with the single restiiction that when they como too close together then fields tend to keep them apart The effect Sec. 69] GENERAL MOTION AND SPATIAL DISTRIBUTION 101 of this limitation must be to produce a shortening of the mean free path and a corresponding increase in the collision rate; this effect must bo large when the molecules are jammed tightly together, as in a liquid or solid, but it must become small as the distances between the molecules are made large as compared with their effective diameters, and it must vanish in the limit of zero density. Thus wo may conclude that in a gas composed of small-field molecules, as this term was defined in the last section, the mean free path varies only with the density and is inversely proportional to it, being, therefore, at a givon temperature inversely proportional to the pressure. The collision rate, on the other hand, is directly pro- portional to the density and also to the square root of the absolute temperature. Those conclusions are of fundamental importance. 69, Distribution of Free Paths. Absorption of a Beam. In addition to the mean free path the distribution of the lengths of the individual paths is a matter of interest. The method of finding this distribution is the same as that for dealing with the important problem of the absorption of a beam of molecules or ions in its passage through a gas, hence we shall develop first a general formula applicable to both problems. For this purpose let us consider a group of similar molocules of any sort that are moving with velocity v through a region where thore is gas. They may havo been shot into it from the outside, or, as a special case, they may bo a group of molecules of the gas itself which wo select for contemplation. As time goes on, those chosen molocules will collide one after the other with molecules of the gas; as each ono does this, wo shall drop it out of the group under consideration. Let the number originally in the group at time t *=> 0 bo No, and at time ( let N of them still bo going without having had a collision. Then during the next interval dt the number NO v dt will collide and drop out of the group, G v denoting the collision rate for a molecule of the group when moving at speed v among molecules composing tho gas. N is thereby changed by the amount dN — — NO v dt. If we divide this equation through by N, we can integrate it, 0 V being a constant as explained above, thus; dN/N — — 0 , dt, honco log N = — 0 V < + const,; choosing tho constant of integration so as to make N <= No at t — 0, we thus find N - iVoe-°"‘ - (97a) if we write l for the length vt of free path that has been covcrecf at 102 KINETIC THEORY OF OASES [Chap III time t by each molecule fiom the staifc The number of molecules that collide between t and t + dl and so terminate a path whoso length lies between l and l + dl is thus | dN\ = -dN = NoQve-o* ‘ dl = dl (97b) These results can be expi essecl also in the following useful form. Let us write <p{l) for the fiaction of the oiiginal A r o molecules that arc still going after tiaversing a distance l without collision and 4>{l) dl for the fraction of all the fiee paths that have a length between l and l + dl. Then <p = N/No and <p(l) dl = |(iA r |/A r o, whence from (97 a, b ), in which we may mlioduce L v fiom (96) in place of Q», __<w _i_ i _i_ <p(J) = e » = e L ", ^.(J) = e (98a, b) Ij v 08 06 04- 0*2 Both <p(l) and \ p(l) aio thus exponential in foim (cf, Fig 24) If we aie dealing with a molecular beam, <p lepiesents the latio by which its intensity 1 s diminished aftei going a distance Z, O v and L v having Values appiopnatc to the motion of a beam moleculo thiough the gas On the other hand, to apply these lesults to fiee paths in a gas we consider a group of molecules that have just collided and aro now moving with speed v Then <p is the fi action of these that go at least a distance l without collision, and i/' is the distribution function fox fiee paths at speed v , (ll ■ ■ ■ m ■ ■ ■ ■ ■ a | ■ u ■ ■ g u i ■ ■ ■ 5 S ■ l 2 3 4 l/L* Fig. 24 — Distribution of freo paths, l *=* length, L v = mean length being the fraction of all of the fiee paths executed in the gas at speed v whose lengths lie in the lango dl It is obvious fiom the foimuhus that very long fiee paths, while not absent, aie comparatively laie, whereas unusually shoit ones aie compaiatively common, the single length of maximum frequency being l = 0 In either case the aveiage of all the free paths comes out equal to L V) as would be expected, for our foimula gives (cf 63a in Sec. 29) j r* 00 l l i(l) dl = ~\ l ■p le dl = L v Problems, 1 Show that, of the fiee paths executed by a moleculo at speed v, 37 per cent exceed L v , but only 14 poi cent exceed 2 L v in length and 0 7 per cent, 5L„ Sec. 60 ] GENERAL MOTION AND SPATIAL DISTRIBUTION 103 2. Show that in throwing a die the probability that just n throws (i.e., acts of throwing) occur between two successive occurrences of a six (or any other given face number) is % Compare this with \p(l) as given by (986) with L„ = G, l = (n — 1) The average number of throws between sixes is, of course, 6, 60. The Mutual Collision Cross Section. Tim relation between molecular dimensions and collision probabilities is perhaps best approached by way of the conception of a mutual collision cross section for any pair of molecules. To define this quantity, suppose we have a beam of molecules all of the same kind, and suppose they are moving with the same speed along parallel paths but are otherwise distributed at random; in this beam let us hold in a fixed position a single molecule, which may be of a different sort. Let all of the molecules be of the small-field type described in Sec. 56. Then a certain fraction of the beam molecules will collide with the fixed one and will be scattered by it out of the beam; if we draw a plane through the fixed molecule perpendicular to Fia. 26.— Sciutoring of n moiooulnr the direction of motion of the beam, ^oum. the molecules that are scattered will be j ust those whoso directions of motion pass through a certain area on this plane. The magnitude of this area is what is called the mutual collision cross section for a beam molecule and the scattering molecule. We shall denote it by S. If both of the colliding molecules are hard elastic spheres, the area in question is a circle; for, if «ri, <r 2 are their respective diameters, the dis- tance between centers at the instant of collision will bo <r„ v = + 02 ), and obviously the beam molecules that collide will be just those whoso paths of approach pass the fixed molecule at a distance less than a„ and so pass through a circle of radius <r„ v on tho plane (cf. Pig. 25). Thus when both molecules are hard spheres S — ITCTnv 2 . If also <n — <rj = <r, iS = ir<r 2 or four times tho cross-sectional area of one molecule. The sphere of radius <r M is often called, tbo mutual sphere of influence fov tdiQ two molecules. 104 KINETIC THEORY OF GASES [Chap HI If the molecules aie not elastic spheres but aie at least of the small-field type, there will still be a definite collision cross section of some magnitude or other. The chance of a collision may depend now upon the relative orientation of the molecules (being smaller, for instance, for disks when meeting edgewise than when meeting flatwise), but if the beam molecules are oiientcd according to some definite rule, for example at random, the number scattered will be the same as the number that aie incident upon some definite aiea £, and this area can then be taken as the mean or equivalent cross section The final result is thus in any case the same as if the molecules had a certain size, and it is convenient to define a quantity o- av by the equation and to call it the mean diametei foi the pair of colliding molecules, or, moxe explicitly, the equivalent elastic-spherical mean diameter , in case the molecules actually are elastic spheres <r* v as so defined is simply the average of then chameteis, and m othei cases it gives a useful idea of their compaiative effectiveness in mutual collisions In geneial, S and <r ftv may, of couise, depend upon the relative speed of tho impinging molecule The collision cross section obviously has simple and important relationships with scattering rates and with collision frequencies Tho number of the beam molecules scattered pei second, if there aro n of them in unit volume and they are moving at speed v> will be 0i = nSv; (99a) for those that collide in a second aie those that aro contained in a cylinder of cross-sectional area S and length v The same collision late, obviously, would result if wo brought the beam molecules to rest and set the scattenng molecule itself moving among them at the same speed in the opposite diiection, the relative motion being then the same as before In doing this we may leave tho beam molecules fiee to lecoil when struck and restoie the scatteier to its original velocity aftei each collision, oi we may hold the other molecules fixed and let the moving one bounce off each time in some new diiection; the collision late will be the same in either case, sinco the other molecules are assumed to be distnbuted at random Accord- ingly 0i, as given by (99a), represents also the collision rate or proba- bility for a molecule moving with constant speed v among a collection of stationary ones winch are all alike and distnbuted at random, Sec. 61 ] GENERAL MOTION AND SPATIAL DISTRIBUTION 106 whether these are free to move or are held fixed The mean free path of the moving one among the others is then clearly L\ = t//0i or If several kinds of stationary molecules are present, we have only to replace (90a, b) by 0i = L v — — — — ) (09c) i n{ being tho density in molecules per cubic centimeter of kind i and Su the mutual collision cross section for a molecule of this kind colliding with the moving one. In the special case of hard elastic spheres of diameters <ri and cr 2 (99a) becomes 0i — i nir(ai + f 2 ) 2 v. (99fi) If all the molecules involved have also the same diameter tr, 0i and the corresponding mean free path are 0 ! = mrv\ Lv = ( 990 , 7 ) We shall have many uses for these formulas in dealing with molec- ular collisions, 61. The Mean Free Path in a Constant-speed Gas. The first extensive calculations of the general mean free path in a mass of gas wore made by Clausius (about 1867). The law of distribution of velocities being unknown to him, he assumed for definiteness that all the molecules move at the same speed. The analysis required by this case forms a convenient stopping stone toward the treatment of the actual case, and we shall therefore take it up ns a preliminary. The molecules will bo supposed, as usual, to bo of the small-fiold type as described in Sec. 56. In order to obtain results of wide applicability, which wo can do with almost no increase in labor, lot us first calculate the collision rate for a single molecule of a certain kind moving among others of a different typo which are all moving with the same speed but chaotically as to direction. Let the vector velocity of tho first be denoted by Vj and tho constant speed of tho others by Vt. The collision probability per second for the first molecule will he given by oq. (99a) above with v replaced by the average of its speed KINETIC THEORY OF OASES 106 [Chap III lelative to the other molecules, which we shall denote by v r) and can therefore be wntten 0i2 = nSv r \ ( 100 ) here n is the molecular density of the others and S is the mutual collision ci oss section for the fust molecule and one of the othcis If this statement is not immediately obvious, it suffices to divide tho other molecules into gioups accoidmg to tho value of then speed v r relative to the first one; then we can add expectations of collision with the vaiious gioups and can wute for the total expectation of collision duung an infinitesimal time dt } in an obvious nota- tion, 0i2 dt = ^7i } Sv r} dt — dt ~ £m> r dt Fig 20 — Relative volooity* The problem of finding 0i 2 thus i educes to the cal- culation of v r Now, under the conditions assumed, the speed of the fiist molecule relative to one of the otheis vanes only with the angle 1 between their respective dnections of motion; and since the other mole- cules are moving equally in all directions, a fraction $ sm 0 dO of thorn [cf eq, (30) in Sec 18] will be moving in dnections making an angle 0 with the direction of Vi Hence for the average value of the lelative speed v r we have [cf (63a) in Sec 29 for the method of averaging] v r =* \ sin 0 do, But t>J - v\ + v\ - 2viv z cos 0 (cf Fig 26) Hence v? ^ h ( y i + A — %V\V 2 cos 0)K sm 0 dO ^ + v\ — 2v x v z cos 0)* 4 | 1 6vif2 9 ~ 0 [(v x + fl 2 ) S — \v X ““ V 2 \% Here |t^i — t> 2 | 3 is written foi (v\ + v\ — 2viv 2 )ft =» [(^i — y 2 ) 2 P* instead of (tu — u 2 ) 3 because the lattei will be negative if v 2 > «i, whereas of the two values of (v{ + v\ — 2v\H cos 0)^, one positive and the other negative, we must choose the positive one because this is the one whose derivative contains the positive value of (v\ + v\ — 2viv z cos 0) w , which occurs m the integial If Vi > t> 2 , \vi — v%\ Sec. 62f GENERAL MOTION AND SPATIAL DISTRIBUTION 107 t>i — V 2 , whereas if Vi < Wa, |tq — «s| = i>t — tu; hence we get different results according as vi or Vi is the greater: j>2 Vr = 1>1 + if Vl > Vi, (101a) v r = Vi jf tn < ti 2 . (101b) OV2 Insertion of the proper one of those values into (100) gives us the desired collision rate. To obtain the special case of the homogeneous gas of Clausius we then put = «a = v, whereupon both values of v r reduce to a, = 4-a. 0 i 2 in (100) then also becomes simply the general collision rate 0 for a molecule of the gas, The mean free path in such a gas is, there- fore, v 3 0 4n.S’ (102a) n being now the total number of molecules per unit volume and S their mutual collision cross section, In case they arc all spheres of diameter <r we have S — ttt 2 and 3 1 4 rnrcr 2 (1026) which is Clausius’ formula. Comparing (102a) with (996), we see that the simultaneous motion of the other molecules has the effect of raising the collision rate, and reducing the mean free path, in the ratio % or %, respectively, obviously because a given molecule is often struck from the side by othors which it would not strike in consequence of its own motion alone. 62. A Molecular Beam in a Maxwellian Gas. The results just obtained can now bo utilized in treating collisions in an actual gas having a moxwellian distribution of velocities. In order to do this, however, we shall again begin with quite a general case; lot us first work out the rate of collision and the mean free path for a moloculo that is moving at a definite speed v through a maxwcllinn gas. Lot the molecules of the gas be all alike but different in kind from the first molecule. Such generality scarcely increases the labor of arriving at our main goal, and the more general result has an interest of its own. 108 KINETIC THEORY OF OASES [Chap III Out of all the possible collisions of the fiist or lay molecule with a molecule of the gas, let us select those in which the second molecule is moving befoic collision with a speed between v* and v f + dv* \ the number of such molecules in unit volume will be, by (61a) in Sec 28, dn = 4nn Av ,2 e~F v '* dv* Since these molecules aio moving equally m all dn ecti ons, the chance pel second of a collision between one of them and the lay molecule will be 0 W as given by (100) with n leplaccd by dn oi Ami ASv r v f2 e'-P 3 '> ,i dv* The integral of this cxpiession ovei all values of v f then gives the total expectation of collision pei second between a lay molecule and the molecules of the gas, which we shall denote by Q vT Inserting values of v r fiom (101a, b) accoidmg to the relative values of v and v* } which aie to be put in place of V\ and v* } lespectively, we can wnte the result thus = 4irnAs{J^r + —y^e-^dv' + J’ *(v' + dv' j The mtegials occuning heio can be simplified and in part evaluated completely by means of scveial intogiations by paits, or with the aid of the table at the end of the book We can thus obtain : Q vT = 4amA8 {( f_ _ j_ 2/3* 4/3 1 2/3 4 2/3 2 + (JL W 2 + or, if we now insert A — /3 3 /ir 54 from (60) in Sec 28 and wiilo x — fiv, y = Pv 1 , e « - [*“” + ( 2t + ;) J/'*’ *»} < 10S “’ 0 vT as given by this expression repiesents the expectation of collision pei second for a lay molecule moving with speed v tin ought a homogeneous maxwclhan gas at absolute tempeiature T t for which jS 2 — 1/(2 RT) in teims of the gas constant R foi a giam; n is the number of molecules pei unit volume m the gas and S } assumed con- stant, is the mutual collision cross section foi one of the molecules of the gas and the lay molecule Unfortunately the expiession cannot be leduced fiuthei in terms of oidinaiy functions, but numciical values for it can easily be found if one has tables of the “ probability integral,” Sbc. 03] GENERAL MOTION AND SPATIAL DISTRIBUTION 109 $(*) = “7= f ^ dy VnJ o [of. (67) in See, 30 and references there]. As a cheek, we may note that if v— > oo, J* e~ v * dy —> r/2 and So > nSv, which is the same as the collision rate when the other molecules are standing still [cf. 0! as given by (99a)]. The mean free path for the ray molecule in the gas is then T — ^ LvV ~ 07 For reference we may note that the generalized result for a mixed gas is now easily written down. If the gas contains several kinds of molecules which have mutual cross sections with the ray molecule denoted by Soi, £02 * * • So, and have molecular densities n% * • • n v and values of 0 denoted by ft, ft • • • ft, (103a) is to bo replaced by a -' - [»-”■ + + s) J 0 V " *] (103t > r*» 1 where av — (3 r v, It is still true, however, that L vT = tf/Gvi'. One possible application of these results is to a homogeneous beam of molecules moving through a foreign gas. Beams of this sort con- sisting of neutral molecules are seldom produced, however, at present in the laboratory; homogeneous beams of ions aro often worked with, but in such cases the velocities are usually so high that tho motion of the molecules of the gas through which tho ions puss can be neglected altogether and the simpler approximate value Oi = nSv, as in (90a), can bo employed in place of tho more complicated form, 63. Mean Free Path and Collision Rate at Constant Speed. Another application of the results just obtained is to those molecules of the gas itself which happen momentarily to bo moving at a par- ticular speed. For this application let us replace (3 by its value 0 = 2/(\Arv) in terms of the moan molecular speed 0 in the gas ; and for convenience of reference lot us also anticipate a little and insert tho values presently to bo found for the general collision rate 0 and moan free path L in a homogeneous gas. Then, from (106a, 6) below, nS m 0 = 0 0(3 2\/2 and, writing 0„ now in place of 0„j', wo have from (103a) for the collision expectancy per second of a molecule of a homogeneous maxwellian gas 110 KINETIC THEORY OF OASES [Chap III that is moving at speed v 01 " <1Ma> wheie x = (2/\4r )(v/v) and $(t) = (2/v^)J^" y4 d?/ as in See 30, 24 22 20 18 16 14 12 10 08 06 02 G 0 05 10 15 20 2 5 30 v/v Fia 27, — Moan free path L v and collision rate O v at speed v , The eoiresponding mean free path is L v — v/O v — x/fiQ v = \/irH/2() v , or, since I/O = L/v; L v . (104b) *-* + (2& + y *(») m terms of the geneial mean free path L From (104a) we see that for a molecule standing hUII (x = 0 ) O v =* 0/\/2 In Fig 27 aie plotted the latios Q v /0 uml La/L as functions of v/v Both cuives use as v increases, but the out* for the mean free path L v starts, of couise, from L v /L “ 0 and become* asymptotic to Lv = \/2h as v — 4- 00 , wheieas that for the collision rate 1 0 V starts fiom the finite value mentioned above and lises indefinitely 64 Mean Free Path and Collision Rate in a Maxwellian Ga«h The geneial collision late for the molecules of a gas in equilibrium can be obtained now simply by avei aging 0 V ovei all values of v> Litth 4 complication is caused if, in order again to kill two buds with a Hingh* stone, we allow at once for possible heterogeneity ot composition Accordingly, we shall calculate first the late at which a maxwellian wet uf molecules of density n\ per unit volume for which ft = collide wilh another maxwellian set of molecular density n% foi which /3 “ 02 , tho Sec. 64] GENERAL MOTION AND SPATIAL DISTRIBUTION 111 differences in p being due, of course, to a difference of molecular weight. At the end we can then obtain the value for a homogeneous gas simply by putting pi ~ /5* The total number of collisions made in a second by the n\ molecules of the first kind with all molecules of tho second kind is ttiOn in terms of the collision rate O 12 for one molecule. Now, of the n\ molecules, 47rtti A dv 1 are moving in the range dv 1 [cf. (Gla) in Sec. 28]. If we multiply this number by O vT) as given by (103a), with v changed to Vij n to n 2 , S to £ 12 , P to p 2 , x to p 2 v it and y for simplicity to ancl then integrate over all values of v\ from 0 to «?, we obtain another expression for the total number of the collisions in question; hence we can write fti0i2 ~ (4Vff?W2Ai£i 2 ) J v\e~W vx * + ( 2j3it ' 1 + M dvi ] dvh The iterated integral occurring in this expression can be evaluated without trouble, provided we first invert the order of integration, a device that is fre- quently useful. The original range of integ- ration is shown by tho shading in Tig, 28, Vi running for a given value of «i from 0 to Vi and then vi from 0 to » . Obviously wo can just as well let Vi run from Vi to » and then j»j from 0 to «5. In this order the integra- tions can be carried out because odd powers of the variable now occur in that integral which has finite limits. Thus we find, with the help of integrations by parts and formula (69a) in Sec, 28 (or tho table of integrals at the end of tho book) : V* dVi - JJ dv i J 0 U ( 2 /^i + = j (zi3 t v\ + dvi -Ilf + S + w] r ** h '** _ M + m , i m ^ 112 KINETIC THEORY OF CASES (Chap III The first pait of the integral m the ongmal expicssion for m0 12) on the othei hand, has the value I •i fh) l Pi 1 4d,03f + /31)Ti' The entile integral is then the sum of these two expressions or i v*( Pi±m | i m+ii \ 2piPi\(Pl + PD»^2(P$ + PI)*) ■s/ir with (PI + Inserting this value in the expicssion obtained foi niO i2 and also insert- ing Ai — Pl/ir^, and then dividing thiough by n h we find. 012 = ~ ^ (PI + 1 38) w = n 2 3»(#f + (105a) in terms of Heie 0i2 represents the collision late poi molecule of a group clisti ibutocl m velocity in a maxwellian manner with mean speed fli, moving through anothei maxwellian gioup oi a different sort whoso density is n 2 and mean speed y 2; &i 2 is the mutual cross section for a moloculo of tho first kind m collision with one of the second, If ?7ioi e than two kinds of molecules arc piesent the appropriate generalized foimula can at once be wntten down, for tho total col- lision rate 0 t for the zth land will simply be the sum of oxpiessions such as (1 05a) lepresenting the effects of collisions with all of tho vanoua kinds of molecules that aie present, including its own kind. By (66a) in Sec 30 l/t> s M f the molecular weight, hence O t and the conespond- ing mean free path, L t - v x /Q iy can convoniontly bo written thus, with obvious meanings for the symbols e, - ^ = <2^(1 + | 0 * ’ * - [2-4 + W]‘ (1056) (106c) Finally, if only one kind, of molecule is present with density n molecules per unit volume, mean speed v and mutual collision cross section S, we need only put S, = S 2 = v and Sn = 8 m (105a) in ordei to obtain the collision rate 0 and the mean free path, L — D/O, in a Sko. 071 GENERAL MOTION AND SPATIAL DISTRIBUTION 113 homogeneous maxwellian gas : 0 = -\Z2nSS, L = For hard elastic spheres of diameter <r these become 0 = \/2 TrnDir 2 , L = - > V2ir w* 1 -S/2 nS (106a, b ) (106c, d ) which are the formulas most commonly given. 68. Magnitude of the Correction for Maxwell’s Law. It is instruc- tive to compare formulas (1066) and (102a). The comparison shows that the introduction of Maxwell's law in place of Clausius’ assumption of equal speeds, after greatly complicating the calculation, only changes the final result in the ratio l/\/2 -r %. — 0.94, or by 6 per cent. The effect of introducing Maxwell’s law is very much greater than this in the case of some phenomena, such as heat conduction, which depend essentially upon differential motion of the molecules; when, however, the motion affects the phenomenon only indirectly, the maxwellian correction amounts quite commonly, as in the present case, only to a few per cent. In view of our ignorance as to the true molecular forces the smallness of the correction often justifies the expedient of saving labor by assuming uniform speed instead of Maxwell’s law. 66. Mode of Determining L and S or d. If we wore now to attempt to compare the results just obtained with experimentally determined magnitudes we should encounter the difficulty that eqs. (106a, b ) contain two new molecular constants, L and S, concerning which the development of the theory up to this point has given us no other information. A second relation between them is, therefore, necessary before cither one can even bo calculated from observed data. The best source of additional information for this purpose lies in a comparison botweon the theoretical and observed values of tho viscos- ity; this will bo discussed in tho next chapter, ancl a table will bo given of values of L and a for a number of tho commoner gases. 67. Collisions in a Real Gas. Up to this point in our discussions of collision phenomena in a gas wo have uniformly made tho assump- tion that the molecules exert forces upon each other only when they come into close proximity. Various considerations indicate, on the contrary, that in reality the mutual force does not quite vanish at any distance, however great. If this is true, then strictly speaking every molecule is in collision with every other one all tho time and there are actually no freo paths at all. The mutual collision cross section for any 114 KINETIC THEORY OF OASES [Chap, III pair appeals, therefore, to be infinite, a molecule hold m a beam os contemplated m Sec 60 throws a shadow fico fiom beam molecules which flaies out to an infinite diameter at infinity, This cncumstance piesents a difficulty that has been met in practice in various ways, In connection with some phenomena defloctions of veiy small size aie ummpoitant and m such cases it may bo sufficient simply to say that, by definition, a collision occuis only when the resulting deflection exceeds a ceifcam arbitiaiily chosen minimum amount A finite cioss section then exists by definition, but, of com so, it will vary somewhat with the choice of the critical deflection. Such a solution of the difficulty is not likely to be widely useful, however, and fiom the theoietical standpoint it is unsatisfactoiy, A better procedure, and one often tacitly employed, is to replace the actual gas in thought by a set of elastic spheies moving classically and of such a size that they aie equivalent to the actual molecules insofar as the particulai phenomenon under discussion is concerned; tho cross- sectional aiea of one of these spheies is then taken as the cross section or, to speak precisely, the equivalent classical-sphei e ooss section of the molecule for the phenomenon in question, the cliamctci of tho sphere representing the equivalent mean diameter foi the two molecules Such a procedure has the disadvantage of yielding, as we shall sco m the next chapter, different equivalent cioss sections for diffusion and foi viscosity and heat conduction, neveitheless, it seems to bo tho best sort of conception for practical use in thinking about tianspoit phenomena When we substitute wave mechanics for classical theory, tho diffi- culty in a sense disappear, for accoiding to wave mechanics the number of molecules thrown out of a beam is always finito However, this advantage over classical theory docs not turn out to bo vory help- ful, for the collision cioss section as defined above is in actual cases enormously influenced by a pi eponderating number of small deflections, and for this reason it is not actually a very useful quantity As a matter of fact, the most inteiesting theoretical lesults obtained from wave mechanics have refeience to finei details of the collision phenom- enon, especially tho angular distribution afteiwaul; this distribution, furthermore, besides its great importance in scatteiing expci imonts of various sorts, is what we shall have to employ when wo como to dovelop an accurate theory of transport phenomena Accordingly, we shall now drop our gross view of the collision process and proceed to consider it as a phenomenon of scatteiing For this purpose we shall return first to tho use of classical theory; then later the results will be desciibed that havo been obtained by wavo mechanics Sbo. 68] GENERAL MOTION AND SPATIAL DISTRIBUTION 116 MOLECULAR SCATTERING 68. The Scattering Coefficient. The phenomenon of molecular scattering can be analyzed quantitatively as follows. As before, let a homogeneous beam of particles approaching along lines that are parallel, but otherwise distributed at random, pass over a given molecule held in a fixed position; but consider, out of all particles incident on unit area of a plane perpendicular to their direction of approach, only the fraction which undergo a particular deflection by the fixed molecule. Let us consider those molecules that acquire a velocity whose direction lies in a definite element of solid angle We can write for the fraction that these form of the total, G cloy) the quantity G thus defined is called the scattering coefficient for collisions of this type. If N particles arrive in the beam per unit area per second, tfG clo) of them will be scattered per second, or NG do> dl is the chance that one is scattered during an infinitesimal time dt } in the direction of clw, G has therefore the dimensions of 1/Nt or the dimensions of area. If thero are N f scattering molecules per unit volume, then NN'G do> of the beam molecules are scattered within dco per second upon their first encounter with a scattering molecule. The same definition can be applied also to the more important case in which the second moleculo, although momentarily at rest, is free to move, provided we require that after each collision it shall be brought to rest before being struck again. Clearly the scattering coefficient will vary in general with the direc- tion of scattering; we can conveniently regard it as a function of polar coordinates whose axis is parallel to the incident beam, and if wo prefer we can write <7(0, ip) dw ~ (7(0, <p) sin 0 dO d<p . G will also depend upon the nature of the colliding molecules and upon their velocity of approach, and it will be different according as the scattering moleculo is held fixed or is left free to move; furthermore, it may depend upon the orientation of the molecules. Questions involving orientation lie far beyond the reach of present experiment, however; all that we can hope to observe is an effect averaged over all orientations of both of the colliding particles. Such average effects will necessarily bo sym- metrical about the direction of the incident beam, and they can there- fore be described^!! terms of a mean scattering coefficient which is a function of 0 alone. Hereafter we shall understand (7 to stand for this mean coefficient and shall regard it as a function (7(0). The moan solid-angle scattering coefficient G{0) must not bo confused with a differently defined coefficient that may bo more useful undor some circumstances. We might defino a polar scattering coefficient F(0) 116 KINETIC THEORY OF GASES [Chap 111 by the requirement that F(6) dO shall lopresent the fraction of the beam molecules incident oil unit aiea which aie scattcied by ono scattenng molecule in all cliiections making an angle between 0 and 0 + do with the direction of the incident beam Then, if we give to da> the form of a ring so that dw — 2ir sm 0 do, wo can wnto foi the molecules scattcied into the range dO eithei G du — 2 irG sm 6 dO or F(0) do , and it follows that F(0) = 2ir G(0) sm 0. (107a) In any case the scatteiing coefficient necessanly boais a snnplo lelation to the collision cross section The mutual collision cross section 5 leprosents the total fraction scatteied out of unit aiea of tho beam at all angles by ono molecule Hence it is S - f Q * F(0) dO = G(0) sin 0 dO = G(0, *>) sm 0 dO d *>, (1076) the last expiession being a more general foim in terms of tho original nonaveraged G Those two equations must hold whether the scatter- ing molecule is held fixed or is left fiee It will be worth while now to calculate tho classical scattenng coefficient foi a few simple cases, 69. Classical Scattering Coefficient for Symmetrical Molecules with Fixed Scattered In the case of field-free haul spheres G ia very Fiq 20 — Deflection by a aoattonng center 0 easy to find, but it is also not very difficult to find a more goneial expression applicable to any spherically symmetrical type of force, To do the latter, let us suppose that the scattenng molecule has its oenter fixed at a point 0 (Fig 29), and that a beam molecule approaches with speed v along a line AB passing at a distance b from 0, Let tho mutual potential energy of the two molecules be t/(r), where r is tho distance between their centers Because of the force the beam mole- cule will be deflected so as to move along a plane curve having as Sec, 09] GENERAL MOTION AND SPATIAL DISTRIBUTION 117 asymptotes the line of approach AB and another line making the angle of deflection, 0, with AB , Let 0i denote the instantaneous angular position about 0 of the approaching molecule, measured positively from a line drawn through 0 parallel to AB but backward toward A } and let the mass of the molecule be m. Then at any moment the molecule has angular momentum about 0 equal to mr 2 6 1, r being its distance from 0, and it has kinetic energy ^ m(r z + r*6 \) ; since the initial values of these quantities are clearly mvb and § mv 2 , respectively, we have, according to the laws of the conservation of angular momentum and of energy, mr 2 0 1 = nivb } $ ni(t 2 + r 2 0 ?) + U(r) ~ ^ mv 2 . Solving the first of these equations for (k and substituting this valuo in the second, we have, therefore, , dO i vb ° L ~ dt ” r 2) and finally, dividing these two equations, 2X7 _ v*b*\H m r 2 ) * dOi = l Fi _ dr r 2 [ wa 2 ? ,!! J Now the path is symmetrical about its point of closest approach to 0, at which wo shall write r — ?'o. Hence wo can write for the total increase in 0i during the entire collision 2 £” (dOi/dr) dr. This is equal also to ir — 0 in the case of repulsion, and the same relation can bo used in the case of attraction provided we lot 9 be negative in that case [of. Fig, (29)], Hence in either case we have for the angle of deflection C M dr 0(b) - «■ - 2 bj n ^ ^2U(r)/mvTr~^W (108a) or, in terms of x «* r 0 /r } since dr/r ~ ~ dx/x, *C X da 0 (b) = 7T — 2j o ||! _ 2U(n/x)/mv^[r7/b T f z ^V' i ' (108b) Since the original lower limit r 0 is that value of r for which r — 0, wo see from the general expression for r given above that it is tho value of v that makes the denominator in the integrand vanish in (108a) or the root of tho equation the denominator in (1086) vanishes for a: = 1 ( 109 ) 118 KINETIC THEORY OF GASES [Chap, III Equation (108a) 01 (1086) gives 6 as a function of 6 From this function 0(b) the scattering coefficient 0(6) oi F(0) as defined m Sec 68 can then be obtained as follows In the homogeneous beam that is contemplated in the definition of G oi F the lines of approach of the beam molecules aie (on the aveiage) umfoimly distubutcd ovei any plane peipendiculai to their common dnection, and in paiticulai over such a plane chawn through 0 , and those lines foi which 6 lies in a given lange db pass tluough an annulai nng on this plane of radius 6 and width db and thus of area 2wb db, so that they form a fraction 27r6 db of the lines ciossmg unit aica This fiaction is also the fiaetion of the molecules that aie scatteied into a lange dO such that dO — |0'(6)| db Hence 2?r6 db ~ F(0) dO — 2?r G(6) sm 0 do — 2itG(0) sin 0 |0'(6)[ db and «*> - ISWSV '<*> = lH) « The absolute-value sign is needed on 0'(b ) because m defining G and F the diffeiential dO is supposed to be positive wheicas 0'(6) may be negative; 0'(b) might also be positive foi some values of b and negative for others, in which case the foimulas would have to be made moio complicated but m a lathei obvious way Fiom these equations G and F can be calculated, piovided the potential function t/(?) is known or assumed 70. Examples of the Scattering Coefficient, There aio two special cases of considerable interest in which the calculations aie easily earned out, Suppose, fust, that both molecules are hard elastic spheres ficc from foice except at contact Then U ~ 0 except during the voiy short mtei val of collision, and dunng the lattei the change in 0\ is negligible, so that the concsponding pai t of the integial in (1086) can be neglected Hence in that integial wc can without appreciable error put U(i) ~ 0 everywhere Fuithcimoie, the distance of closest appioach is in this case ?o = <ri 2 — (<r\ + cT 2 )/ 2 , the mean of the diametois of a beam molecule and the fixed one Accordingly, (1086) gives Sec. 70] GENERAL MOTION aND SPATIAL DLi'l &.8UT10N 110 m 1 . 0.0 2 alt cos ^ sm ^ sin 0 = iff* <rh- (110c) As a check we may note that this value of G substituted in (1076) gives a cross section S = mhC* sin 0 dO = jr<rf a , (HOd) which is obviously correot. The fact that according to (110c) G is independent of 0 moans that uniformly distributed spheres incident upon a fixed sphere arc scattered by it equally in all directions, a result which is, of course, more easily obtained directly. As a second example, let us consider the general case of slight deflections, characterized by the condition that 2 U f mv 2 is small through- out the collision and 0 is consequently also small. (With some forms of the function U a small 0 might, of course, occur without 2U/mv‘ 1 being at all times small.) Then, if we expand in powers of U and drop all after the first, wo can write, from (1086), after substituting for r§/6 a from (109), 2 Wl)¥l 4 . mv* JL ^ 2U(n) m r - a :*)-» U(n) - m«*(l Wg) l c 1 )* 4 • • ■ j 1 w* — X 2 j dx dx or approximately 0 = 2 p f/Q-p) - U(n/x) mv 2 Jo (1 — a ,s ) M dx. As a special case, supposo U(r) — C/r n , where n is some positive integer and C is a constant, positive or negative. Then, if we sub- stitute x — sin v C, wo have J*L f"Y_i ain> ' A dt mv 2 r$J o \cos 2 £ cos 2 (y ' after integration by parts, Here the integrated expression equals — 1 at £ = 0 if n » 1 but otherwise vanishes at the limits, for wo can write it sin f — sin ^ 1 f _ sin 2 f — s j n a(tt-i) f ^ (n — 2 ) cos f ■ ' » cos £ ~ cos f (sin f + sin ,, ~' 1 £) sin £ + sin ” - * 1 £ KINETIC THEORY OF GASES 120 [Ciiap lit after substituting sin 2 f — 1 — cos 2 f, highei powcis of cos f not being wntten Let *■ - /„ 2 > 4 sm” $dt; 1 • 3 -Al 1 t- isl for odd n, VI ‘ 1*3 (n — 1) 7T 2 4 foi oven n ($[ = 1 if n ~ 1 but otheiwise — 0) Then it is easily scon that for any positive integial n } r 0 being leplacccl by b because ( WO aro kcopifig only the first powei of the small numbei C, Q _ 2 nK n C mv 2 b n For the scatteung coefficient wc have then fiom (110a), letting 0 stand now foi its absolute value, mv 2 b^ 2 ^ 1 ( 2nK n \C\ \ v " 1 2 n 2 K n C sin 0 n\ rm l ) ' 0 » Bin 0 (110c) after eliminating b in favor of 0 Foimula (HOe) indicates a spinelike concentiation of the ncatloicd molecules in dnections near that of incidence, the fact that it contrib- J v /* i , ? ? o GsmOdO the infinite pait J d0/0 n « (n/20”)|o moroly lllustiates the infinity of the classica 1 cioss section that was l of erred to above Foi n — 6, which is suggested by wave moohanics for the attractive or van der Waals molecular field of all molecules, G « 1/(^0** sin 0) and, by (107a), F(0) oc 1 Pioblems 1 Show duectly that elastic spheics falling with equal and parallel velocities but otheiwise at landom on a fixed sphero aro reflected equally in all dnections 2 Show that if n = 1, l o , U — C/i as in a Coulomb field, 6 = 2 tan "‘ Jk’ G = iSiT* am 4 \o/2) ’ acouiale, y (These formulas are easily obtained directly, oi from (108b), by elimi- nating with the help of (109) fiist C and then, m the lcsult, ro. Tho case n ~ 2 can likewise be worked out completely ) 71. Relative Scattering The piecoding calculation had reference to the veiy simple situation m which the beam molecules all have tho same velocity and the scatteiing molecule is held fixed In actual experiments, on the othei hand, the beam passes thiough a clustci of ti»o. 71) GENERAL MOTION AND SPATIAL DISTRIBUTION 121 molecules that are not only free to move but are already in motion, perhaps forming a gas in thermal equilibrium; and the beam molecules themselves may be distributed in velocity either in a maxwellian man- ner or otherwise. The general treatment of such eases is complicated and will not be given here; it may be worth while, however, to take one or two easy steps toward it by investigating the effect of the motion of the scattering molecule alone. This effect can be divided into two parts, one arising from the initial motion of the scattering molecule, the other caused by its acceleration during the collision process. The first effect reduces to a simple problem in change of axis. It turns out, furthermore, that the second effect can be handled in a similar way; wo have only to make use of the well-known theorem concerning the two-body problem of planetary theory which states that, when just two particles move under the action of mutual central forces, the motion of either ono relative to the other is the same as would be its actual motion if the other one were held fixed and if at the same time its own mass were reduced in a certain ratio. The truth of this theorem is so quickly seen from the differential equations of motion that we shall prove it here. Let mi, nh be the masses of the two particles and ri and r 2 their vector distances from the origin, Then, if f(r) is the magnitude of the force that each particle exerts upon the other, expressed as a function of r and measured positively as a repulsion, Newton's second law of motion leads to the vectorial equations: d 2 d 2 m i ^ r, = f(r) r 0 , m* ^ r 2 = -/(?') ro, ro being a unit vector drawn from the second particle toward the first. From these two equations we find for r — ri — r 2 , the vector position of the first particle relative to tho second, dh dtfx d% _ mi + m 3 ,, * di 2 ~ dt 2 ~ dt 2 mim% } °* This is exactly the same equation that would hold for the first particle if the second one were fixed while tho first had a mass, not mi, but m f — + w»). Accordingly the theory of the scattering by a fixed molecule can bo utilized in arriving at formulas appropriate to the more general case in which the scattering molecule is free to move; we have only to apply that theory to tho relative motions and to make the proper change in the mass, 122 KINETIC THEORY OF GASES [ClUP III Suppose, now, a scattering molecule of mass moves with velocity V 2 into a uniform beam of molecules of mass TOi moving with velocity v u Then lelative to the scatteung molecule, the beam molecules appionch as a unifoim beam moving with vector velocity v = Vi — Vs (cf. Fig 30) Let us denote by 0 r the angle of deflection of a beam molorulo in this xelativo motion, 01 the angle tlnough which the velocity of tlio first molecule lelative to the second is turned by a collision Then the scattering coefficient foi 0 r will be, by (110a, b), GrW b |#(b)l Sill Jr Fr(Or) - 27Tb \0'M’ (111a, b ) & denoting the peipendiculai distance of the scattciing molecule fiom the initial line of lelative appioach of the beam molecule, 0 r (b) is tho same function 6(b) as is given by (108a) or (108&), in terms of u M determined by (109), but with m icplaced in all three equations by m f = mi7n 2 / (mi + m 2 ) and with v now 1 epiesenting the magnitude of Llio relative velocity, The collision does not alter the magnitude of v } as we showed m Sec 24, using only the same principles of momentum and energy that were employed above in calculating the deflection 72 Classical Scattering Coefficient for Free Symmetrical Molecules, By means of the coefficient O r (0 r ) or F r (0 r ) for scattoung in the relative motion we can then find the coefficient G(0) or F(0) for scattering m terms of the total motion of tho beam molecules, it is only necessary to find the l elation between 0 and 0 r The goncial foimulns thus obtained are lathei intneate, however, so wc shall treat in detail only the simplest case, that in which the scattciing molecule is initially at rest The lesultmg foimula should give some idea of the aveiago relation between G(8) and G r (0 r ) for a beam passing thiough any gas that is flee from mass motion To have the scatteung molecule initially at rest, wc put va — 0 in the analysis of the last section, Then the beam velocity Vi is the same as the initial lelative velocity v, while &, which was defined above in Sec, 72] GENERAL MOTION AND SPATIAL DISTRIBUTION 123 terms of the relative motion, represents also the distance of the second molecule from the actual initial line of approach of the first. Further- more, the entire motion occurs in this case in a single plane, Let the velocities of the two molecules after the collision bo and v£, respectively, and let the subscripts ||, X denote components respec- tively parallel and perpendicular to the beam velocity Vi (cf. Fig. 31). Then, the relative velocity v' after collision being still of magnitude v m, “ 2 m, » 2 m^, Fia, 31. — Scattering by a freo molooulo initially at rest; volocity di tig ram a but inclined at the angle 0 r to the beam direction, we can write for its components try' *=» Vi\\ — v%\\ « v cos 9 ri Vx ~ ~ ^ Conservation of momentum during the collision requires now that m\Vi\\ + W2t»2i| “ flhvi = vhv f mvtx + m%v^x — O* From these equations and the obvious relation, tan 0 — vJxAi lb wo readily find that tan 0 sin Of tan $ 0 r mi . A — + COS Of 1712 i + - 1) 2Vm, j HOC 3 i 0 r ( 112 ) It is then easy to find V(0), lor F(0)\d0\ » F r (0 r )\(lO r l Calculating dO/dO r from (112), wc thus find m = 2 + w? _ i ‘ 1 -f — COS 0r K(0r). (113) This gives us F(0) as a function of b in terms of the value of F r givon by (111b), and then by (107a) 0(0) - As a special case of these formulas, wo may no to that if Wi 853 i)h> by (112) 0 ~\0 r > and by (113) F « 2F r < Thus in this caso, as ranges from 0 to ir } 0 is confined to the first quadrant, so that the beam 124 KINETIC THEORY OF OASES [Chap III molecule retains at least a slight foiwaid component except in tho ono case of a central impact If, on the other hand, mj < nit, tan 0 passes through oo and 9 increases continually with moi easing O r up to a maximum of tt foi a central impact [case (a) m Fig. 31] Finally, if mi > a calculation using (112) shows that 0 attains a maximum value when cos 9 r — —mt/mi, beyond which 0 do menses to zeio again [case ( b ) in Fig 31], at the tinning point F(0) and G(0) become infinite, while conesponding to any othoi angle tlieie aic two values, say, I'\ and or 6’ i and Gt, and the paiticles scattcied into a given lango dO consist of two groups moving at difteient speeds and foim a fiaclion [F^fl) + F 2 (0)] d.6 oi 2v[Gi{0) + Gt{6)] sin 0 dO of tho whole Tho two groups for the same 6 aie shown by tho two diagrams foi case (b) m Fig 31 The speed of the scatteied beam molecule is easily found to be v[ - ( cjh ' 2 + » lX ' 2 )* = + m ( 2 + ^ mi ™ 2 008 °^ H The ends of the vectors v[ and v 2 can be shown to ho for vaiymg b on two semiciicles as indicated in the figiue. Problem, Show directly that for field-fice hard clastic splines of equal mass, when the scattenng ono is initially at lest but loft fice to move G(0) = a\ a cos 6, F(0 ) = i nrf 2 sin 20, where cr ia = + <rs) and 0 g 0 g v/2, 73. The Experimental Determination of the Collision Cross Section * Veiy mteieating cxponmonts have begun to be loportod in recent years which furnish diiectly values of tho collision cross sec- tion and of the scattenng coefficient. The general arrangement in such experiments consists of a source chamber emitting a stream of molecules, out of which there is selected by means of a pair ol collimat- ing slits a nariow beam moving in a fairly dofinito direction, as shown schematically in Fig 32, The beam then passes through a region into which various sorts of scattenng gas can be introduced, and the number of molecules that pass through it oi are scatteied m various directions is determined by means of some sort of dovico for detecting and measuring molecular beams Tho density is kept veiy low throughout by pumping, and often also in pait by keeping the wails sufficiently cool to condense all molecules that stnko thorn , if tho souico * Cf R G Fbasbh, "Moleculai Rays,” 1931 Sec, 74] GEN DUAL MOTION AND SPATIAIj DISTRIBUTION 125 is an oven in which a substance is being evaporated at high tempera- tures, the outer walls of the tube may bo at room temperature, whereas in other cases they may need to be cooled with liquid air. For accuracy it is important that collisions of beam molecules with each other be rare occurrences, and also that those beam molecules which have col- lided more than once with the scattering gas do not enter the detector in disturbing numbers. When determinations of the total collision cross section are to bo made, the detector is placed in line with the direct beam (at D i in the figure) and the intensity of the beam is noted both with and without Fra. 32. — Arrangement for measuring collision cross scoLions. the presen’ce of the scattering gas. Let h, / 2 be the numbers of beam molecules received per second by the detector in theso two cases and lot x denote the distance traversed by the beam through the scattering gas, which can bo made effectively equal to tho distance from St to Du Then, if the beam molecules aro moving with uniform velocity v, wo have, by (98a) in Sec. 59, from, which tho mean free path L v of the beam molecules moving through tho gas can be calculated; the mutual collision cross section is then S — l/(nL v ), n being the number of scattering moloculcs per cubic centimeter [cf. (995)]. Unfortunately, however, it is difficult in practice to select out a beam of uniform speed, and consequently up to the present only a maxwellian distribution in the beam has boon worked with, tho results being then taken to represent scattering at the mean speed with an accuracy that is sufficient in view of the rather large experimental errors. 74. Knauer’s Observations on Scattering.’ Some extensivo obser- vations made in this way have been reported recently by Kmiuer.* His results on the passage of several gases through mercury vapor illustrate beautifully the variation in the apparent moan free path with ♦ Knatjbii, Zeits.J. Physik, 80, 80 (1033); 80, 660 (1034). 126 IONBTIC THEORY OF GASES [Chap III the cnteiion adopted for a collision Since it is always possible in such expenments foi a molecule to be deflected veiy slightly and yet entei the detecting device and be counted as an undeflected molecule, Kiiftiiei estimated for each of his detect ois the aveiage maximum angle tluough which a beam molecule could be deflected by collision with a meicuiy atom and still entei the detectoi His values foi the mean fiee path in ccntimeteis of hydiogen and helium in merciuy vapor at a piessiue of 1 dync/cm 2 aio shown below foi vaiious values of the limit- ing angle, which was fixed oxpenmentally by the size of the slit placed in fiont of the detectoi ; foi companson wc have added undei the heading “by i?” the eqiuvalcnt-spheie value as calculated fiom the molecular diameteis given m the table in Sec 86 below and the equation L = — 4[^7r(cri + c^) 2 ]^ 1 [Cf (99 d) ] The temperatures given axe those of the beam, the mer- cury vapoi was at loom tempciatuie ii» Its He Limiting angle 0 9° D By v 0 9° D D Absolute temp 120° 1 4 ; 2 7 2 5 3 0 5 5 4 7 295° 1 7 ' 3 5 3,0 3 4 7 2 5 8 ! 12 4 If these icsults arc loally typical of the behavior of uncharged gaseous molecules, the enoimous vanation of the appaient cioss section with the limiting angle seems almost to lulo out, in such cases, any application whatevoi of the conception of a collision cross section, or even of the idea of a progiessive absoiption of a beam of molecules A beam passing thiough a scattenng gas must undeigo a weakening as judged by any test, but these results indicate that it also straggles moie and moie duung its passago and that the line of distinction between stiagglmg and weakening is decidedly indefinite* The most probable cause of the large diffeicncc between the mean free paths obscivod by Knauer and the values deduced from viscosity data is picsumably to be found m the occurrence of an enormous number of small deflections This is stiilangly borne out, indeed, by Knauoi’s own piinmpal results, which had to do with the scattering coefficient itself In Fig 33 aro shown on a log-log scale his values of f(0), piopoitional to our 0(d), foi II 2 and lie passing CXIO| f<e> 1 6 (XIO' 1 ) 2 (xio- 3 ) Sec. 76] GENERAL MOTION AND SPATIAL DISTRIBUTION 127 through mercury vapor at 295°K; one curve is also plotted for f(0) sin 9,’ proportional to our F(6), in order to give an idea of the total distribution with respect to 0 . Because the mercury atom is very heavy and consequently moves very slowly, w r e might perhaps expect 0 to be somewhere near constant, as was found above to be the case theoretically for spheres incident at random upon a fixed sphere; but on the contrary the observations exhibit an enormous increase with decreasing 0, The upper parts of the curves correspond roughly to an increase in f(0) by a factor of 10 for an increase in 6 by a factor of 4, which would make G proportional to the (log 10/log 4)th or 1.66th power of 1/0; according to (110c), such a variation would result from a classical force varying as the inverse fourth power of the distanco between the two molecules (2 /n — 0.66, n = 3, U cc, r~ a and the force «r 4 ), At larger angles, however, the curve is I steeper, suggesting a higher power. 6 The continued rapid decrease of Cat (xrcr 2 ) large angles, o.g., from 46° to 00°, is 11 surprising. It seems unlikely that J the quantum effects to bo described in the next section could be large in the present case, although they would no doubt amount to some- thing; the do Broglie wave length of the hydrogen or helium molecule i ri «. should be only 1.0 or 0.8 angstrom, respectively, as against a mean collision diameter, calculated from the viscosity data, of 3.4 or 3.2 angstroms. The final explanation of these phenomena must await the results of theoretical calculations for the actual typo of collision involved, and also, perhaps, tlio obtaining of more accurate data in this difficult but fascinating field. 76. The Wave Mechanics of a Particle. The classical calculations of scattering coefficients that we have just made can only bo regarded ns a preliminary exploration of the possibilities. For any accurate treatment of molecular scattering the use of wave mechanics is essen- tial. The general wave-mechanical theory of gases is reserved for a special chapter (Chap. X), but the approximate method that is usually employed in handling collision phenomena is simple enough <v r r II !! H 2> f(0) ■1 ill Jn! T- IS ■ — ■ m II • ■ wsm ■ ■ ■ ■i S 5 Si IK ■ s Si 10 V r N. V \ \ : V ' X. 5 m 1 Si — — —— 1 --v~ A 1 * 0.6 I 2 4 6 1 2 4 6 1 14 (X10) (XI 00) Scattering Angle 0 In Degrees 33.- — SciUtoriiiR by moratory, (Knaucr.) 128 KINETIC THEORY OF CASES [Chap, III so that a descnption of it, and of the Jesuits that have been obtained by its use, can be given heie For the undoi standing of this method the material m Chap, X is not necessary The new mechanics stalls out from ladically novel conceptions concerning the fundamental piopeities of matter, In classical theory it was supposed that a molecule 01 othei paiticle could be imagined to move along a shaiply defined path, possessing at every instant of time a definite position in space and a definite velocity, just as a thiown ball can be seen by the eye to tiace out a dofimto ciuvc in the air, Accoiding to the new thooiy, this is not so; slmiply defined tiajectoiics do not leally exist, and the motion of molecules can be described acciuatoly only m terms of probabilities In place of the definite position with caitesian cooidmates x } y t z t which a molecule formcily was supposed to possess at a given instant t t we have in the new theory, in its nomelativistio fonn with spin omitted, a probability density P(x y y, z ) t); this has the significance that. P(x , y, z } t) dx dy dz leprcscnts the probability that, if a very accurate obseivation of the molecule’s position wore made at the time t y this obseivation would leveal the molecule within the element of volume dxdydz So long as no such obseivation has been made, howevei, we aio unable to say, and mean anything physical by the statement, that the molecule is definitely at one point or another or is moving m any particulai way We might, to be suic, icpeat our observations of the position at shoit intervals of time in an effort to follow the molecule along a definite path, which is the method actually employed by astronomers m obseiving the motions of the planets; but if we did that to so small a body as a molecule, according to piescnt knowledge each observation would disturb tho motion so greatly that the path observed in this way would be an irregular zigzag devoid of significance It is only in dealing with much heavier bodies, 01 with molecules moving at much higher speeds than those of thermal agitation, that repeated obseivakons can be imagined to reveal an approximately smooth motion along a classical tiajectory. This revolutionary change m kinematic al ideas compels a cor- responding change in dynamical laws, In tho place of Newton’s laws of motion we have in wave mechanics a law concerning the propagation through space of the piobability density This law is expressed by an equation of wave propagation analogous to the equations that hold for the propagation of sound or light and is most simply stated in terms, not of the probability itself, but of tho so-called probability amplitude, which we shall denote by y, 2, t) The lattei is usually a complex numbei and the squaie of its absolute Sec. 75J GENERAL MOTION AND SPATIAL DISTRIBUTION 129 value equals the probability density, so that P = \p\ 2 , much as the energy density in a light beam is proportional to the squares of the electric and magnetic intensities. The wave equation for p, in tho case of a particle of mass m moving in a region in which its potential energy is U(x, y, z), as first proposed by Sohrfidinger in 1926, is h d<p _ A 2 / (Pp , <Pp , d*A 2 Vi m W i ' aj/ 2 + c )zy + Up = 0 (114) where i = \/— 1 and h is Planck’s constant or 6.62 X 10~ 27 in o.g.s. units. * We shall not attempt here a detailed mathematical treatment of this equation but shall only describe some of tho properties of its solutions. The behavior of P is closely similar to the behavior of ordinary waves in a dispersive medium, for simple harmonic waves of \p having different wave lengths travel at different speeds even in free space where U — 0. As a result, if we attempt to localize a molecule closely by giving to p initial values that vanish outside of a small limited region, then the "wave packet" of values of p so formed rapidly spreads out, in consequence of the varying speeds of the various harmonic wave-trains into which p can be resolved, just as an initially concentrated disturbance on a water surface spreads out for the same reason. Such a concentrated distribution of \p represents a molecule that is for the moment definitely localized in position but has no very definite velocity; for if wo determine its position after a considerable lapse of timo there is obviously a wide range of locations in which we may find it. The only way to prevent such a spreading of the packet is to start with a very large patch of waves which are very nearly harmonic or sinusoidal in form, e.g., P = M V, where v is the frequency and X the wave length, and the coefficient f(x } y> z) is almost constant over a largish region and sinks to zero outside of it. It follows from the theory that such a wave group, like a similar group of waves on water, will keep together for a com- paratively long time, moving as a unit with a fairly definite velocity v; this velocity and the corresponding momentum p are related to the * Birgo [Phys, Rcv. t 49, 204 (1930)] gives h/e - 1,37588 X 10“r j , which with e = 4,805 X 10 -10 (Sec, 16) makes h =* 0,018 X 1CT 97 , Insertion of this value of e and tho derived value of in in the Rydborg constant gives h » 0.032 X IQ"* 27 ; the discrepancy is not yet understood. 130 KINETIC THEORY OE GASES [Chap HI wave length X by the do Bioglic equation mv\ — p\ = (US) A packet of this type thus icpiesents tho contrary case of a molooulo moving with a fairly definite velocity but with a gioal lndofimtonoas of position* coirespondmg to the wide legion thioughout which ^ and the probability density P =* |^| 2 = I / I 2 have appieoiablo values 76. The Indetermmation Principle. Intcimcdiate sizes of wave packets correspond to intermediate degrees of indofinitoncRH of position and of velocity The general pnnciple involved hoic is Iloifoonborg'rt indetermination (or uncertainty) pnnciplo, which can bo stated for the case m hand as follows Let A** A* denote the loot-moan-sqiuuo expectations of vanation of a cooidinatc x and of tho coi responding component of momentum £ from their mean expectations of valuo, that is A* - [ J'l (v - 3)W dxf, l = 1 dx, with a corresponding definition for A* Then 4.A, £ A. (Ufl) Let us see what is lequued by this pnnciplo as applied to some actual molecules Suppose for a nitrogen moloculo of mass m » 4 65 X 10" 23 g, we allow A* to be 3 X 10“® cm oi about onc-llurtoonlh of the molecular diametei , then the least degree of indofimlcneas in tho ^component of velocity that we can have is A y M A f/m - h/(4wm A*) = 3780 cm/sec or roughly one-twelfth of the mean speed of a mtiogon moloculo at 15°C For a hydrogen molecule of mass 3.35 X 10~ 24 g, if wc make A„ = 5 4 X 10~° cm oi a fifth of its diameter, wo havo A v = 2 9 X 10 4 cm/soc or a sixth of the mean speed at 15°C. Decreasing A* incionsofl A v in the same ratio, and vice versa Thus upon tho introduction of quantum refinements the classical pictuio becomes definitely bliured for nitrogen and rather badly so foi hydiogen In tlio case of a fioo electron, with mass 9 12 X 10~ 28 g and mean speed 1 05 X 10 7 cm/soc at 15°0, even if we take A* = 5 X 1CT 8 cm, which exceeds the diam- eter of most molecules, A„ comes out at least equal to 1 16 X 10 7 cm/sco or larger than the mean speed itself, thus the classical picfciuo fails Sue. 77] GENERAL MOTION AND SPATIAL DISTRIBUTION 131 completely for the collision of an electron with a gas molecule when moving at ordinary thermal speeds or with an energy of a few hun- dredths of a volt. At lower temperatures the situation is still worse, but at high temperatures, or, in other words, at higher velocities it becomes better. These considerations make it clear that treatments of collisions in a gas by means of classical theory can possess a high degroo of validity only for very heavy molecules or at very high temporatures. On the other hand, it is important to note that the general principles of the conservation of momentum and of energy still hold exactly in wave mechanics in so far as tho momentum or the energy possesses a definite value under the circumstances of any particular case. 77. Wave Mechanics and Molecular Collisions. Tho method of treating a collision between two molecules in wave mechanics runs as follows. It turns out that tho problem can bo reduced, just as in classical theory, to a problem in tho motion of ono molecule relative to tho other, tho motion of thoir common center of mass being treated separately. A molecule approaching another with a definite relative velocity v is then represented by an infinite train of plane waves of $ having a definite wave length, This use of monochromatic wave trains cor- responds exactly to tho use of infinito sinusoidal wave- trains in optics, in treating such problems as the dispersion of light by a prism. In such a train of waves ji/'l 2 is uniform, so that equal probabilities aro assigned to all positions of tho molecule. This fact obviously cor- responds to tho assumption mado concerning tho beam of molecules that was employed in defining Q in See. 08; in fact, tho train of waves is usually for convenience regarded as representing such a beam rather than a singlo molecule, just as an infinite train of monochromatic light waves is commonly regarded as representing a continued flow of radiant onergy rather than a single photon. These incident waves aro then found bo bo partially scattered by tho second molecule in all directions, mathematically because of tho term Uf in tho wave equation (114); and the intensity of tho waves scattered in any given direction, as compared with tho intensity of tho incident waves, gives tho number of scattered molecules crossing unit area in that direction, as compared witli the number crossing unit area in tho incident beam, and so leads to a knowledge of 0. Without carrying out any calculations, many of the qualitative features of the scattering process can lie inferred immediately from these facts by the samo kind of reasoning about waves that succeeds so well in optics. If tho molecular wave length is small compared with 132 Kinetic theory or gases [Cha'p 111 distances within which the scattering potential V vanes appieciably, then it can be shown that the seatteung follows appi oxnnately tho classical laws. Thus the classical foim of mechanics coi responds to geometneal optics, the classical paths being the analogue of the lays in tho optical case When, howevci, tho wave length exceeds the limit mentioned, the process called diffraction begins to play an appicciablo 10 I 0 , just as m the optical case; ancl finally for sufficiently long waves thoie is little of the classical pictmo loft, just as the laws of geometrical optics fail completely foi veiy long waves ol light Since according to (115) the wave length goes down as the momentum mv mci oases, heavy pai tides behave moie nearly classically at a given speed than do light ones, and tho bolmvioi of any paiticle appioximatcs to the classical typo when its speed is made gioat enough. The wave length associated with a molecule moving at a definite speed thus plays a decisive 10 I 0 in collision processes Such wave Lengths for a number of common molecules moving at then moan speed when in a gas at 15°C are given in the table in See 86 below They range fiom 0 1 to 1 2 in units of 10 -8 cm Tho gcneial formula for molecules of molecular weight M moving with tho mean speed i) pioper to a gas at absolute tompeiatuio T is X h 2 74 X 10" 7 Mm oti s/MT ° m where mo — 1 661 X 10 -24 g or the mass of a molecule with M — 1 [cf eq (23)] and h — 6 62 X 10 -57 ; tho numerical foimula is obtained from (66a) m Sec 30 using R,\ r = 83 15 X 10° as in (20a). 78. Wave -mechanical Scattering Coefficients. Not veiy many scatteiing coefficients have as yet been calculated by wave mechanics, principally because tho molecular fields aie not sufficiently well known Some very mteiostmg results havo, howovor, boon published recently by Massey and Mohi * Working with assumed laws of foico they found results which in pait differed gioally from classical values, even m tho case of hard elastic sphcies; this scorned somewhat Buiprising at tho time but rtught leally havo boon anticipated fiom the optical analogy A liaid sphere scattering a piano beam of molecules corresponds to a spheio reflecting peifectly fiom its suifaco a piano beam of light Now it has been known for a hundred years that there is in tho contoi of the geometrical shadow of such a spheio a bright, spot foi mod by waves which all meet thoie in phase afloi being diffracted around the edge, Tho outer boundary of this bright spot * Massey and Moiih, Roy Sac Proc , 141, 434 (1033), 144, 188 (1034) Sbo. 78 ] GENERAL MOTION AND SPATIAL DISTRIBUTION 133 occurs at an angle corresponding roughly to a retardation of ono wave length between the two sides of the sphere or, approximately, at an angular distance from the center of the shadow 0 o = X/<r in terms of the wave length X and the diameter of the sphere <r. As the distance from the sphere is increased, 0„ remains fixed and the bright spot, therefore, spreads out, until finally it becomes much larger than the gcomotrical shadow itself and tho latter is practically obliterated. This phenomenon appears clearly in tho results of Massey and Mohr. Figure 34 shows at A their value of G for a sphere impinging upon a dissimilar fixed sphere at a speed corresponding to a wave length X = <ria/3, for which 0 C - X/V 12 * H “ 19°, C 12 being the mean of tho diameters of the two spheres. For 0 > 0„, G approximates to tho classical value, G = j^cr^ [cf. (110c)], but it exhibits oscillations as 0 increases, owing to intorforenco effects; for 0„ < X/<ris, on tho other hand, wo observe a rapid rise which continues until at 0 « 0 Q roaches a maximum almost COO times tho classical value. Tho total mutual collision cross section, S = 2 G sin 0 ilO, is found to vary with X, but it is never as small as classical theory makes it; for X «= 0 it is twice tho classical value or 2 mrft, and then it rises to 2.0 irtr^ at X = | tran and finally to 4w& at X ~ co . These numbers refer, however, only to spheres dimmilar in nature. If they are exactly alike, as would be the ease for the molecules of a truly homogeneous gas, a curious lack of complete individuality makes itself felt and modifies the diffraction effects. Tho three values of 8 corresponding to those just cited become then 2 ir<r X 2 i 2.4 iro-,\, 8.0 and for the case X = *ru/3 the coefficient G follows curve B in Fig. 34. Experimental verification of oscillations such as tlioao shown by tho latter curve would bo extremely interesting, but it will also lie difficult to obtain because in averaging the theoretical curve over a maxwellian distribution these oscillations will bo effectively obliterated. 134 KINETIC THEORY OF OASES [Chap til The same foimnlas should hold, according to theoiy, oven for heavy masses such as billiaid balls, but only unclei conditions suffi- ciently extreme to make diffi action effects appreciable The doubled collision cross section, as compared with the classical value, foi 1 da- tively shoit waves cannot be mierpicled as meaning anything so astounding as that two billiaid balls aie able to deflect each other without touching; for the doubling anses fiom an excess of exticmoly minute deflections, and a ball definitely known to have missed the other one would, by the mdetcimmation pnneiple, noccssanly have a sufficient mdefiniteness in its dncction of motion to pi event us fiom saying whothoi it had undoigono a vciy minute deflection or not Massey and Molu also made calculations foi some foico fields that fall off rapidly but extend nominally to infinite distances, and showed that wave mechanics leads in all cases to a finite value for the cioss section The analogous optical phenomenon is that an infinite pane of glass, in which the rofiactivc index uses (01 falls off) continuously but moie and more slowly in cveiy dncction away from a certain point, must, accoiding to geomctucal optics, cast an infinite shadow at infinity, whereas physical optics shows that the effects of such deviations in the index will, at gicat distances from Iho pane, bo largely wiped out by diffraction, and tho total amount of daikonmg in the shadow will thoiofoie bo finite It seems doubtful, howcvei, whether either tins fimtonoss of tho cioss section in general m the doubling of the cioss section foi sphcics possesses any leal significance for kinetic theory because, as has been said, they rcpicsent effects of very small deflections Tho applications that Massey and Mohr mado of these icsults to the theory of viscosity will bo discussed in connection with that topic* CHAPTER, IV VISCOSITY, THERMAL CONDUCTION, DIFFUSION In tho preceding chapters wo have dealt. almost exclusively with a gas which from Um macroscopic viewpoint in in ooinploto equilibrium. In tho present chapter wo shall now take up some of tho principal phenomena exhibited by Rases under oireu instances such that, while they may perhaps bo in a steady state, yet they are not in equilibrium in tho strict sense of tho term. Tho topics of gaseous viscosity, the conduction of heal, and diffusion will bo taken up in ordor, Tho meth- ods of handling those three phenomena arc so similar that they ai’o most conveniently discussed as a group; thoy arc often referred to under the name of transport phenomena. Tho conduction of electricity through gases is another very similar topic, hut it involves so many novel features that it is best reserved for a special chapter. Throughout tho discussion wo have kept a double goal in view. On the one hand, wo endeavor to derive known properties of gases from simple and broad theoretical assumptions, and such must always he the primary goal in the development of any theory. On the other hand, the comparison of the results of theory with experimental data has also yielded much information concerning molecular magnitudes. Tho combination of those two viewpoints is especially characteristic of kinetic theory and we shall encounter many more examples of it. During the past century the theory was on trial, and ovory new expla- nation of a gaseous property constituted a fresh triumph and a wel- come addition to the evidence for its truth. During tlm present century, however, such a wealth of direct evidence has been secured in favor of the basic, assumptions that the theory is now universally regarded us well established and the emphasis in research has definitely shifted to the problem of discovering the properties of tho molecules. It is still useful, nevertheless, to consider in what way tho general properties of gases arise as consequences of the properties and motions of tho molecules, and it will always ho worth while, us fresh data accumulate, to make sure that no contradiction develops anywhere between the theoretical conclusions and the experimental facts such as might forces a radical revision of our fundamental ideas, 135 136 KINETIC THEORY OP OASES [Chap IV A. VISCOSITY 79, Viscosity. The phenomenon of viscosity occurs m a fluid when it is undergoing sheanng motion To stait with the simplest case possible, suppose the gas is m mass motion with a velocity everywhere the same in diiection but vaiying in magnitude fiom point to point, and let this spatial variation of the magnitude be most lapicl m a coitam ducction poiponcliculai to that of the velocity itself, while ovei any plane pcipcndiculai to this direction of most lapid variation the velocity is constant The maximum late of variation is then called the velocity gradient Under these conditions it is found experimentally that the stiess which acts in the gas acioss any plane perpendicular, to tho direction of the ve- locity giachent is not of the natuie of a simple pressure noimal to the plane but contains also a tangential 01 shearing com- ponent, whose dnoction is always such as to tend to equalize tho velocities at diffcienl points, and when tho effect is small, as it is m all gases and m mobile liquids at not too high velocities, the shearing component of the stiess is proportional to the velocity Fio 36 — Illustrating visoosity g ra client, To obtain a mathematical foi imitation of this idea, let us take the a-axis in the diiection of the assumed velocity gradient and tho y - axis paiallel to the diiection of tho velocity itself (cf Pig 35). Then ll we draw in tho fluid a plane suifaco pci pondicular to tho u-nxis ancl therefore parallel to the volocity, the fluid lying on each side of this surface oxoHs a shearing force acting in a dnoction parallel to tho y-axis upon tho fluid lying on tho other side of tho suifaco, If wc denote the (mass) velocity by vo> the volocity gradient is dvo v /dx ; and if wc then denote by P xv the shearing component of force in tho y-diiec- tion which the medium on the side of tho suifaco toward excits upon each unit aiea of that on tho side toward — x } this foico boing called positive when it acts toward +y 7 wc can wnte (117) The factor of proportionality tj in this equation is called the coefficient of viscosity of the fluid or, for short, its \iscosity :i37 Smc. 80] VISCOSITY, THERMAL CONDUCTION, DIFFUSION At the same timo, of course, by Newton's third law the fluid lying beyond the piano toward —x exerts a force — P xu per unit area on the fluid lying on tho side toward -ha.* The problem for kinetic theory is then to infer tho value of the coefficient y and tho nature of its properties from the assumed or known fundamental properties of the molecules, Before taking up this subject, however, it may bo worth while to interject a short dis- cussion of fluid stresses in general; the uninterested reader can easily omit this and pass at once to tho following section. 80, Fluid Stresses in General. If a small plane bo drawn any- where in a medium which for tho purpose in hand can be treated as continuous, and if one side of this plane bo labeled positive and tho other negative, then tho medium lying on the positive side of tho plane will bo exerting a certain vector force upon tho medium lying on tho negative sido; the amount of this force per unit of area is called the traction acrosB the plane. At tho same timo, of course, tho medium on the negative side exerts an exactly equal but opposite vector forco upon that on tho positive sido. The traction can in all oases bo resolved into ono component acting perpendicular to tho piano and a second "tangential” component acting in a direction parallel to it. In general, both components vary as tho orientation of the piano is altered; but it can bo shown that if tho values of the traction are known for any three mutually perpendicular positions of the piano the traction can l)o expressed in terms of those three values when tho piano has any other orientation. Now it is characteristic of a fluid, ns opposed to a solid, that when it is at rest the traction across any plane in it is wholly normal to tho piano and is thus of tho imturo of a pressure, either positive or nega- tive; and furthermore, that this pressure at a given point in the fluid is indopondont of tho orientation of tho surface across which it acta, which is commonly expressed by .saying that the pressure is equal in all directions. On tho other hand, when tho fluid is moving, the stresses become altered in consequence of tho relative motion of its parts. Even the normal component of the traction now varieH, in general, as the test * lb is not generally romurkocl in trout meg on lcinotio theory that tangential stress forces of oqual inagniturlo must likewise act in tho *-dirootUm across surfaces drawn perpendicular to the v-axis; if lhoy did not, a oubo of fluid with faces per- pendicular to these two rtirootioiiB would obviously bo sot into rotation. The interested roador will find it an excellent oxoromo to conslmet tho theory of thoso stresses, in parallel with tho treatment of tho others that is hero given. 138 KINETIC THEORY OF GASES [Chaf IV plane is rotated; so that the pressuie is no longei equal in all directions, although, of course, it is always the same in each of two diametrically opposite directions An analysis of the most geneial type of con- tinuous motion shows, however, that the motion can be lesolved, in the neighboihood of any point, into three shearing motions m tlnee perpendicular planes, plus a motion of compression 01 dilatation occuiring at an equal late in all directions. The diffeienccs in pres- sure icfened to above and the tangential stresses then anse as the sum of the three tangential shewing sti esses, each related to one of the shearing components of the motion in teims of the viscosity ij as described above, togetlioi with a positive 01 negative component of pressure propoitional to the rate of compression or dilatation The constant of propoitionality for this latter pait of the stress constitutes a second frictional constant charactemtic of the moving fluid. Noth- ing is known experimentally, however, in legal d to its value, and theory indicates that in a gas it should bo small if not actually zeio; accordingly we shall give it no further consideration 81, Simple Theory of Viscosity. The physical explanation of vis- cosity in a gas becomes obvious at once if, lotuining to the case y described in Sec. 79 we consider the vector . momentum carried by the molecules across a i 0 macroscopic ally small piano suiface S drawn v o1 | s I perpendicular to the s-axis, l o , to the dnection — 2L of the velocity giadient (cf Fig 36) In the ^ je' first chapter, when wo made oui calculation of dr " ~~X~ the pressuio we considered only the component of the momentum that is normal to jS, now wo EHa turn our attention to the component tangential J " to S. Wo shall suppose, as usual, that the molecules influence each other only when extremely closo together, so that we can neglect, for the piesont, tlioso short-lived situations in which there is mutual force-action between two molecules situated on opposite sides of the plane. Suppose, for definiteness, that dvo/dx is positivo Then molecules crossing S from lef t to right oomo from i ogions where the mass velocity »o is less than it is in the region into which they go and so tend to arrive in their new positions with less than the piopor amount of y-momontum, whereas at the same time thoso crossing to tho left come from regions where v 0 is greater and so tend to carry out an excessive share of such momentum, In this way the gas lying to the right of 5 tends on the whole to suffer a loss of {/-momentum, and this is equivalent to the action of a foice on it diiectcd lowaid — y, whilo Hko. 81) VISCOSITY, THERMAL CONDUCTION, DIFFUSION 139 tho gas on tho loft tends similarly to gain momentum and so experi- ences a forco toward +2/. In order to dovolop this idea quantitatively we need to know how much momentum is carried by the molecules. Now the velocity of a molecule is determined as a result of its last collision. We are led therefore to consider the distribution of velocities among those molecules which collide in a given clement of volume dr, A natural assumption to make, and tho one that was universally made in the early days of tho theory, is that these molecules have an average velocity after their collisions in dr equal to tho mass velocity of the gas at that point. If a: denotes the distance of dr from S and v«» the value of tho mass velocity at S, its value at dr can be written v<, a + xv' 0l whore »o stands for the value of dvt/dx at S, provided we may suppose Uo to vary only inappreciably over a distance comparable with the molecular moan free path. Let us suppose that the molecules all ha vo mass in, Then those that 'collido in dr carry away from it, according to tho assumption just made, an average y-momentum of amount -\- xyj). Tho total momentum carried toward +x across unit area of 8 per second will thoreforo be obtained if wo take tho average of this expression for all molecules that cross toward +*, and multiply this average by tho number of molecules that cross in this direction, which is given by (72a) in Soc. 37. The resulting expression for tho momentum is }■ + OTo) = 1 nfmi(vn, + &>' 0 ). Hero n is tho number of molecules in unit volume and 5 their mean speed, and wo have boon able to wnto xvl = SrJ because uj, like Oo«j is tiie same for each molecule. Wo need now to find S. This quantity must obviously bo con- nected somehow with the mean freo path. Now after a molecule crosses S, its clmnco of collision in going any given distance is tho name as if its last collision had boon rnado at S; hence tho molecules must go, before colliding again, a moan distance beyond 8 equal to their ordinary mean free path L v > By symmetry, howover, they must also have como on tho average, before reaching S, an equal dis- tance from the point of their last collision. Thus for molecules that cross 8 in a direction inclined at a given angle 0 to the normal drawn toward tho quantity 3! is simply the component of U perpendicular to S, and for those molecules !E » -A, cos 0. By (716) in Sec. 37 tho number that cross S with given speed v and with 0 in a given range dO can 1)0 written w' sin 0 cos 0 do whore »' is a certain number mdc- KiNi-.nr tii mu v or u,\.srs ii’mc IV MO prlldcul of 0 Till' IV\(*l«Hi‘ nf J fnl all nutli'fllli ‘tlllll uo'n With ‘.JMiil 1 1 is thciefmc [rf (tlUfd in Sri* 120 1 X * f K,9 ( L, CHS (l)n' “III II CHS (I lilt . f * "V “ill II 1 1> I II >111 $ l , J 0 lid us fur the picsenl igumc tin* \iiiliHmn of / , with >|sed sod lopllice /,„ by ft, I ill' K •li**l til ini'Hli ilio path Tht m r Ink the name vului't i \>, fui nil "I ill'' iih'I' * iih 'I odinidv. lb" i\pt" Himi uliliuiicil jiisl ahme fur (lie uilc ul luui'(«i <»l u luoimiilmn !■> mulccilIcH ntmsuiK fiuiii left (u ll«ht bet huh ‘ In tlie Hiiiue way tine limls thal nmlei iilen i ro»»uiai toward tlm left carry {/•momentum out nf the leniou Iviiik mi the riplil uf N »i the min and mibImelinK tliia expieshion funn the prc\ iou*> one we have, liimllv. for I he nrt trnimfer of // lnuiiiciituiu hciom N toward I r, prr unit area per hccoiiiI, & iiwil7,ej. UlHi lly the llcl'mitlon of the CoelHcieiil nf xhicodlV. ' K M« «»j«r«Mid in t;q. (1 17), Una net. tianufer must al«m equnl the al«>«'ii*c nf n inimiH ni|$n in (117) la due to (lie fail that there /'*„ t»fer> in toiidi r toward - .r. II dice >1 ' £ 11)111*1, i > ^ pil,, ittlli for a liomogeiieoitri gim of density p Tim formula wan uhtnuod hv Maxwell in 180(1 Krom a formula mieh aa Ibis, in which p, p, mid f arc «H known, valued of Utn molecular magnitude I, can Ih* rnhutalcd Ihfnre proceeding to do Hum, however, wo imml first nixeditiale the loattnt todc of llio eiroiH aiiaiitK fiom the rather violent rnup)dir«tioi»« thnl lmv< been introduced in the course nf our deduction It will !*• found that boeauRO of tlniHc eirora the numerical factor 'a rcfpiir*" f otmidernble correction. VrabUm Develop the eorroMpouding theory for n two dine tmmml gaa, allowing that the number rricwing unit [eugth pwr wood h r*t g, and thal i ** ir LfA and i) J pi 1 1 , Sue. 82] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 141 82. The Mean Free Path across a Fixed Plane. Ail interesting objection to the reasoning just givon lias sometimes been raised on tho ground that, since tho mean froe path between two successive col- lisions is certainly L, the mean path from the Inst collision up to tho plane should be, not L as we inferred it to be, but only L/2, leaving an equal amount L/2 for tho mean path from the plane up to tho noxt collision. This reasoning rests, however, upon a tacit assumption that is not justified. Those freo paths that are executed in such a position that they intersect tho plane S constitute ft special group seine: ted out of all tho free paths executed in the gas, and it is not safe to assume without proof that their average length will bo tho same as the average length of all the free paths. As a matter of fact, a long free path stands a much better chaneo of happening to intersect the piano than does a short one (cf. Fig. 37, in which S'S” is tho trace of tho piano, and freo paths of two different lengths aro drawn). Wo can writo for the probability that a path of length l intersects a givon piano, cl, o being a factor of proportionality independent of l. Then, if dl paths are executed per S" Fia. 37. — Froo pntlitf nofti 1 (V piano. second with lengths in a range dl, cNl\p(l) dl among those will intersect. tho plane, and tho average length of all that intersect will bo ? fcNmVdl f 0 m ty(Q dl jcNlHl) dl ~~f 0 *lMl) dl Tho average length l of paths intersecting a plane thus depends in general upon tho form of their distribution function, ^(l). For molecules moving at a given speed v, according to (986) in See. 59 \j, a fl-Mi, and evaluating tho integrals in the formula just obtained wo find l « 2L V . Thus in a maxwellian gas the moan free path of those molecules that cross a piano at givon speed is just twico the mean free path for all molecules in the gas that aro moving at that speed. Tho ratio of 1 to L for all molecules regardless of their speed can be written down in tho form of an integral and comes out a little different from 2. In all cases, howovor, the moan distance to the plane, arid also tho mean distance traversed beyond it, are each just half of I itself. As a more concrete example illustrative of the same principle, sup- pose a large bundle of straws of varying lengths is tossed high in tho 142 KINimr Til hour Ob'UAHKH IC’iui* IV air and allowed In fall upon tin* Hour in laudoni diHlulmttoii Then llio average length tit llnwe nIihwk wlueh happen lo full u« »»».*»< n given nark in tho (lour will be gi eider limn llu* n\ »‘i »K 4 ‘ I' "Kill fur the wind*’ bundle; if the lengths me dMllbiiled exponentially, it will be pM twice an gieal. Pioblnnn i. In flic llliiHltalioii of the stmws ju >| given, bud (lie ratio of the nvcmge leiigllw of tho.e lying items* tin* i mek (u the average length of all of lliein, when the slinvvs have lengths di« tributed equally belween 0 and l u I o» * (/. 2. When a dm ia llnmvii, (he mean niuuber of (Iuouh u e , of at N of throwing) between sixes in, of cnuise, (I Show tlml, if mi oh>oivei looks in at iriegular inlervalH and watches each lime until a mx m thrown, he will watch on the aveinge for 11 (blow i, and if he ii«k*< each lima how many tluowa have occuiied since the bed m, he will learn that this nix lien on the aveinge II litmus I no k, the no rage *»f (he total numbers of llitowa helween sixes a« (lum observed hj him being 12, 83. Correction for tho Velocity Spread. One of the error < m the doduotum of formula (1 10) lien in (lie icwumpliou (hid (he menu free path is llio Haniii for ntulerulcs moving al all speeds I. el tn. mviMigtde tho magnitude of Hits ermr In lining an we shall «lill rrfain (he assumption concerning molecular veloeiliea, amplifjing it into tlm moro apecille assertion that those molecules whieii lm\e pud mllided in any cloment of volume ilr ptmaenH fm the inomeiil (he velonty dmlribution cliaraetemlin of a gas (Iml is iii equilibrium bid moving with llio muss velocity n 0 . Tho distribution in veloeity of these molecules (hid have ju«( collided ia caaily found Using (lllu) in See. 2H, we have «« the number of inoicculeH that collide in a second m dt wdh a sjieed of tliormal agitation lying in a given innge i/c' htfurr iollu.um ‘Iirfii dr)0„vi r'V n being tho total number of mnlrciilcs per unit voltime mid (»« the collision rata for molecules moving with this speed Hi tire we are iwuming oquihliriimi of velocities lo exist, mi etptal number of other moloouloH nuiHl in the same time ticrpiire tin* speed e' Uf (h«'we Intier, wlucii according to tho nsHuiuptinu j not. Minted have (hernial v rlorUm** equally distributed in direction, a fuudimi w1k °, >,a ,l ' fl will have (hdr 'i sr valocitieM lying in an element of solid angle mu Odttdv jef <2fh|, let tho axis of points bn taken llirough dr tiormnl to ,S* and let the leforrme piano from which y ia meiumird be taken parallel to the aj plane (of Fig 38), tho mass velocity being assumed a* Indore lo have the Sec. 83] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 143 direction of y with a gradient in the direction of x. Of those mole- cules again, by (98a), the fraction e -°" r/v ' will eventually cross S without having beon stopped by a collision, r being the distance along the path from dr to S, Finally, the {/-component of v' is v' sin 0 cos <p, * so that, if wo write again v a a + xvg for tho mass velocity at dr, v{, standing for dv 0 /dx and x for tho coordinate of dr with tho origin on S, tho total {/-momentum of a moleoulc is mv u = m(v<, s + xv' Q -|- v' sin 0 cos <p). ♦ Tho product of all theso quantities is nmAQ v 'v" 2 e~ ll,u ' > {v< l .‘} + xv' 0 + v' sin 0 cos ^)e _r0u,/ ”' sin 0 dO dip dv' dr; (120 a) and tho integral of this expression over 0 < <p < %r, 0 < 0 < r/2, 0 < o' < oo then represents tho total amount of {/-momentum carried across S per second toward +a by mole culcs whoso last collision occurred in dr. If wo then also integrate dr over a cylinder of unit cross section standing normally on S, wo lrnvo tho total transfer per second across unit of aroa of S) to be sure, somo molecules that actually originate within this cylinder will ultimately Icavo it and cross S outside of its base, but for every ono that does this another ono moving in tho same direction will originate outside the cylinder and, after entering it, cross through its base. To carry out tho specified integration over dr it suffices to replace dr by dx and thon intograto over the range — <» < x < 0, for tho result of integrating with respect to <p, 0 and v' is independent of y and z, and tho integration ovor tho dy and dz in dr merely introduces tho unit cross-sectional aroa of tho Via. 88. cylinder. Since r (-*) , cos 0 tho complete integral thus obtained can bo written in iterated form thus (v being substituted for v 1 in tho definite integral) : nmAf 0 *O v vW> d»f* /z sin 0 dof** dip /•0 J * fowr + at* + v (An 0 cos <p)o v 0 dx t (1206) * Roaolvo v first into v sin 0, v cos 0; thon only v ain 0 has ft y-oompononfc, and its magnitude* is v sin 0 gob <p< KIN line THEORY Ob' OASES (<‘u\r IV 144 Fur tin' momentum earned aeionh >S m the opposite dne< lion we Ihen obtain an expiessum dilTeniiK fium ( his only m Hint x i uu*. limn 0 In -l-w and the exponent of i is 0 „i/i> ens 0 bemuse in linn i »*>(« r - |-r com 0, Tlin dilTeienee between (lie Iwo expulsions I Inis oblained is Ihen I lie net mini of //-moment nin in I hr nas In (be ukIiI of *S‘, pei mill men of M pel nerond, and diwdhiK Ibis dilTeienee by i# we have I lie eoellieienl, of viseosily ij Now /’() 11 ** 1 (On* -I ill J tfl X m n f (.I'iii | V HIM 0 HIM yi)* mmply Ivy lopInoniK ? l>y .1 ; honor in Iho Militimlnm nf tin* \\sn oKpnwmns Huh pm l nf the mlogml i>\<m ^ in (I20M ohum*!* llm mi xonpoiidinR term in tin 1 nmmd On ( In* ntlmi liuud, r U JT xvh ' 1 "* 11 dx w k4 II t >M l* 0 fl j» a,',e ! 4*US a 0 u» ’ ho that in the wditmelion linn leun beeomeM mnlli]iiied by 2 introdueiiiK the furthei values, I d<p 2ir, •/II lienee, fj' J Hin 0 eoH a 0 dO i 4, wo find finally for 17, nflor dividing ml — oj, f * v* i) * J mm A r /Mfl tliK ,/o Here let ua rnpluan « by Ihe new \nnable of iuloKiatiuii x r.> //e runt let UK alno inaerl, A -> fiom (00) and nm p, the denmty m Kianm; the liwull to 1) M *1 /> I' » H <b bet uh then inlrodiiee fl <-i (2/vV)( l/fi) or l/ft • y/rP/2 from (tlflji) in Her. Ill), and for 0„ let um inseil, ila value from (Kiln) hi Her (W, inliodueniK in thin laUer expieHmon U * f '//, fiom (ilfil The n Mill in: » dr 1121 ) Sbc. 84] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 146 For tho definite integral occurring here Boltzmann found by a numeri- cal quadrature 0.838264/4 = 0.209506.* Inserting this value wo obtain tho formula found by Boltzmann in 1881, v = 0.350 pHL. (122) This differs from the result of tho simple theory as expressed in eq. (119) only in that tho numerical factor is 0.350 in place of 0,333. Thus with all this work wo have only changed the numerical coefficient by 5 per cent. We have hero another typical example of the smallness of the correction that is required by the maxwcllian distribution of velocities in those cases in which tho spread of velocities merely plays the role of a disturbing element in the situation. 84. Further Correction of the Viscosity Formula. Even jjo, the calculation still contains another error of a vory similar sort which wo have not yet mentioned. Wo have treated the collision rate G„< as a constant, whereas in reality it will vary during tho flight of a molecule because of the variation in the mass velocity of tho gas through which the moleculo is passing. The necessary calculation to allow for this effect is straightforward, but it is tedious and no details of it will bo given here, for a reason that will appear presently; the final result is to replace Boltzmann's formula (122) by V - 0.310 pdL. (123) This is the valuo of i? that follows rigorously from tho assumption that the velocity distribution of tho molecules that have just collided at any point is maxwcllian. This assumption itself is open to grave question, howover, Tho molecules that assemble momentarily at a given point have come from regions of tho gas having different mass velocities and so must exhibit some departure from a maxwcllian dis- tribution. Our own calculation indicates, in fact, that molecules which’ havo come from a region of higher mass velocity have an average excess of velocity in that direction, and after they collide with others it is quite thinkable that they might retain somo of this excess so that tho transfer of j/-momontum would bo greater than wo have found it to bo. A completely accurate theory of viscosity could undoubtedly bo developed by introducing further corrections for effects of this sort. Tho most satisfactory theory lias actually been achieved, however, by viewing tho wholo molecular process from quite a different standpoint. Accordingly, wo shall now abandon the lino of attack that wo havo been following and make a fresh start. * Cf. L. Boltzmann, "GnBtlicorio,” vol. I, p. 78. MO KINETIC TIIEnur Ob' (USES |l mr IV 86. New View of the Molecular Process. If we look hi the mo|< i - nhvr processes from a cerium angle, we hi mi* nt the mhn .img mu elusion that tlio force of visenNity iieluully Inin its entile union id n certain type of depnitme fiom (lie nmvwelhiui dMuhulion •<( v. 1... >n< < For, if x i« any molerubii magnitude vvlmtevei, tin mle ol timmfi i of x arrows unit aica of n plune w eleaily il'lV ill'll wlieie ii j is till' coinponenl of (lie moleeuhu velueitv popi into nlar to tile plant', taken positive m the diieehoit eho'en ns lletl «»1 pMlive not transitu ; I, ho summation extends over all mulct ule«» m uml volume 111 tlio iuimeihule neighborhood ol Hie plane. Negative udunof e, Horvu auloinalically in the mint In subtract (lie not of \ (Iml in earned hackwaul. (C!f., e,g,, our hint ileiliietioit of »j m ftr Hi, or tlio calculation of the picsmun in See. fi.) In (1211 vve have h gi uernl expression for Ihti rate of tnuinfer of any limit eiilar iimgtitlude To obtain tlio viscosity, we have then only In *nihstitule hi tI2H, in our pluvious notation, \* * mv v ami i't e„, if we abu mippome tie* velocity Kiatlicnl to lie unity, the mte of tiaupffcr of lummiilmo thna obtained is numouttally et|iml to the coelliuenl of \im*ily »j We IhtiH obtain »t <1251 Thin Hum would obviously vanish owing In the symmetry «,f /i y| if tlio distribution wem exactly nmxwelli/nil The vmcomly can be calculated al oiiee fiom (IgAi if we tun hint out what the distribution of velocities iielually m lit n dc tiring g,t?i Now thin distribution must be a eeilaiu lype of sternly tmlutmu of the BolUimmn dilTerenlial equal ion, as lepteweiiteil by eq <Htl »l«ae Accordingly, ilollmuaiin himself attouipled to *u|\e Huh cijunltoit for the case of a Hliearing gas; and a feasible method of d«.ing ihp by moans of successive uppioximuthms Ima tlnburnteil bv I mkug * A Homowhat cllfforent method of appnmeh, Imlmted by rinu-»nin nml Maxwell, seems, however, to be corner fu follow ami hn« l»c< i< enrrir.1 through by Chapman. For oblainiiig urnirale reiutlN th«-«* methods arc ho far superior to the free-path attack that the latter might well be relegated to the status of a historical tutimiiy, were it not that it throws a peculiarly vivid light upon the complexity of mutee ular phenomena in a gas; furthermore, simple < ideulntime. hke our ♦Knhkoo, “Kiiwtmrhn Tlicurlo tier Vcrgfuiitn la itiRMilg vonluimlen Piiworlnlitin, U|tMuln, Hit 7. Sec. 80] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 147 first one often serve a good purpose because they indicate quickly the order of an effect and the quantities upon which it depends. In our treatment of heat conduction wo shall actually employ an approximate form of the Maxwell-Chapman method. The necessary calculations are rather lengthy, however, and the features of the method are less well brought out in the case of viscosity; for theso reasons the application of this method to viscosity will be loft as a problem (Sec. 101) for the student and only the final result will be stated hero. 80. Final Viscosity Formula. Magnitudes of L and <J, The approximate form of the now method, which is equivalent to that employed in Chapman’s first paper, when applied to tho treatment of viscosity yields for the numerical factor in (123) above 0.491 in place of 0.310 [cf. (1566) below], Tho rigorous development of the same method by means of successive approximations, as worked out by Chapman* gives, in placo of (123), y = 0.491(1 + c)j>vL VC) L„ c = (126a) hero iSvc is a sort of mean equivalent cross section for viscosity and heal conduction , which is given in terms of tho scattering coefficient by eq. (163) in See. 100 below, and e is a numbor which is probably very small for any actual molecular field. For a repulsive force « 1/r", e = 0 when n — 5 and rises ns n in ore ns on only to 0.016 for n -■ w ; this last case corresponds to hard elastie spheres, for which accordingly we can write i) <=■ 0.499 pvL, (1206) since for such spheres S v0 = irtr a and L va then becomes tho ordinary mean free path L. This latter formula was found also by Knulcog.t As usual, p is tho density in grams per cubic centimeter and 0 is tho mean molecular speed; and tho results are accurate only for an indefi- nitely rare gas. The reason for such a largo increase in tho numerical coefficient, from 0.310 to 0.499, lies in tho persistence of velocities which was mentioned above. Molecules exhibit a certain average tendency to continuo moving in their original direction after a collision; in tho usual case of elastic spheres tho averago component in tho original direction after collision can be shown to amount to some 40 per cent * Chapman, Phil. Trims., 211A, 433 (1012); 210A, 270 (1010); 217A, 116 (1018). t Enskog. Inc, cil, 148 KINETIC THEORY OF CASES tCHAP IV of the velocity before collision, In oiu analysis above we assumed that those molecules which collide in a given element of volume dr and then move off toward the right, for example, had a ti an s verse or y - component of velocity equal to the avciagc ^/-component foi all molecules in dr, but m leality these paiticulai molecules originally enteied dr predominantly from the left and so amved theie with arc average tiansveise velocity chaiactenstic of a region lying to the left, and pait of this diffeience as compared with the geneial aveiago in dr persists after collision and is earned along fai fchei by the molecules as they move off towaid the right The effect is clear ly the same as if these molecules had collided somewheie to the left of dr, and so amounts to a vntual mciease in L, with a resulting increase in the transfer of momentum Equation (1266) is probably to be legal ded as the correct foimula for a rarefied homogeneous gas of clastic spheiical molecules Since everything m the equation is alieady known except L, we can employ the equation to calculate values of the mean fiee path and thou from these by means of (lOGd), L ~ 1 wo can calculate values of the molecular diameter cr We havo hole probably the most reliable source of information in regard to those quantities that can be obtained from the oidmaiy piopeities of gases Values of L and <r so calculated for a number of gases are given in tho table on p, 149 They can be regarded, of com so, only as equiva- lent elashc-sphere mean fiee paths and diameteis, since no mblecules really are hard spheies, but they are vciy Useful for many soils of approximate calculations If the exact law of molecular foico were known it might be pieferablc to calculate the cioss section from (126a) and then to calculate from it an equivalent diamotei <Tva - (SvoM*, but the value of the “diameter” so found would probably differ only moderately from the elastic-sphere value The temperature has been chosen as 15°C lather than the moie eustomaiy 0°O bocauso the former lies closei to common laboratory tempoiatuies; tho only exception is Hg, for which rj f L, and <r are given at 219 4°C The power of the temperature to which is approximately proportional near 15°C is given as n\ thus rj T n , L oc at constant density, and < t « I n a u cases L refers to atmosphono pressure, being arbitrarily reduced to this pressure in the case of H 2 0 and Hg The sources from which the data for rj weie taken arc listed below the tablo. 87. Variation of Viscosity with Density, Our formulas predict several interesting general properties that the viscosity should have. Sec. 87] VISCOSITY , THERMAL CONDUCTION DIFFUSION 149 Some Molecular Data M V L O X Unit (10~ 7 dyne sec/ cm) (10 _(} cm) (10-« cm) (io-8 om) H, 2.01G 871 (1)(2)(1C) 11.77 2.74, 0.09 1.136 Ill (heavy hyd.) 4.027 871 (17) 11.77 ■ 2.74 0.69(?) 0.8040 Helium 4.002 1043 (3) 18.62 2.18 0.64 0.8065 CI-I 4 (methane) 10.03 1077 (1) 5,10 4.14 0.88 0.4030 ni-i 3 17.03 070 (4) 4.51 4.43 1.09 0.3010 HaO 18.02 920*(14) 4.18 4.00 (1.07) 0.3801 Neon 20.18 3095 (3)(12) 13.22 2.69 0.67 0.3502 N, 28.02 1734 (3) (5) 0.28 3.76 0.77 CaH,, (ethylene) 28.03 998 (4) 3,01 C 2 II 0 (otliano).. 30,05 900 (1) 3.15 O s 32.00 2003 (1)(0) 6.70 3.61 0.81 0.2852 HOI 30.40 1397 (16) 4.44 •4.46 Rffiyn 0.2672 Argon 39.94 2190 (3) (7) 0.60 ■ 0.2563 co 8 44.00 1448 (3)(8)(10) 4.10 ■ Krypton CIIjBr (methyl 82.9 2431 (9) 5.12 (.85) 0.1772 bromide) .... 04.94 1310 (10) 2.68 5.86 0.1G6G Xenon 130,2 2230 (18) 3.76 4.86 0.02 Hg.. ! 200.0 4700 (13) (21D.4°C) • 8.32 (219.4°C) 4 , 20 . (219, 4°C) 0.1139 Air Electron 28.90 5.400 xio-< 1790 (11) 6.40 3.72 0.79 0.2908 08.80 * Calculated from a monfturomonl on antiirntod vapor. M ® molecular weight, rj = viscosity, L = mean free path at fttmoephorio pressure, o- = olftstic-sphoro oquivalont diameter of tho molcoulo, X 1=3 molecular wave length at tho mean speed tf; n is the exponent in: rj « T n f L oc at con- stant density or L « at constant prosauro, a- « near 16*0. The tomporaturo is 15 D C except for ij, L t a- for Hg, Sources of material: (1) Trauta Mid Sorg, Ann. PKynik, 10, 81 (1981)} S) Traiit* nnd Stau rt r i5?Vf., % 787 (1020)} (8) Trau 1 7 , nnd 55ink, VWeh, 7, 427 (1080); (4) Tran l z nnd Hoborling, ibid., 10, 1GB (1031); (6) Tnuit* nnd Baumann, ibid., 2, 733 (1020)5 (0) TrnuU and Mclator, ibid., 7, 400 (1030); (7) Trnut* and Binkolo, ibid., 5, 501 (1030); (8) Trail 1 8 and Kurz, M., 0, 081 (1081); (0) Naalnt and ltcmfll, Omx. chim . ital, 68, 433 (1028); (10) Tltnnl, Bull, Chm. 8oo> Japan, B, 08 (1030); (11) Kollstrttm, Vafur*, 136, 082 (1035); (12) Edwards, Hoy. Soo . Proc ., 110, 578, 1028; (13) Brniino, BahoIi ami WonUol, Zeits. phytt. Chain., (A) 137, 447 (1928); (14) Brauno and Linlco, ibid, (A) 148, 105 (1030); (15) Jung and Sohmiok, ibid, ( li ) 7, 130 (1030); (10) SulJierlaml and Manas, Cmiarf. Jour, lies., 6, 428 (1032); (17) Van Cleave nnd Manas, ibid, 12, 57 (1035); (18) Tran (z nmllloborHng, Ann* Phytik, 20, 118 (1034). Suppose we keep the temperature constant. Then fJ remains con- stant, and so does pL, according to our conclusions in Sec. 68. Hence our formulas indicate that, for a rarefied gas composed of molecules interacting only when very close together, the viscosity should be 150 KINETIC THEORY OF OASES [Chap IV independent of the density This lesult, contrasting stiongly with our natuial expectation that less gas ought to mean less viscous cliag, was deduced fiom theoiy by Maxwell in 1860 and was found actually to be confiunecl m some caieful cxpenmcnls that he peifoimecl upon an In its day this constituted a paiticulaily sinking success for kinetic theoiy Latei woilc indicates that this law holds foi tho viscosity of all gases which aie sufficiently laio, but it fails, natuially, at high densities, f 01 example, the viscosity ol CO2al40°Cis 1 57 X 10” 1 c g,s units at 1 atmosplioic and uses only to 1 69 X 10~ 4 at 23 8 atmos- pheies, but theieaftei it mci eases moio and moie lapidly lip to 4 83 X 10 “* 4 at 100 atmosphci es, being not f ai fiom pi opoi tional to pat this latter pressuie, Theoiotical tieatmcnts of tho vaiiation at consideiablo densities clo not yet exist As the density is made vanishingly low, howover, the effects of viscosity must eventually decrease, and it is easy to see under wlmt conditions this should occur When tho mean fiec path finally becomes comparable with the spatial dimensions of the phenomenon undei obsoivation, the basis of oui calculation fails and dopaituioH fiom oui formulas may be expected to set in When tho density is made so extiemoly low that mtoi molecular collisions uio actually infrequent, the, effects of the individual molecules must be simply additive, and the viscosity should then bo dncclly pioporlional to tho density; we shall letiun to this subject in a special chapter on low- pressuie phenomena (Chap VIII) and shall find that tho thooiolionl piediction is confhmcd 88. Variation of Viscosity with Temperature. Suppose now, on the othei hand, wo keep tho density constant and vary tho tcmpcin- tuie, Under these conditions, as we have soon in Soo, 58, the moan free path L should be a constant, piovidecl tho gas is raic and composed of molecules that interact only when close together Tho moan speed v> however, is pi opoi tional to the squaio root of tho absolute tom- peratuie T* Hence, in such a gas, according to our approximate formulas (119) oi (123), tho coefficient of viscosity should bo pio- poi tional to T^\ and accoidmg to (1266) the fully coirecloci foimula leads to the same conclusion at least for haul clastic spheics. Now it is a fact that the viscosity of gases does use in all eason with a rise of tcmpeiatxuc; and this qualitative confirmation of tho theory is paiticulaily mte testing because the viscosity of liquids ih observed to change m exactly the opposite direction, and a decrease is what we might naturally expect as a result of tho increased mobility of the molecules at higher tompoiatiucs The incicase is in all gasc# moie lapid than tlic square root of T i howoveu In Fig, 39 arc plotted 151 Sec. 88] VISCOSITY, THERMAL CONDUCTION , DIFFUSION values of rj against T for six common gases, and in Fig. 40 log ij is plotted against log T for the same gases. If it were true that rj « TY\ Fio. 40. — Log tj vs. log T, (In tho ourvo marked (IIj) log [i)(l + 14 /T)\ Is plotted against log the latter curves would be straight lines with a slope of but in reality tho curves everywhere slope upward more steeply than this* 152 KINETIC THEORY OF OASES [Chap IV It is not suipiising, howevoi, that a theoiy based on hard-sphcneal molecules should pi edict the light variation of viscosity with density but not with tempeiature, foi lowering the density does not change the chaiactei of the collisions but only lengthens the intervening fico paths, whereas laisnig the tempeiatuie inci eases the violence of tho impacts and so may easily alter their effect upon the moleculai motions A moie lapid mciease in ^ with using tempeiatuie than that indicated by the hai d-spheie theoiy is in fact just what we should expect if tho molecules weie in leality somewhat soft, tho lepulsive foico m a collision developing at a lapid but finite late as two of them appioach each other, foi then as the velocities become gicatei, a molecule appioachmg another along a given path will bo less doflootcd upon striking The net lesult of a rise in tempeiatuie must Ihciofoio bo an lnciease in the persistence of velocities and a losultmg increaso in the numeiical factoi that occuis m the fozmula foi tho viscosity 89. Viscosity and Temperature with an Inverse-power Force. Further quantitative piogicss toward a theory of tho vaiiation with tempeiatuie can be made only on the basis of a now assumption concerning the moleculai fields. In tho absonce of definite knowledge a favonto assumption has been that the force action between two molecules is piopoitional to some inverse powei of their distance f apai fc, say to i ~ e It happens that in this case tho vaiiation of viscosity with tempeiatuie can bo found by tho method of similitude (or by the ncaily equivalent but moie abstiact* method of dimensions, m teims of which tho lesult was fiist obtained by Loid Rayleigh in 1000). It is possible to do this because wc know alicady that the viscosity is a function of the tompeiaturo alone To apply the method of similitude, consider a particular motion that is executed by the molecules, and imagmo this motion to bo modi- fied in such a way that the moleculai paths ictain their shape but have their lineal dimensions all changed m the same latio X, and sup- pose that the times taken to Iraveise eoncsponding paits of tho paths are likewise changed lout in some other latio r, Then if wo compaio this modified motion with tho original one wo sco that velocities at eoncsponding points of the molecular paths have boon changed m tho ratio X/r, and the acceleiations and the forces necessary to produce these have, theioforc, been changed in tho latio X/r 2 , tho masses being unaltciod The factor however, to which the molecular forces aie assumed to be proportional, has now beon changed m the ratio X'*' 1 , * In using dimensions, a special aigmnent is necossaiy to show that must bo propoiliowil to a powei of T } cf P T Bridgman, “Dimensional Analyse " Sbo. 80] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 163 Hence tho modified motion will bo a dynamically possible ono only provided X/t 8 = X~', i.o., provided r „ (127) It is now readily seen that tho energy has been changed in ft dofinite ratio depending only upon X and t and so has boon changed in valuo to that chamctoristic of some other temperature. Accordingly, by modifying in this way all of the motions executed at temperature T we obtain the gas moving at a now temperature. Lot us find out in what ratio tho temperature and tho viscous forties have boon changed. In tho modified motion, because of tho changes in tho velocities, times and linear dimensions, wo have X/r times as much momentum trans- ferred in a time r times as long, across an area X s times ns great; and the velocity gradient is obviously (X/r)/X = 1/r times as great. Hence, from tho definition of tho viscosity in terms of momentum transfer divided by velocity gradient, wo see that n has been changed in tho ratio [(X/r)/rX 5 ]/(l/r) = 1/Xr, or, by (127), in tho ratio On the other hand, T “ y 5 and so lias been changed in the ratio X 8 /r 8 or X 1 "*. Now it is easily scon by multiplying out that Hence, if wo sot n - 5 + ~n> (128) T" has boon changed in tho same ratio as has y, Accordingly wo may conclude that y « f n with tins valuo of n. It is immaterial to this conclusion whothor tho force bo attractive or ropulsivo, According to (128) y should vary as a power n of T somewhat higher than M and log y should plot against log T as a straight lino with a slope equal to this higher power. Tho experimental data plotted in Fig. <10 above actually show good agreement with this con- clusion for helium and neon except at high temperatures, tho best values of n being 0.04 and 0,07, respectively, and the agreement for hydrogen is fair between 200°K and 600°K with n « 0.00; but tho data for argon, Nj, and OOa can bo fitted to such an equation only 164 KINETIC THEORY OF OASES [Chur IV over shoit ranges of tempeiature The values of n jy»t stated, and the best values near 288°K for the other thioc gases, aic collected in n table at the end of the next section opposite the heading ?i', together with the approximate coi responding values of s as calculated fiom (128), listed opposite the heading s' The success of cq (128) for helium, noon, and hydiogon seems at least to justify the conclusion that in those gases the forces between two molecules diop off very lapiclly with distance; the variation is doubtless not exactly accoidmg to any powei of the distance, but it can piobably be concluded that m making othei theoretical calculations wc may icasonably expect to obtain appioximalely con cot results by assuming the power to be the value of n given in the table The much highei values of n obtained foi the olliei Unco gases mentioned may perhaps bo taken as an indication that these molecules dopait much moie widely than do the othei llnce fiom the piopeitics of haul spheics Not veiy much significance can be attached, howovci, to the agreement over a limited range of the experimental curve for i\ with some power of T } since any analytical cuivo whatever can be fitted to some power of the independent vaiiablo as accurately as desned ovoi a sufficiently shoit ningo of values 90. Viscosity and Temperature on Sutherland’s Hypothesis. Quite a diftcicnt hypothesis m logaicl to the molecular foices was proposed by Sutherland * lie retains the assumption of haul spheics but adds a weak atti action between them, falling oft rather rapidly with distance. Such an attraction acts in two ways to pioduco shortening of tho mean free path In tho first place, molecules which pass eacli other at close langc without actually touching oxpoiionco a small deflection, and tins obviously has much the same effect upon tho molecular motion as would a glancing collision In tho second place, this deflec- tion will also cause some molecules to come into actual contact when they would othei wise havo passed by each othei without touching, and will thus mcicaso the actual collision late. Some possible pallia of the center of mass of one moloculo i dative to anothor accoidmg to these conceptions aie indicated loughly in Fig. 41. Of tho two effects, Sutherland supposes that the second is much tho more important, and he actually neglects the fust entholy, apparently no simple justifica- tion for this disenmination can be given, but it has been justified subsequently in certain cases by tho more comploto analyses of Chap- man and Enskog * SvrHBBLANDj Phil Maq x 36, 507 (1803). Bmo. 00| vmcom'Y, TIUiRMAL conduction, nm' union Ifif) The increase in contact collisions is easily calculated by utilizing some of the results obtained relative to the ooeflieient of (Mattering. To do tliis, let us select for consideration two groups of molecules moving with vector velocities Vi and v«, and consider the motion of the first group relative to the second, in the manner of the analysis in Sec. 71. If a molecule of the first group approaches one of the second group along an initial line distant b from the second moleeule, the distance between their centers at the instant when they are closest together will ho r 0 , ns given by «q. (109) In See. Ill), with m replaced by T w»a), as explained in Sec. 71, provided they do not collide; if, Imwovor, n as given by tins equation is less than the menu diameter of llm two molecules or <rn » -\- try), a collision must Fiu, 41.— liypnlhwta nf niolmmlar iiiM’MtflUm. occur boforo r sinks to r () . Now, for an attractive force (17 < 0) that falls oil with increasing r, or for any sufficiently weak force, according to (109), r a and b vary in the same souse, lienee all values of b result in collision up to that limiting value b hn, which makes r« « an- The mutual collision cross soetion is tluiH 8 «« irbl, or, putting b » bn and r# « «riij in (109), wo havo 8 m 2(«i-i -|- vh) J/(n.) j mpaatH (129«) where A’o »irrj » •>ror I 3 a and denotes the cross section as it would he if the field worn absent. If both molecules have the same diameter a and the aamo masH m r r/(o)‘| 8 » <S\|l - 4 J- ( 1296 ) 156 KINETIC THEORY OF OASES [Chap. IV This result shows that for haid spheres an atti active force-field ((7 < 0) increases the collision cross section, wheieas a ropulsivc force (17 > 0) decreases it If all molecules had the same speed , we could now at onco substitulo S fiom (1295) in (102a.) in Sec 61 and obtain for the mean fieo path in a homogeneous gas an expression of the form L - (130) wheie I/o = 3/(4wSo), T is the absolute tempeiatuio, and C stands foi T the constant quantity — 417(<r) — 5, and then for tho viscosity wo should have, by comparison with (126a, l) in Sec 86, aipalo _ a!/4' J * ~ 1 + (G/T) “ 1 + (C/T)’ (131) in which a% has a value at least veiy close to its value of 0*499 m (1266), and tho numeiator can bo wntten aT H in tcims of a new con- stant a provided we assume that pL 0 is constant <For 1000/^> 0 both flonlos aro reduced in tho ratio li 40, but with a aoparft to origin for onoh cur vo ) In a maxwellian gas it is easily shown that a result of tho same foi m must hold, but perhaps with some change in the valuo of C, This is so because, in tho first place, the lelativo distubution of tho molecules os to velocity is tho same at all temperatures, and, in tho second placo, each group contributes independently to tho collision rato. Equation (131) is known as Sutherland’s foimula. It is found to hold, when suitable empirical values of a and C are lnserlod, for many gases over a considerable range of temperature; it is, in fact, moio widely useful than a formula liko (128), containing a powei of T, TUo Sec, 01] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 167 data require in all cases a positive C corresponding to an attractive field. Of course, the equation necessarily holds also ns a two-constant empirical formula for all gases over any sufficiently short range of temperature, the necessary values of C varying with tho choice of the range; but this latter fact, of course, proves nothing in regard to molecular forces. When the observations are extended over tho full range of temperature that is available in modern experimentation, tho formula fails badly in all cases, for either high or low temperatures or both. A straight-lino test of Sutherland's formula is readily made by plotting T^/n against l/T. In Fig. 42 a plot of this sort is shown for the same data that are plotted in another way in Figs. 36 and 40. It is at once apparent that the formula is hopeless for liydrogon and is useful only over limited ranges for the others. The values of C that corre- spond to the three straight lines that are drawn on the plot and to corresponding tangents at 288°K drawn to the other three curves are given in the tablo below, opposite tho heading O'. Constants in tub Viscosity 1'’ohmui,ah For n‘ and s'bcoSoo. 80 , for » mid C, Sco, fit. O' in described just above. 91. Viscosity and Temperature : Other Hypotheses. In attempt- ing to obtain an expression for tho variation of viscosity with tem- perature that will give a still better fit with observation, wo can proceed either empirically or theoretically. An obvious empirical stop would bo to combine tho features of both (128) and (131) into tho modified formula* v - rvic/if m Since this formula contains three disposable constants, a, n, and C, it oan nocessarily be made to fit more closely, or over a much wider range, than can either of tho original formulas, In order to test it, * Of. Timm mid Binkhi.u, Ann, Physik, B, fifll (1030). i< MM* I\ J.C8 KIN l Tir Til I Oil r OI II 1 s I K ilio mum* ilnln fot m\ hums dial tin* plofli il ut I m ‘HI w t I, a Kfioil valiii* of (' \wh ilrlrnmiiml l>y (util ami llmu l"K | *i^l 1 J was plnlli'il iikiiiii*-I low ami ultra a fair **liamhl lim* li>«l b>«n hccuiciI in tlu*< wav, (In* \alm* of » u«>* lakni li<<m (li< li<|t. iln* line Tin* Imi* olilaimal in Hih \wiv fo» l'Mli‘'K* a ( • 'IrawiMM I *« H>, poiniH lioiiiK hIiow ii I iv 1 1 ii** a* * ’I In* \aluiiiiif a ainl I tlm* * , l*(niu>il tvio colliu'li'il in I In* laltli* on p lfiV 'I In* («a inula ran I ** 1 III I* *1 w illuii itluml I |h*i itii I mri (111*1*111111* i!iii|/<< for la hum. mmi. h\<lr«m.*n nml liUl Del’ll, bill I'Vi’li (III** I In <*i* i iiiiNlaiil fouiiiilu i a it uni l«* III t • < I uili** fiii'ii inly In tla* ilala for iukoii ami nulmii tla.M-lr Fill llii'i* jiuiKii'tia ran Imi til v I"' imnlr i \r«*p( by iiUrmlti* hi« umro apmaliznl a»Hiiiii(tlioiiH rtimrinniK (In* m»»t« * ulur hrMt* I **nnnl« 1 i t i l*iu *11 ISttliw tmilrr KMrnt Mi‘ii j*hi# (182) may ho logmdod nn ouuon|mmlhiK hhirIiIv iIm* pmim Kitnl miliud molooulon imwow* littlh I Ih* nl ( rn< (no in*l<l |Hi*iuljifr<l h\ Mnthr*r Imul and, a( nimlII dmlanron, a milium? pimiltir U* fhnt u* xtlmh n ropnlnivo Hold givon lino; (ho ii'lntiw* puflm wwdd lliMt l*o nn in Pig. *18 mllior limn a* m Kik, *11 A mmpln furm u{ hm U n Md would Im ono in wluoli tin* fmoo i 1ml ih r\i*r(r d l»x mho tiuth * iih* m« thn othor wlnm thoir eonloiH mo n dMunoo r npnrl, nml tin* i«»rrr«|w*mlmn polonliul onorgy l f , lwt\o (ho \nlm«n r n i I*" r Ut K 1 \r* Un I n:m X» and X w hoing ohiihIuiiIh Cult nlHliim* 11 * 111 # h a Hr Id* taking mi 8 for roimnuH nf nmtlinnmfioul rumonionno, xxoro mmlo l*y Jmn *,* with Iho rowiilt that fnr n lolulivrlv Honk fUtmoUug firld nf 1 * n ' T *«i •’ l (.S'/ 7 “(‘ * Jon I'll*, Hoy Kite Trite, 100, -111 fl|l2h ii:tn Sec. 91 ] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 169 This is a throe-constant formula but is somewhat different in form from the empirical formula (132), Jones showed that with suitable values of the constants it represents the viscosity of argon within 0.6 per cent from — 183°C to +183°G; the fit was found to be equally good, however, either with n = 21 and S = 62,45 or with n = 14 }{ and S — 38.02, so that no very definite conclusion could be drawn in regard to the molecular forces. It seems doubtful whether much further progress can be made by trying to invent arbitrary force-fields that will reproduce such data as those on viscosity. Tho possible forms of field are too many, the necessary mathematical labor is too oxtonsivo, and tho precision in the experimental data that is required in order to distinguish between different hypotheses is too hard to sccuro. More interest, perhaps, will attach in' the future to attempts at calculation of tho viscosity in terms of universal constants by moans of wave mechanics without the introduction of special postulated constants, or to attempts to cor- relate viscosity with other molecular properties. A few calculations by wave mechanics have been made by Massey and Mohr with interesting results.* They first work out the case of an artificial model, assuming the molecules to bo small elastic spheres. In this caso the enormous number of small deflections duo to diffraction of tho molecular waves, which was described above in connection with molecular scattering (Bee. 78), simulate in their effect a force-field, for witli rising temperature and increasing molecular velocity these deflections are confined progressively to smaller angular ranges and so have less effect. As a result of tins phenomenon tho clastic-sphere model gives a variation of t; with temperature that is more rapid than by assuming tho right diameter for tho sphere, tho experimentally observed variation could actually .bo reproduced rather well in the case of helium, and somowhat leas successfully for hydrogen. In a second paporf Massey and Mohr apply to helium an approxi- mate form of the fundamental method of wave mechanics, in which everything is deduced without fresh hypothesis from the basic proper- ties of electrons and nuclei. A good approximation to the field of a helium atom is known from Slater’s work; it is stated in cqs. (177a) and (1776) in Sec. 121 below. Using this field the authors calculate a cross section for viscosity which is % of our <5„», and, substituting its value in their equivalent of our oq. (126a) in Sec. 86 above, they obtain tho value of tho viscosity. Comparing their results with observed values they find; * Massey find Mouu, liny, Soc, Proa,, 141, 434 (1033), t Massey and Mown, Roy. Soc, Pro c„ 144, 188 (1934). 160 KINETIC THEORY OR OASES [Cjiai*. IV T (abs ) 16 0° 20 2° 88 8° 203 1° 204 6“ 7i (obs ) 29 46 36 03 91 8 1 166 4 100 4 7j (calc ) 36 43 98 106 218 The unit of viscosity employed heie is the miciopoisc 01 10~ a c g s unit Such a degree of agiecment between theoiy and expoumont would not oidmaiily be consideied vciy good, but it must bo lcmombeiod tliat in this case the theoretical calculations themselves aio only approxi- mate, and the agieement becomes leally improssivo when wo further recall that in all of the theoretical woik no special assumptions lefor- ung to helium aie introduced except that its nucleus is much heavier than an electron and carnes a charge numci ically twice as groat. 92. Viscosity of Mixed Gases. Up to this point tho gas has boon assumed to be homogeneous. We must now eonsidoi how tho theory is to be extended to a gas consisting of several lands of molecules. When a laiefied gas composed of seveial dilfeiont kinds of molecules is set m slicanng motion, tho viscous tiansfei of momentum across any plane will be the simple sum of the transfers by tho difforont groups of molecules The analysis of the “simple theoiy” in See. 81, if valid for a pure gas, should accordingly be applicable to each group sc pa- lately We are thus led to write foi the viscosity, as a generalization of (119) in Sec 81, , = the sums extending over the vaiious sorts of molecules; L\ stands for the' mean fiee path of land t m the mixed gas, whetoas L, is what tho mean free path of this kind would be if it alono woio present at its actual density p„ and y, - }p,L, or the viscosity of kind i according to (119) Taking L t from (106b) in Sec 64 and L[ from (105c), wo have L [ _ \/2n t S, L ' 2 %S „[1 + (MJM,)]* in which M, is the molecular weight of molecules i and S, tho equivalent cross section for collisions of these molecules with oach other, while /S,, is the mutual cross section when they collide with kind j. Tho expression foi r> can then be tin own into tho fonn Sec. 92] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 161 -2 Vi r, + %% ^ l , Stf[l + Wi/MiW f<# „_ , (135a) in which the rji represent the viscosities of the separate pure con- stituents and the coefficients f< 7 - should be .independent of the composi- tion; and Uj can be interpreted, if desired, as concentrations by volume, or in terms of moles per unit volume, In the case of a binary mixture this becomes V 5=2 Vi + v* i+f»? ■ i +h.£ Ui 111 (136b) The complete analysis of Chapman shows, however, that so simple a formula cannot be quite accurate, presumably because the persistence of velocities is different for the different lands of molecules and in different types of collisions; he obtains for a binary mixture a formula that can be summarized in the form* i) lflifti 4~ wnpte qaQafl-a a in? + brum + ami ’ (136a) 1 ) 12 , ai, a 2 , and b being four new constants depending on the molecular masses, the law of force, ancl the temperature. Chapman fitted f a quadratic formula of this last type to some observations made by Schmitt J on mixtures of ai'gon and helium and of oxygen and hydrogen ancl found good agreement (mostly within 1 per cent) ; the data and the theoretical curves are shown in Fig, 44, abscissas representing per cents of the lighter gas in the mixture. Schmitt himself, however, hacl found good agreement of his extensive results with Thiesen’s formula, which is equivalent to our (136b). More recent work by Trautz and his collaborators § is in agreement with the double-quadratic type of formula, eq. (136a), but it is not clear how definitely the simpler form (135b) is ruled out, For a given pair of gases formula (136a), regarded as an empirical one, has four disposable constants in addition to the viscosities of the pure con- stituents, as against only two in (135b), and very precise work is ♦Chapman, Phil, Trans,, 217A, 116 (1018). t Chapman, Phil. Trans,, 211A, 433 (1012). t Schmitt, Ann. Physik, 30, 393 (1909), § Tiiautz el al, Ann. Physik, 3, 400 (1920); 7, 400 (1930); 11, 000 (1031). 162 KINETIC THEORY OF GASES [Chap. IV necessary in order to show definitely that all four are needed. For practical purposes the simple quadratic expression, ( n i\ 2 i *712^1^2 , (u 2 V ’ “ + ~g~ + - (lm) in which rju is a new constant to be determined empirically and ni/n , n 2 /n are the fractional densities in terms of volumes or moles, appears to suffice. So far as we can see in advance without knowing the relative values of the constants, the viscosity for a given mixture might lie either between its values for the pure constituents, or above or below both of the latter. Cases are actually known in which the addition of a gas with smaller viscosity raises the viscosity of the mixture; an example is furnished by Schmitt's data for helium and argon as shown in Fig. 44 above, according to which, for example, the substitution of about 40 per cent of helium for an equal amount of argon raises rj frond 2.22 X 10~ 4 for pure argon to a maximum of 2.30 X 10“ 4 , from which it then sinks to 1.97 X lO - " 4 for pure helium. In other cases, how- ever, the viscosity of the mixture has an intermediate value for all compositions. B. CONDUCTION OF HEAT 93. The Kinetic Theory of Heat Conduction. When inequalities of temperature exist in a gas, heat is transferred by molecular action from hotter regions to colder; this process is called conduction and is 163 Sec. 94] VISCOSITY , THERMAL CONDUCTION , DIFFUSION independent of any transfer of energy that may be going on simul- taneously by means of radiation. The heat flows in the direction in which the temperature falls most rapidly; the maximum rate of decrease of temperature is called the temperature gradient. The amount of heat conducted in a given time is found by experiment to be accurately proportional to the temperature gradient so long as the latter is sensibly uniform over any distance equal in length to a molecular mean free path; and the amount of heat that is transferred per second across unit area of any small plane drawn perpendicular to the direction of heat flow, divided by the temperature gradient, is called the thermal conductivity of the gas. We shall denote the con- ductivity by K. The gas is assumed to be at rest and in a steady state (although, of course, not in complete equilibrium). If the ar-axis is taken so as to be parallel to the direction that the temperature gradient has at any given point in the gas, then across a small plane drawn through that point perpendicular to the z-axis an amount of heat II will pass per unit area per second given by T being the temperature. The qualitative explanation of this phenomenon in terms of kinetic theory is obvious. Molecules moving from warmer regions into colder ones carry with them more heat energy than those moving in the opposite direction, and the consequence is a net transfer of heat toward the colder regions, To calculate the conductivity, therefore, we need to study the energy carried from place to place by the mole- cules, just as in treating the pressure and the viscosity we studied various components of their momentum. 94. Simple Theory of the Conductivity. For a first simple attempt at a theory, we can utilize the calculation already made for viscosity merely by substituting in it the average heat energy i of a molecule in place of rm; 0 , the '//-momentum of mass motion. If we do this throughout in the argument leading up to eq. (118) in Sec. 81, we obtain at once for the stream density II of heat energy, in analogy with that equation, H 1 „ r de l mh T,' n being the number of molecules per unit volume, v their mean speed, and L their moan free path. This equation, in comparison with the one written down just above, shows us that, since 164 KINETIC THEORY W GASES tClIAf IV de _ di dT dz ~ dT dz' Now de/dT is connected with the specific heat of the gas] foi when it is heated at constant volume all of the heat supplied must go into an ancieaso m the eneigy of the molecules, so that ncU/dT = pc v , p being the density in giams and c v the specific heat of a gram at constant volume measuied in mechanical units, Hence our simple theory gives for the theimal conductivity of a homogeneous gas K ~ \ pvLci , (137) or, by comparison with (119) in Sec 81, K = r,c v (138) These equations aie, of couise, lostnctcd to a larofied gas, as will be all of oui lesults on theimal conductivity, and this foi two reasons In the fust place, m calculating the tiansfer of heat acioss a suifaco we shall, as we have done lntheito in ti eating the momentum, con- sidei only bodily convection by molecules that actually cioss the suifaco, lgnonng all foice-aclions between two molecules wlnlo lying on opposite sides of it. In the second place, the heat energy can bo expressed as the sum of the heat energies of tho scpai ato molecules or as rie only so long as there is not on the average an appreciable amount of mutual molecular energy Equation (138) is interesting and impoitant, both bccauso it involves nothing hut dneelly obsoivable quantities, and also bccauso it predicts on the basis of theoiy a relation between physical mag- nitudes of two diffcient soits, mechanical and thermal, which wo might have expected to be quite unielatecl. To test tho equation, the oxpeiimental value of tho lalio Ii/y\Cv for a number of gases is given in the table in Sec. 103 below In expressing K and c\< it is obviously immatoiial wliat unit of heat is employed so long as tho same unit is employed for both, A glanco at the table shows that the piedicled relation is m agieomcnt with the data at least as logards order of magnitude, the maximum value of K/ijc v boing only 2 6. Equation (138) constitutes, theiefoio, a distinct success for tho theoiy, and it is reasonable to hope that further lofmomcnt of tho calculation will introduce a numoucal factor into it winch will lead to a good agreement with observation. In infilling the theory we might now follow the trail blazed by O E Moyer some sixty years ago and proceed to introduce corrections Biro. 06] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 105 for Maxwoll’H law, etc., as we dicl for the viscosity. The result, however, is oven less satisfactory hero than it was there, for the reason that conduction is a distinctly more complicated process. Accordingly, wo shall shift at once to the viewpoint required by the other method of attack, which was initiated by Clausius and Maxwell and finally carried through to completion by Chapman. Incidentally wo shall bo able to insert for trial the assumption made by Moyer and obtain thereby his result as well, for the sake of comparison. 96. Thermal Conductivity of Symmetrical Small-fleld Molecules. First Step. Lot us consider as usual a rarefied gas composed of small- field molecules, which act upon each other only when they come very dose together. Lot us suppose also that they possess energy of translation only, so that tho energy of a moloculo is t - £ mv 2 . This condition requires that whonovor two of them do interact tho forces must pass through their centers of mass; this can also bo expressed by describing tho molecules as spherically symmetrical. Small, hard clastic spheres obviously constitute a special case satisfying these requirements. A general expression for the flow of hont in such a gas is obtained from (124) ill Sec. 86 if wo substitute in that expression X — \ mv 2 . Let us also change «x to a*, tho rc-axis being taken in tho direction of the temperature gradient. Wo obtain thus for tho amount of energy transferred by molecular convection across unit area per second, which also equals —K clT/dx by tho definition of tho thermal conductivity K, 11 - -*§■■" 2 S’""”'' (ISO) tho sum extending over all molecules in unit volume (more precisely, over all in a given volume, tho result being then divided by tho volume). In tho special oaso of a homogeneous gas containing n molecules in unit volumo each of mass m, this equation booomea ~ \ wl 2 Vtt ' 5 “ § nm j M’/fr) dK, (140) where /(v) is tho distribution function for velocities and dn =* dv x dv u dv t and tho integral extends ovor all velocities. Tho conduction of heat in a homogeneous gas thus depends directly upon the value of the quantity 2v x v 2 summed over all mole- oules in tho neighborhood of a given point. If the distribution were exactly maxwellian, this sqm would bo zero, in consequence of thq 166 KINETIC THEORY OF GASES [Chap. IV symmetry of /, and there would be no conduction at all. The new method which we shall follow consists, accordingly, in seeking that slightly modified form of / which comes into existence in consequence of the temperature gradient; from this we then calculate the value of Xv z v 2 , and from this in turn the value of K is obtained. For convenience let us split up / visibly into a maxwellian term and a correction term, thus : /O) =/o(v) + /,(v), /o(v) = Ae-^\ (141a, 6) where A and /3 2 are so chosen that ffodK = Jf di c = 1, J* v 2 fo dx = JV/ die, the last integral representing v 2 . Thus /o(v) represents that max- wellian distribution which would correspond to the temperature of the gas as determined by its actual value of whereas f a (y) represents a small correction term. By inserting / from (141a) into the last two equations we can also obtain equations equivalent to them in a form containing/, alone, thus: Jfsdn = 0, J* v 2 fa dn = 0. (142) Furthermore, to make the mass velocity zero, as it is assumed to be in defining the conductivity, we must have, as an additional condition upon f, j'vf dx = 0 or, in terms of components and of /„ J v x f, die = J v y f s die - J'vj, dx = 0. (143) Finally, when we substitute/ from (141a) in (140) / 0 cancels out of the integral by symmetry, and we are left witti K d -1 K dx v x v 2 f a dx. (144) This equation expresses K in terms of the correction term /, alone. For the determination of / we have now the Boltzmann equation, (87) in Sec. 51, in which we put F = 0 here because we are dealing with the simple force-free case, also d(nf)/dy, d(nf)/dz because con- ditions are here assumed to vary only in the direction of x, and d(nf)dt because by assumption we are dealing with a steady state of heat flow. The equation thus reduces here to two terms only and can be written, after inserting / from (141a), Sue. 95] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 167 d , ,v , d , , N _ r 9(nf) l v *iz + Vx ^ ' at L ^ -1 °°n the term in square brackets representing the effect of collisions. _ In this equation, moreover, the term in nf, can be dropped in comparison with the one in n/ 0 , since nf, is a small quantity of t e same order as the temperature gradient; nf, will probably vary with x because the temperature does, but the rate of this variation wi again be small as compared with nf, itself and ^ will, therefore, be a small quantity of the second order, whereas is of the first, The dropping of this term is required, as a matter of fact, for the sake of consistency, for the whole theory as ordinarily developed is limited to the first-order effects of the temperature gradient. On the right side of the equation, on the other hand, we can rep ace / or fo + /. by f. alone; for the collisions can alter neither the total number of molecules nor, in consequence of the conservation of energy, the value of v>, and so have no tendency to change the value of / o, as is at once evident from our definition of it. According y, e equa tion reduces for our purpose to the following : (145) If wo then insert fo from (1415), the second member of (145) becomes 'dx {nf o) v»(^ . dn , dA A Tx + n li nAv 2 $4 But P <* l/T [cf. eq. (56)] and A « p [cf. eq. (60)]; hence* J -—p P daT 1 dT i an. 31 d - T T — — o oO P l_ dA A dx 3 ldT, 2 T dx T dx' A dx 2 (3 2 dx Furthermore, since the pressure must be uniform in order to leave the gas at rest, nT must be independent of x and so 1 rfn __ __1_ dT n dx T dx Inserting all these values into (145) and noting that « is ^affected by collisions and so can be taken outside of the square bracket on the left, we find finally • jp.g., W * - -log T t const-, and differentiation yields the result 168 KINETIC THEORY OF GASES [Chap. IV CU6) This equation is now to be solved for /«. 96. Thermal Conductivity on Meyer’s Assumption. Proceeding from this point, we can obtain Meyer's result if we now assume with hirp that those molecules which collide during each element of time have a distribution after collision of the maxwellian type. This assumption implies that all molecules after collision pass into the dis- tribution represented by / 0 , none of them entering that represented by The quantity [n df 8 /dt\ co n thus contains no gains but only losses. In estimating these losses, moreover, we can ignore col- lisions of the f a distribution with itself because their effect is a small quantity of the second order in the temperature gradient. Hence, using Meyer's assumption, we can write for the left-hand member of eq. (146) — O v nf s , where e* is the collision rate for a molecule moving at speed v through the maxwellian gas represented by nf 0 . If we then solve the resulting equation for /*, we find This value of f 8 can now be substituted in (144), and the resulting integral can then be evaluated by numerical integration, 0„ being taken from eq. (104a). The final result of doing this can be written, if for comparison that value of the viscosity is brought forward which was derived above on the basis of Meyer's assumption, i.e., 7} = 0.310 pvLy as given in eq. (123) in Sec. 84: K = 1.10 rjCv. This equation agrees scarcely better with the experimental data than does eq. (138) obtained from the simple theory. A serious objection to the procedure here outlined, however, lies in the fact that the form employed for/ a does not make the net flow of molecules vanish, as is readily discovered by evaluating j* v x f 9 d/c. There would thus be a thermal-transpiration effect, manifesting itself by a mass flow of the gas in one direction. Now there may well be a tendency for such a mass flow to be set up as a consequence of inequalities of temperature, but the absence of mass motion which is specified in the definition of the conductivity, and which is actually enforced upon the gas by the experimental arrangements when a Sec,. 97] VISCOSITY, THERMAL CONDUCTION , DIFFUSION 169 measurement of the conductivity is made, requires that any such mass motion resulting from molecular diffusion must be exactly offset by an equal mass flow in the opposite direction. In his pioneer calculation Meyer himself followed a procedure different in one respect from that just indicated; he brought the net flow of molecules to zero by making a suitable choice of the value of dn/dx , instead of choosing this latter quantity so as to secure uni- formity of pressure. Had we done this we should have obtained K = 1.540 rjcvj a much better result and almost the same as Meyer’s own final one, K = 1.603 r\c v . The equality of pressure seems, however, to be an absolute essential in order to keep the gas at rest, and Meyer’s procedure, therefore, can scarcely be defended. The net flow of molecules must in nature be balanced out in consequence of the existence of a form of f 3 different from the one that we have here assumed. We shall return, therefore, to our differential equation for f s and seek a solution of it that keeps the net flow of molecules zero. 97. Thermal Conductivity: Second Step. The most direct pro- cedure in attempting to solve eq. (146) for f s would be to insert in it the value of [<9//<9^] 00 ii given by eq. (43) in Sec. 25 and then to endeavor to solve the resulting integrodifferential equation for f s . This rather formidable undertaking has seldom been attempted, however. The Maxwell-Chapman method proceeds by a sort of flank attack. The expression in the right-hand member of (145) or of (146) represents, as is clear from its origin in the process of deriving the original differential equation for nf, a steady inflow of molecules dis- tributed in velocity in the manner exhibited by that expression in (146) ; thus, molecular convection is continually tending to build up a distribution of the form, nf a = av x (y 2 - /3V)e-^ 2 , (147) a being a constant of proportionality.' Here negative values of /' can be regarded simply as representing missing molecules that would have been there had the distribution been uniform and maxwellian throughout. This distribution is then at the same time being con- tinually transformed by collisions 4n the direction of a maxwellian one, and the existing steady form of f s is that form for which these two opposing influences are in balance. Now Enskog showed 5,4 that we could infer a good deal in regard to the final form that / a must assume, from the nature of the inflow * Enskog, loc. cit. [Chap. IV 170 KINETIC THEORY OF GASES distribution/' and from the symmetries of the situation. The function /' is of the type /: = v x F(v 2 ), (148a) and we can show that under the influence of collisions this type of function is invariant except forlTpossible change in the form of the function F. To show this, consider the way in which collisions between the f' and the maxwellian / 0 distributions change the distribution of veloci- ties during a time dt ; collisions of /' with itself need not be included because /' is small. Select any element di <! in velocity space about a velocity v' making an angle a' with the ?vaxis (cf. Fig. 45) and another element da about a velocity v making an angle a with v x and an angle 0 with v'. Then the latter element t d/c will contain^/' die molecules out of the /' distribution, and their collisions with /o willjthrow into d/c' during dt a number dK / of molecules which will be proportional to n/' da, to die' and to dt, but which will otherwise depend, because of the properties of the scattering process, only upon the magnitudes of the velocities v and v' and upon the angle 9 between their directions. Let us write for the number thus thrown into d/c' nf ' g(v , v ', 9) d/c' d/c dt. The total number of molecules thrown out of /' into d/c' by all colli- sions will then be dN = n d/c' dt Jf ' g(v, v', 9) d/c integrated over all values of v. We wish now to show that the new distribution represented by diV/d/c' as a function of v' is again of the form (148a) . Let us introduce polars with the direction of v' as their axis and write in terms of these die = v 2 sin 6 d9 d<p dv. In expression (148a) for /' we can write Vx = v cos a; and by projection we readily find that, if the polar azimuth is measured from the plane containing v' and the Vs-axis, cos a = cos 9 cos a ' + sin 9 cos <p sin a'. Hence dN = n d/c' dtj* J J v 8 (cos $ cos a' + sin d cos <p sin a') F(v 2 ) g(v, v', 6) sin 6 dd dip dv Sec. 97 ] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 171 integrated over all of velocity space. The <p integration gives at once cos <pd<p = 0, J* 2 dy =$ 2ir, whence dN = n da' dt cos a! J* JV F(v 2 ) g(y, v ' , 8) sin 0 cos 6 dd dv. The integral now represents a function solely of v f ; and we can write cos a f = v'Jv'. Thus dN has the form, v' x times a function of v f , and, accordingly, has the same general form as had/'. We may conclude, therefore, that collisions may at most tend to change the form of the function F(y 2 ). A plausible first guess is now that perhaps there is, after all, no change even in F(v 2 ), and that the final form of / * is the same as that of the inflow distribution/' itself, which is given by (147). If this is so, we can write /. = Cv x ({- - /5V)e~^ 2 , (1486) where C is a constant to be determined. Such a value of /, satisfies the requirement that the net flow of molecules must vanish, for with this form of /, J v x f dtc = J Vxf 8 da = C — /3V)<r"0 2v2 die = 0 (cf. table of integrals at the end of the book). In the hope of obtaining a good approximate value let us calculate the conductivity using this assumption for/*. The constant C we shall choose so as to secure the best approximate fit to eq. (146), which is the equation determining/*. •The novel feature in Chapman’s work was the method of making such an approximate fit by multiplying the equation through by a suitable function and then integrating over all velocity space; this procedure greatly facilitated the handling of the left-hand member. For this purpose it is natural to employ the same function, v x v 2 , whose sum determines the conduction of heat [cf . (140)]. Treated in this way the left-hand member of (146) becomes This expression is easily seen to represent the rate of change by colli- sion of Xv x v 2 summed over all molecules in unit volume, which can also be written f v z v 2 f da or J v x v 2 f a da ; this rate of change we shall 172 KINETIC THEORY OF GASES [Chap. IV denote by D%v x v 2 . The entire integrated equation can then be written D2v x v* = ~ - fi*v*)<r*** dK. (149) This equation may be regarded as a special case of a general trans- port equation worked out Iby Maxwell for any molecular magnitude Q , in analogy with the Boltzmann differential equation for / itself, and Chapman's method was originally developed with that equation of Maxwell's in mind. Its general form, for a gas at rest and free from external force, is i {nQ) + lx (n ^ } + Ty ^ + l (n ^ } = [I (r4) L ; the bar denoted an average for all molecules in the neighborhood of any given point. In our case Q — v x v 2 . In order to proceed we have now to calculate 1)2 v x v 2 . 98. Effect of One Collision upon Zv^v 2 . In calculating D2v x v 2 we can ignore collisions between any two molecules of the / 0 dis- tribution; for such collisions, being the same as the collisions in a max- wellian gas, can have no tendency on the whole to alter 2v x v 2 , which remains permanently zero in the maxwellian case. Collisions between two molecules both belonging to /*, moreover, can likewise be ignored as a second-order effect. Thus there remain for consideration only collisions between f a and /o. Let us begin by considering a single collision between a molecule of f 8 moving with velocity v x and one of /o moving with velocity v 2 and find the effect of this collision upon the contribution made by these two molecules to 'LvxV 2 . To obtain general expressions for the components of velocity of these two molecules after the collision, let us introduce the velocity of their common center of mass u and write w for the vector relative velocity between the first and the second. We shall assume for the moment that the molecules have unequal masses m x and m 2; in order to make the formulas useful in other connections. Then U = jUiVx + jU 2 V 2 , w = v x - v 2 , mi m% mi + m2 H 2 ^ i ’ m\ + m 2 in terms of which (cf. Fig. 46) Vi = U + M2W, V 2 = U — jUiW. (151a) Sec. 99] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 173 The effect of a collision is now merely to rotate the relative velocity w through some angle B into a new position W without changing its magnitude, while u remains unaltered. Hence for the new velocities after collision we can write Vi = u + /xoW, V 2 = u — gxW; W = w. (151 b) (Cf. Sees. 23 to 24. Of course, W is usually not in the same plane with Vi, v 2 , and w.) The results of squaring Vi and V 2 can now be handled more neatly if we employ the notation of vector analysis. For this purpose we need only the scalar' product , which for two vectors A and B is denoted by A • B and defined thus: A * B = AH cos {A, B) - A X B X + A y B y + A s B g) A and B denoting the magnitudes of A and B and (A, B), the angle between their directions; the last expression is obtained by noting that if l m n } V m! n* are the direction cosines of the direc- tions of A and B, respectively, then cos (A, B) = IV + mm' +• nn', and IA = A*, VB = B X} etc. As special cases A • A = A 2 and (A + By - (A + B) . (A + B) - A 2 + B * + 2 A . B, The change in Xv x iA produced by the collision is then, obviously, A2tw a - V lm V\ + VuVl - v lx vl - v 2x vl Substituting from (151a) and (1516), in which we now put mi = because we are at present dealing with a homogeneous gas, and then multiplying out and using the vector formulas, we obtain from the last equation for the effect of one collision azvxv* - (u x . + \ w x ) (u + \ w y + (u* - i w m )( u - 1 wy - (u m + I to«)(u + \ w y ~~ (u x - 1 w w )( u - j w y or A'LvrV 2 » W x ii' W — w*u * w. 99. Average of the Effect on 2v x v 2 . As the next step in the cal- culation wo can now conveniently average the expression just obtained for A2v x v 2 over all positions of the plane containing the angle 0 between w and W. All positions of this plane are equally probable when Vi KINETIC THEORY OF GASES 174 [Chap. IV and v 2 and hence w have given values, in consequence of the symmetry of the scattering process. Let us write V7 = w cos 0 + r where r is perpendicular to w (cf. Fig. 47). Then we can imagine the average in question to be taken as the vector r revolves at a uniform rate in a plane perpendicular to w while retaining the constant mag- nitude r = W sin 6 = w sin 8. Inserting the value just written for W, we have as the effect of one collision = (w x cos 6 + r x )u • (w cos 6 + r) - w x u • w « — w x vl • w sin 2 8 + r x u • w cos 8 + w x u • r cos 8 + r x u • r after multiplying out. Here the second and third terms on the right average to zero because any component of r does. The last term, r x u • r, is quadratic in r, however, and requires special study. Let us resolve r into two rectangu- lar components in fixed directions perpendicular to w, writing r = ri + r 2 ; then r x u •_ r = r lx u • r x + r lx n • r 2 + r 2x u • r x + r 2 *u • r 2 - Now for a given value of r\ x negative and positive values of r 2 occur in pairs with equal frequency (e.g., OA and OA f in Fig. 47), and simi- larly for r 2x and rr, hence the middle two terms average again to zero. Furthermore, if we denote by u 2 components of u in the directions of r i and r 2 , respectively, and by Zi, l 2 the cosines of the angles between these directions and the cc-axis, rixU • ri = r lx riUi = hr\ui) and r\ = r 2 cos 2 (r, ri) = w 2 sin 2 8 cos 2 (r, r x ) and averages J w 2 sin 2 $ as r revolves (since as an angle <p varies uniformly the average of cos 2 <p is cos 2 <p d<p/j* Q 2ir d<p = %)- The term r 2x n *r 2 can be simi- larly treated. Hence the average of r x u • r reduces to | w 2 (liUi + l 2 u 2 ) sin 2 6. This expression can be converted back into cartesian components in the following way. If u w denotes the component of u in the direc- Sec. 100] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 175 tion of w and l w the cosine of the angle between this direction and the a>axis, whence and U x =: l\Ui -f- I 2 U 1 *4“ IwUw* l\Ui “j“ ^2^2 — Wx Iw'U'w j wl w = W X} WU w = wu cos (u, w) = u • w. For the average of r x u • r we have, therefore, % w 2 (liUi + I 2 U 2 ) sin 2 0 = i (u x ^ 2 — ty*u • w) sin 2 0. Thus, finally, writing AXv x v 2 for the average of A'ZvxV 2 as the result of a collision between molecules of velocity Vi and v 2 , we have ~A2v x v 2 = (£ u x w 2 — f w x n • w) sin 2 0. 100. Total Effect of Collisions on Sv^v 2 . The next step is to sum A'LvxV 2 over all collisions between the f a and/o distributions. Of the/ a molecules, nf a (vi) die 1 in unit volume have velocities lying in a range da 1 about Vi, and of the / 0 ones n/ 0 (v 2 ) d* 2 lie around v 2 ; of the former, my/XvO d/c x cross unit area per second in their motion relative to the second group. Hence in terms of the scattering coefficient G defined in Sec. 68 above [cf. eq. (107a)] there are 2t n 2 w G(w , 0) f 8 (v 1 ) / 0 (v 2 ) sin 0 die 1 ck 2 collisions in unit volume between f s and/o molecules which result in a rotation of the relative velocity through an angle between 0 and 0 + dd; G(w, 0) is written here because G will in general depend upon the rela- tive velocity. Multiplying this expression by A2v x v 2 and integrating the result over v x and v 2 and over 0, we have then for the total change in 2v x v 2 made by all collisions in a second: D'LVxV 2 = 2wn 2 J J f*w(A2v x v 2 ) G(w , 0) f a (v 1 ) /o(v 2 ) sin 0 dd die 1 <2k 2 . (152a) This integral is most easily evaluated if we change variables from v x and v 2 to u and w. From (151a), in which again we put /z = $, vix = u x + %w x , V 2 x = u tt — \ w X) etc.; * If we think for the moment in terms of new axes in the directions of Ti, r 2 and w, a unit vector in the original ^-direction has components h, h, l w , hence its scalar product with u, which equals the old u X} also equals hu\ + hu^A- l w u w . 176 hence KINETIC THEORY OF GASES [Chap. IV *>! = Vi . Vi = u 2 + i w 2 + u • w, v\ = m 2 + I w 2 — u • w, and in any integral dv ix dv 2x = \J\ du x dw x , where J stands for the Jacobian A similar result holds for the y- and ^-components. Hence, insert- ing also the value found above for A2v x v 2 and the expressions for /« and/o obtained by changing v to v x in (1486) and to v 2 in (1416), D2v ^ 2 = 3m2CA ffJffffo w $UxW 2 ~ W x (u x W x + UyWy + U,w,)] [“* + 2 wj[i - 0 2 (« 2 + i w 2 + U X W X + UyWy 4- U ,«>,)] G(w, 8)e^( 2 »w sin 3 e d9 dUx dUy dUz dWx dWy dw ^ (1526) u • w being here written out in cartesians. The reduction of this septuple integral is a straightforward but somewhat lengthy process and we shall omit most of the details. After separating it into simple parts by multiplying out, those few terms which do not at once go to zero by symmetry are easily integrated with respect to u with" the help of the table of integrals in the back of the book. The resulting expressions can then be reduced to an integration over the two vari- ables w and 0 by introducing in place of w x> w v , w x polar coordinates in w ; V V ’ and then carrying out the integrations over 6 ' and he whole integral thus finally comes to depend upon a single irreducible one, which we shall write in the form S vc — £ J" o G{w, 9) sin 3 6 dd = L [Jo G ( 2 ^M X> 9 ) Sin8 9 d9 w i e -yiP*w* dw x 7 e~ xl dx (153) »‘cZen f ta Ihe constant factor has been 0 - Jo* as in rilTfift el “ lic s P hercs . for which cross section i- Th ^ ’ S ” reduces to * he ordi »"!’ collision cJcXw Tarn TZllZ ani “ K ^ The final result obtained in this manner, after inserting A = 03 / n h m Sec. 101] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 177 D2VxV2 = ^yr^S vc C. (154a) It is interesting to compare with this rate of destruction of Lv x v l the value of the latter quantity itself; as found from (144) and (148b) this is 2v x v 2 ~ n ^J v x v2 (^, ~ dn = - 5# 4 jS 7 nC. If we insert the mean speed v = 2/ (\/ x/S) we have, therefore, DPvxV* _ 16 8 v V 15 Vl UVSvc ~ 15 17/ (154b) where L vc = 1/ (\/2nS vc ) and would represent the mean free path if S vc had the significance of an ordinary collision cross section. According to this last equation, if the quantity Zv x v 2 underwent no change except by collision, and if /„ retained always the form that we have assumed for it, then 2v x v 2 would die out exponentially, about half of it disappearing in a time L vc /v or roughly the time taken for each molecule to make one collision. Maxwell showed that in the special case of molecules, repelling according to the inverse fifth power of the distance Lv x v 2 would decay at the rate just found regardless of the form of /„. In any case this result serves to give us some idea of the great rapidity of the process discussed in Sec. 32 in Chap. II, by which departures from Maxwell’s law are ordinarily smoothed out. As an example, in air at 15°C and 1 atmosphere (L = 6.40 X lO -0 cm, v = 4.59 X 10 4 cm/sec) equilibrium is practically restored in a billionth of a second, but at a pressure of 10 -3 cm Hg nearly a thousandth of a second is required, and at 10“ 6 cm Hg nearly a second. 101. Thermal Conductivity: Final Approximate Formula. The value which D~Zv x v 2 must have to keep the state of the gas a steady one is given by eq. (149). This equation contains the same integral as occurs in the expression just given for and so reduces to „ Zir^nAdT (154c) Insertion of this value of D 2u x y 2 in (154a) would now give us C, but our main objective is reached more quickly if we insert it into (154b) and so obtain, after putting A = , 2v x v 2 75 V2 dT 64 8 t vS vc T dx ' 178 KINETIC THEORY OF GASES [Chap. IV This value of Xv x v 2 can then finally be inserted in (140) in order to obtain the value of the thermal conductivity K. In doing this it is convenient also to replace one factor, 1/jS 2 , by (s/ttv /2) 2 in terms of the mean speed v and the remaining 1//3 2 by 2p/3 [cf. eq. (656) in Sec. 30]. If we wish, we can also go farther and eliminate T, as we did pre- viously, by means of the fact that the specific heat at constant volume of the monatomic gas under consideration is simply its mean trans- itory ^energy per gram divided by the absolute temperature c v = h v*/T) and it may also be interesting to insert again which would be the mean free path if the molecules were spheres. In this way may be obtained the following expressions for the conductivity: v _ 25 t mvP 25 r _ T 128 s *cT ~ "64 pvLvcCv ’ ( 1550 ) where p = nm, the density in grams. Comparing this value of K with that obtained for the viscosity ?? by the same approximate method, which is stated in eq. (1566) below, we find also: K - i vcv. (1556) These formulas may now be compared with those furnished by the simple theory of heat conduction. The numerical factor in (155a) has the value 1.23 as against H in (137) in Sec. 94 above, so that our value of K is nearly four times that furnished by the simple theory (provided we may overlook the difference between L and L vc , which would be identical for hard elastic spheres anyhow). In a similar way (1556) differs from (138) in Sec. 94 by a factor of %. Part of the latter increase is clearly due to greater persistence of Ev x v 2 after collision than of the quantity 2v x v y , upon which viscosity depends, for the coefficient m the “relaxation! rate” is only in eq (1546) above as against % in eq. (156a) below. The remainder of the increase up to A appears to be due to correlation between high values of the kinetic energy, § mv\ and high rates of convection for high values of v. e . ProWm _ A PPly the preceding method to the treatment of viscosity. The principal equations obtained in doing this are given below, each preceded by the number of its analogue in brackets: [ 140 ] dV$y ~ V W = m2VxV *> Sec. 102] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 179 [146] df. dt I coll = 2nA/3 2 v x v y e-P iv * dv Qy dx at a point where v Qy = 0 — elsewhere v y must be replaced by ( v y — Vov) in this and the following equation; [148b] f» = Cv x v y er^\ A2v x v y = I (W x Wy — w x w y ), AhVxVy — — f w x w y sin 2 9, [154 b] ' VSnMI. , *>. 2v x Vy 5 5L V c (156a) 5t mv 5? r V ~ 32 V2 S xc ~ Z2 pvLv °- (1565) Here 77 is the coefficient of viscosity, p the density, and v the mean molecular speed, whereas L vc = l/(\/2nS vc ) and S vc is the collision cross section for viscosity and heat conduction as defined in terms of the scattering coefficient by eq. (153) above. The value of the coefficient in (1566) is 57r/32 = 0.491. 102. Final Correction of the Conductivity Formula. The formulas just obtained represent still only a first approximation to the correct ones, since we employed a special assumption as to the modified veloc- ity distribution, using f a as given in (1486) instead of the more general form given in (148a). The error arising from this assumption was examined by Chapman in a later paper.* His method was equivalent to expanding F(v 2 ) in powers of v 2 , then substituting the form of f a thus obtained in eq. (146), and finally integrating this equation after multiplying it in turn by various functions, whose individual rates of destruction by collisions were calculated much as we did that for hv x v 2 . In this way he obtained an infinite number of linear equations for the determination of the coefficients in the expansion of F(y 2 ), as a generalization of our single equation for the determination of C. The final result of his rather laborious calculations was only to replace our (155a, 6 ) by *-& + •>! = + ^ <15&> where a and 8 are very small numbers. In the case of repulsion according to the inverse fifth power of the distance, which by mathematical luck is easy to handle and was exactly solved long ago bv Maxwell, a and 8 are both exactly zero, so that in this case our * Chapman 1 . Phil. Trans., A 216, 279 (1916). 180 KINETIC THEORY OF GASES [Chap. IV approximate formulas are exactly right; for repulsion according to higher powers a and 5 are small positive constants, rising for hard elastic spheres to their maximum values: a = 0.026, 8 = 0.010 (0.009 according to Enskog). It seems safe, therefore, to conclude that (155c) will hold with very small positive values of a and 8 for classical point- mass molecules having any type of field that is likely to occur in nature. In all cases, therefore, our own approximate values as given in (155a) and (1556) must themselves be very nearly correct. 103. Comparison with Observed Conductivities. It remains now to compare these formulas with the results of experiment. Accord- ingly, observed values of the quantities involved are given for a number of gases in the table below. Some of the viscosity values are extra- polated from data in the sources cited under the table following Sec. 86 the remainder and the other data are taken from various tables. The temperature is in all cases 0°C. Except where stated, c v is cal- culated as c p /y. V K Cv y K rjCv \ (9y - 5) Unit (10“ 7 10-3 cal cal c.g.s.) cm sec deg g deg Helium 1875 0.344 0.753 1.660 2.44 2.485 Neon 2986 0.1104 0.150 1.64 2.47 2.44 Argon 2100 0.0387 0.0763 1.67 2.42 2.51 H 2 840 0.416 2.40 1.410 2.06 1.92 n 2 1664 0.0566 0.178 1.406 1.91 1.91 O 2 1918 0.0573 0.156 1.395 1.92 1.89 H a O at 100°C 1215 0.0551 0.366 1.32 1.24 1.72 OO 2 1377 0.0340 0.151 1.31 1.64 1.70 nh 3 915 0: 0514 0.401 1.32 1.40 1.72 CH 4 , methane 1027 0.0718 0.400 1.31 1.75 1.70 C 2 H 4 , ethylene 948 0.0404 0.282 1 . 25 1.51 1.56 CaHe, ethane 854 0.0428 0.325 1.23 1.54 1.62 mh ' 17 x —-7 waaucuvny, ev = specific heat at constant rT Cp CVl Cp ~ specific heat at constant pressure, all at 0°C except in the case oi xijv/ * We note at once that the ratio K/rjc v is close to the approximate theoretxca 1 value of 2.500 for the three monatomic gases mentioned in The dlscre Pf nc y ln the case of argon, however, probably exceeds the experimental error, although in the case of the conductivity eimerimpn+^^ri 6 ^ ather lar S e - Argon seems to deserve further experimental and perhaps theoretical study. Sec. 104 ] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 181 For all other gases K/i)C V is much less than 2.5. A possible cause of its departure from this value presents itself immediately, however, in the fact that the energy of complex molecules must include other forms than mere kinetic energy of translation. This point we shall proceed at once to investigate. 104. Conduction of Heat by Complex Molecules. In the theory of conductivity as developed up to this point account has been taken only of the translatory energy of the molecules. Now if the internal energy always stood in a fixed ratio to the translatory, our final formula for thermal conductivity as expressed in terms of the specific heat by eq. (1556) or (155c) ought still to hold, since the presence of the internal energy would raise both the heat energy and the conductive flow of heat in the same proportion and would therefore have the effect of multiplying K and Cv by the same factor. Closer consideration of the processes involved in heat conduction raises doubts on this score, however; for we have found the conductivity to be greatly enhanced by the fact that those molecules which move about most actively are also the ones that carry the largest amount of translatory energy, but it is by no means certain that they will also carry more than an average share of internal energy. It might well be that the internal energy is propagated at a less rapid rate than is the translatory, and any differ- ence of this sort would obviously have the effect of lowering the con- ductivity without altering the specific heat. The question can be settled theoretically only by investigating the rapidity with which collisions produce the interchange of energy between its various forms. Not much progress has been made as yet, however, along such lines. The theory of conduction has scarcely advanced beyond a simple suggestion made by Eucken* inT913. Starting from the observation that the ratio of the rate of propaga- tion of energy in the conduction of heat to the rate of propagation of momentum in the production of viscosity might well have different values for the different types of molecular energy, Eucken suggested that we might perhaps come close to the truth if we assumed the ratio of these rates for the internal energy to have the value predicted for it by the simple theory of both phenomena, in which no allowance is made for persistence of velocities, whereas for the translatory part of the energy the more elaborate theory should hold. To develop this idea in quantitative form, let us divide the specific heat at constant volume c v into a part c V t representing changes in the translational energy and a part Cvi arising from the internal energy; we can divide K into two corresponding terms representing the respec- * Eucken, Phys. Zeita., 14 , 324 ( 1913 ). 182 KINETIC THEORY OF GASES [Chap. IV tive rates of transmission of these two kinds of energy. Then, accord- ing to Eucken, for the internal part of K eq. (138) in Sec. 94 should hold, at least approximately, with Cv replaced by Cv%j whereas for the translational part eq. (1556) should be very nearly true with Cv replaced by cvt, and for the total conductivity we should have K = (•$ cvt + Cvi)y- Now, Cvi = c v — Cvt , and we shall find later that for a perfect gas cvt-$R [cf. (203) in Sec. 142 below]; hence we can also write K = (f R + Cv)y* But for a perfect gas R, = c p — c v = (7 — 1 )cv in terms of y , the ratio of the specific heats at constant pressure and at constant volume (cf. Sec. 141). Hence, according to Eucken, K = i (9y - 5)rjc v . (157) To test this formula values of \ (9y — 5) are shown in the table of conductivities on page 180, in the column after the observed values of K/i)C V . The agreement in general is surprisingly good, in view of the crudeness of our reasoning, and seems to justify the conclusion that there must be a great deal of truth in Eucken's assumption. 105. Properties of the Conductivity. The first value of K given in eq. (155a) indicates that at a given temperature the conductivity should, like the viscosity, be independent of the density , for S vc should not vary with the frequency of the collisions so long as their individual character remains unaltered. The complete theory of Chapman and Enskog leads to the same result. This rather surprising conclusion was drawn from the theory by Maxwell in 1866 and was soon verified by Stefan (1872) and others. It cannot be expected to hold at high pressures, of course, and it must fail when the pressure becomes so low that the mean free path is com- parable with distances in which the temperature gradient varies appreciably, or with the dimensions of the gas-filled space. The range of pressure allowed by such conditions is sufficient, however, to make the fact of importance in some types of vacuum work. In a 10-cm tube containing air, for example, the pressure must be reduced to less than 0.001 mm to obtain much benefit in the way of diminished loss of heat by conduction through the air. On the other hand, all of our results predict an increase of the con- ductivity with temperature. In the case of monatomic gases, in fact, the first expression given for K in (155a) or (155c) suggests that it should vary in exactly the same way as does the viscosity, since in those expressions v 2 /T is constant and the remaining variables are the same as the ones that appear in the first expression given in (1566) for the Sec. 105] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 183 viscosity. In a monatomic gas, therefore, the thermal conductivity and the viscosity should be proportional to each other. Now the experimental study of conductivity at various temperatures has not been carried out to the same extent as has that of viscosity, because of the greater difficulty of making reliable measurements, but the existing data do indicate a fairly close parallelism between the temperature coefficients of these two quantities. The conductivity of all gases increases, therefore, like their viscosity, at least more rapidly than the square root of the absolute temperature. For example, the observations of S. Weber on neon* yielded the result that within 2 per cent its thermal conductivity is proportional to !T 0 - 7 between — 181°C and 106°C, whereas the viscosity, according to the value of n r in the table at the end of Sec. 90 above, is roughly proportional to T 0 * 67 over a similar range. Rather extensive observa- tions of conductivities at various temperatures were made by Euckenjf some of his results and some values of interesting ratios given by him are shown in the table on page 184. If the conductivity K and the viscosity tj varied in the same way with the temperature, the ratio K T rj27s/K27zriT would be unity. According to Eucken’s data this ratio shows some variation with T, even for the monatomic gas helium, but the variation in the ratio is at least much less than the total changes in K and tj themselves. The same statement holds for the ratio K/rjCv. Finally, if we turn to the conductivity of a mixture of different kinds of gas, we readily reach the conclusion that the simple theory suggests formulas which can be obtained from (135a) or (1356) in Sec. 92 merely by replacing 97 by K throughout. In the fully corrected theory of monatomic gases these formulas are replaced by very complicated expressions which we shall not write down. For a mixture of polya- tomic molecules no accurate theory exists. Perhaps all practical needs can be met by a simple quadratic expression, which for a binary mixture can be written in terms of the conductivities K h K 2 of the two constituent gases when pure, ni/n and n 2 /n being their fractional concentrations in terms of volume or moles and K 12 a new constant to be determined empirically. Experimentally it is found that in many cases the still sim pler mixture rule holds, K = K t — + K 2 —] but in other cases it does not. n n * Webjbb, Commun. Leiden Suppl., no. 42a (1918). f Euckun, Phys. Zeits., 12, 1101 (1911); 14, 323 (1913). 184 KINETIC THEORY OF GASES [Chap. IY Gas T (abs.) Kt K 27 Z 0 ill ( K/rjcv)T (K /y)Cv) 27S° 373° 1.193 0.961 He 194.6 0.788 0.983 81.6 0.441 0.930 0.93 21.0 0.155 0.843 0.84 373 1.311 1.008 Ar 194.6 0.750 1.007 90.6 0,364 1.034 1.03 194.6 0.774 0.980 1.07 h 2 81.5 0.335 0.754 1.15 21.0 0.0813 .. 0.760 1.21 373 1.264 0.996 n 2 194.6 0.758 1.003 81.6 0.322 0.965 1.04 373 1.303 1.006 0 2 194.6 0.745 0.988 1.02 81.6 0.302 0.900 (0.91) 0 p 373 1.495 1.109 194.6 - - - 0.656 0.88 1.02 CH 4 (methane) 194.6 91.5 0.702 0.314 0.912 0.924 0.99 1.01 C2H4 (ethylene) 200 0.626 0.803 C 2 H 6 (ethane) 200 0.631 0.825 Eucken’s data: K = conductivity, 77 = viscosity, cv = specific heat at constant volume; the subscript specifies the absolute temperature T. For example, mixtures of ammonia and air or of steam and air may have a thermal conductivity 5 to 10 per cent above that given by the simple mixture rule. * C. DIFFUSION 106. Diffusion. When a gas contains two or more different kinds of molecules whose relative densities vary from point to point, a process called diffusion is observed to occur in such a way as continually to diminish the inequalities of composition. The explanation of this phenomenon by kinetic theory is immediately obvious : in consequence * Gruss and Schmick, Wiss. Abh. Siemens- Konzern, 7, 202 (1928). Sec. 107] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 185 of thermal agitation more molecules of a given kind travel from regions rich in that kind to regions of scarcity, than travel in the opposite direction, and this process tends to smooth out inequalities of distribu- tion. The net flow of each kind of molecule will obviously occur in the direction of its negative density gradient, i.e., in the direction in which the density decreases most rapidly. In preparation for a theoretical investigation of this phenomenon we shall first review briefly the customary manner of describing it in quantitative terms. We shall confine ourselves, however, to binary mixtures containing just two kinds of molecules. To express the rate of diffusion it might be thought that, in general, two coefficients would be necessary, each having reference to the motion of one constituent; for example, in a mixture of hydrogen and carbon dioxide, the hydrogen would be expected to diffuse much more rapidly through the carbon dioxide than does the latter through the hydrogen because the hydrogen molecules have thermal velocities more than four times greater. From one point of view this expectation is perfectly correct. If, however, nothing more than these two processes were involved in the phenom- enon of diffusion, a greater volume of hydrogen would be transferred in one direction than of carbon dioxide in the opposite, and the gas would thereby be caused to move bodily toward one side. The experimental conditions under which diffusion is studied usually preclude such bodily motion of the whole gas; and in other cases it is more convenient to treat such a mass motion as a separate phenomenon, to be handled by the usual methods of hydrodynamics. Accordingly, a pure case of gaseous diffusion is arbitrarily defined to be one in which any tendency of either constituent to move with exces- sive rapidity toward one side is offset by a mass current of the whole in the opposite direction, this current being of such a magnitude that the total net transfer of gas, as measured in terms of volume, is zero. The mass velocity of the gas as a whole is then considered to be zero. The transfer of volume is thus balanced out by definition, but there will usually be a net transfer of mass in one direction. In defining the mass velocity of a diffusing gas we are compelled to choose between a criterion in terms of volume and one in terms of mass, and the advan- tage seems to lie with the former. 107. The Coefficient of Diffusion. In perfect gases, to which our attention will be confined, volume and number of molecules are pro- portional to each other by Avogadro's law, so that zero transfer of volume means zero net transfer of molecules by number. Accordingly it is convenient to define the coefficient of diffusipn for a mixture of two gases in the following way. 186 KINETIC THEORY OF GASES [Chap. IV Let the numbers of the two kinds of molecules per unit volume be Wl _ and respectively, and the total number in unit volume' l ~ n l + n *- Then for equilibrium the pressure, and hence n must be uniform. Accordingly, if the 2-axis be drawn in the direction of the concentration gradient, we must have JL dpi = _±d fis dx dx mi dx m 2 dx' Vm f enSitieS in g ? mS of the two cons tituent gases and in "" “ A SimpIe ° f tMs 8 ° rt “ ex ^rbite<i Now let r l; r 2 denote the (algebraic) net number of molecules of n ea °h kind that pass per second toward +z across unit area of a macroscopic- ally small plane drawn perpendicular to the concentration gradient, and lot the conditions be such that r 2 = —r u _ — ? f th ® gas is not P erf ect it would per- 48 -~ Concentration gradients. na P s be more convenient to require that the two directions should V. lh \ ralio « f the riunib '' rs transferred in volume on the two sides of the”!!™ CtIUal increases in of experimental importance however 6 ) Then J StinCt “ n is scarceI y *■ " f ° r * dx ‘ d ? ^2 dx ( 158 ) JZSZSS7 , :* d T iti “ r *> r - « •»» we understand n, to denote coneentre)i'o‘“ d “ d condi . tiona I )r0 tided The numbers of grams of the two on +•+ DS ex P ressed m those terms, area per second, on the other hand transferred across unit and in genera! iese “* **•* ***** fr. nsuaUy employed thesc equations are not from the current densities Y, which are n f , , ntage . ous i to P ass at once differential equation containing the cono / dlrectly ob «ervable, to a of doing this is similar to S emo Wd ° n T The “othod differential equation for nf. In the simnle f ^ V- ^ obta ' nbl H the tions vary only in the direction of 2 ] e tr 2 ? the concentr a- sent the values of the stream densitv of th* a ^ + ( dT ^ dz ') dx repre- planes perpendicular to the 2-axis * first gas at points on two * axis and separated by a distance dx. Sac. 107] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 187 Then this kind of gas is accumulating between the planes at a rate equal to the negative difference between these two expressions, or at the rate molecules per second per unit area of the planes, by (158) . This rate of accumulation is also represented, however, by (dn x /dt) dx, the volume of the space included between the planes being dx for each unit of the area. Hence we have dfti _ ^ ( T) d n A dt ~dx\ dxj’ or, if H is constant or practically so —i — n d 2 fti dt ~ U ~dR r When n x varies in three dimensions, the latter equation takes the more general form dni _ r, / d 2 Wi d 2 n x d 2 n x \ dt "\dx* + dy 2 + ~w)’ In all cases there exists likewise a similar equation for n 2 . The diffu- sion coefficient H is often defined as the coefficient that occurs in these differential equations, under such conditions that the net transfer of molecules in any direction vanishes, just as the thermal conductivity is sometimes defined as the constant that occurs in the analogous and mathematically identical equation for the temperature in an unequally heated body. Problems. 1. Show that if Hi denotes the coefficient of single diffusion of one gas through another and D 2 that of the second through the first (i.e., —Hi is the number of molecules of the first gas crossing unit area per second under such circumstances that on the whole none of the second gas crosses at all), then Hi = -H, D 2 — — D. n 2 n x 2. A slender jar has some alcohol in the bottom while across the top a light breeze blows gently enough so as not to disturb the air inside the jar but rapidly enough to keep the density of alcohol vapor practically zero at the top. The temperature is 40°C, the pressure is 1 atmosphere. Find the rate at which the surface of the alcohol will 188 KINETIC THEORY OF GASES [Chap. XV sink in consequence of evaporation and diffusion of the vapor upward, when its surface lies 20 cm below the top of the jar. Assume the air to be saturated just over the alcohol. Vapor pressure of alcohol at 40°C, 134 mm; density of the saturated alcohol vapor, 3.15 X 10““ 4 g/cc, of liquid alcohol, 0.772 g/cc; coefficient of diffusion, D, of alcohol vapor and air through each other at 40°C, 0.137 in c.g.s. units (variation with concentration to be neglected). Ans. The stream density of alcohol vapor is uniform and given by F = (^y) * og ~~ w ^ ere n i = molecular density of the alcohol vapor, nio its value just above the liquid, n — density of the mixture of air and vapor, h = 20 cm; ni sa — ^1 — —^j 1 * j, x = height above the surface. The alcohol sinks 2.7 mm a day. 108. Simple Theory of Diffusion. Just as we obtained a simple theory of heat conduction by making the proper substitution in the treatment of viscosity as given in Sec. 81, so by another suitable modi- fication we can obtain at once a simple theory of diffusion. For this purpose we assume that in a mixture of two gases equilib- rium exists in all respects, except that the densities of the two constitu- ents vary in a certain direction, which we take to be that of the rc-axis. Then if ni denotes the density of the first constituent at a surface S drawn perpendicular to the z-axis (cf. Fig. 36), its density in an element of volume dr located at a small distance x from S will be ni + x dni/dx and thus greater than the density at S in the ratio 1 + {x/ni) (dni/dx). The number of molecules of this constituent gas that collide in dr and thereafter cross S will accordingly be increased, as a result of the density gradient, in the same ratio, and the total number crossing unit area of S per second toward +x will be, not | n x vi as given by eq. (72a) in Sec. 37, but 1 + x dni ni dx _ 1 _ . 1 _ _ dni Vi denoting as usual the mean molecular speed for this kind of molecule and x standing for the average value of x at their last point of collision. We then insert x = — § L h as in Sec. 81, Li being the mean free path of the molecules of the first kind in the mixed gas, and then subtract the corresponding expression for the molecular stream that crosses toward —x, in which x — +§ L%; the result is an expression analogous to eq. (118) in Sec. 81 for the net molecular stream density of the first kind of gas: rj = § v x x or 'Sac. 108] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 189 TV _ _ 1 , r dny Ti ~ i ViLi w For the second kind of molecule we find similarly r' 1 2 1 - r dn 2 - r ^ irx ' and it must be remembered that, as noted above, uniformity of pressure requires that dn 2 /dx = —drii/dx. Since, however, T'* ^ r£ in general, to have the standard conditions * for pure diffusion we must now superpose upon this molecular process the mass motion of the whole gas that was mentioned in the last section. The mass velocity v 0x must be such that there will be no net flow of molecules across S, or such that U + TJ + nv ox — 0, where n = n x + n 2 and denotes the total number of molecules in unit volume, the term nv 0x representing the flow of molecules due to the mass motion. Using previous equations, we thus find Vo* = ^ (v x Li - v%Li) ^ (5i L* - v x L x ) The total numbers of molecules of the separate kinds crossing unit area of S per second, taken positive when they cross toward +x, are then, since r.-r 1 + Wfc --‘(a Wl + 5M.)§. r 2 = ri + n 2 v o* = v,u + - v x l}\ ™ 2 ; y \ W J CtX and comparing these equations with (158) on page 186, we have for the coefficient of diffusion D ~ l (? Siii + 5 5aL2 ) ■ (^ 9 ) This is often called Meyer's formula for the diffusion coefficient; in his book, however, he gives the factor as tt/8 instead of }{ after includ- ing a correction for Maxwell's law. * The formula predicts substantial variation of the diffusion coefficient with the composition of the gas, * 0. E, Meyer, “ Kinetic Theory of Gases” (transl. pub. by Longmans. Green, 1899). 190 KINETIC THEORY OF GASES [Chap. IV which is represented by the relative values of n i and n 2 , for the quanti- ties V 1 L 1 and vJL 2 are likely to be quite different if the molecular masses differ greatly because lighter molecules will both move faster and tend to have longer free paths. Experiment shows, on the contrary, that the coefficient of diffusion is almost independent of composition. The simple free-path theory thus fails badly in this instance. Accordingly we shall pass at once without further comment to an approximate treatment of diffusion by the Maxwell-Chapman method. 109. Approximate Coefficient of Diffusion for Spherically Sym- metrical Molecules. The calculation of the coefficient of diffusion by the method of correcting the velocity distribution function parallels so closely the calculation already made for thermal conductivity in Secs. 95 to 101 that we may save space here by leaning heavily upon that treatment; it will suffice, in fact, merely to indicate the differences. The quantity which determines the net flow of the molecules themselves in one direction, in the same way as 2vi_v 2 determines the flow of their translatory energy, is obviously simply v± denoting the component of the molecular velocity v perpendicular to the plane across which the flow is being calculated. Hence, if we take the z-axis in the direction of the composition gradient in a binary mixture of gases, the stream densities of the two kinds of molecules can be written, as an amplification of (158), Tj = — D — = r 2 = ~D^tz ~ (160a, b ) the sums extending over all molecules of the appropriate kind in unit volume. We proceed to study these sums. Resolving the distribution function for each kind of molecule into a maxwellian part plus a small correction term, we write /l(Vi) = jfoi + fal, fox = Ai 6“^ i2ui2 , fziyz) == /02 +/a 2 , /02 = A 2 6 ~^ V2 \ We then obtain differential equations for f al and / fi2 in analogy with (145) and (146) in Sec. 95, of which the first is djnrfsi) dt = Vi* w; (wi/oi) coll ox Aiv lx e~^ drii dx’ (161) since we assume the temperature to be constant and the gas to be at rest, so that only Ui and n 2 vary with x . The right-hand member of this equation then suggests as approximate forms for the correction terms fal = CiVi x e~^\ fa 2 = C 2 V 2 ,e^\ Sec. 109 ] VISCOSITY, THERMAL CONDUCTION, DIFFUSION as a result of which 191 ‘J 2th. = niCi v^-^d , d = V 7T* 4 JliCl 2 /9J S«2x = ntCz pi ' To prevent net flow of the gas, we must have Evi x ■+ Ev 2x = 0. Hence Ci and C 2 must satisfy the relation Ci _ n iffl Ci n 2 Pi (162) Finally, multiplying the differential equation (161) through by Vy x and integrating over Vi, we have as the condition for a steady state, in analogy with (149), <163) since Ai = jSJ/V 4 . Here D2v lx represents the rate of change of by collisions; and there is a corresponding equation for DSv 2 ». Turning then to the calculation of D'Lv lx , we first make the interest- ing observation that collisions of the molecules of the first kind with each other can have no effect whatever upon Swi* because they leave unaltered the total ^-component of the momentum of these molecules, which is miSvia,. We can thus say that each kind of gas is hindered in its diffusing motion only by the other gas. This observation was made long ago by Stefan, who developed upon this basis a theory which predicted no variation of the diffusion coefficient with composition and so agreed much better with the facts than did Meyer's theory. Maxwell adopted Stefan's theory but seems to have failed to explain its true basis in such a way as to make it generally understood. The formulas for v x , v 2 , Vi, V 2 in Sec. 98 are, accordingly, to be applied only to collisions between a molecule of the first kind and one of the second; and we must now retain in them ju x and jjl 2 with the values given by (150). One finds at once that in a single collision the change in averaged over <p 9 is A2z; lx = Vix — Vi x = x — w x ). Writing then, as in Sec. 99, W = w cos 9 + r, we note next that the average of r will vanish as before because of the symmetry of the scattering process, and accordingly the average of A%v ±x per collision will be simply A2v lx = — /z 2 w*(l — cos 6) in terms of the angle d through which the relative velocity w is turned. 192 KINETIC THEORY OF GASES [Chap. IV In summing over all collisions between unlike molecules, then, we can omit those that occur between the maxwellian distributions foi and/ 02 , which because of their symmetry cannot affect Stoi*, and also those between / s i and/ 32 , whose effect must be of the second order in the concentration gradient and therefore negligible; there remain thus only collisions between f 8l and / 02 and between / 0 1 and f 8 2 . For the contribution of the former to the rate of change of %v Xx we find easily, in analogy with (152a) and (1526) in Sec. 100, Di2v lx = 2irnin<L J* f fj w ( &2v lx ) G{w f 0) f a i(vi) f 02 (^ 2 ) sin 0 dd dn x d> k 2 , or Dihvix = ~~2wnin2fi2CiA2j* j* J* j* j* J j* Q w x w(u x + &2W X ) G(w , $)e~W vli ~-W V2 *(l — cos 0) sin 0 dd du x du y du z dw x dw v dw t . To express the exponent of e in this expression in terms of u and w, we note first that, by (56) in Sec. 28 and (150) in Sec. 98, P<2p\ — M lP\ mim 2 2(mi + m 2 )fcT* (164) whence, by (151a) in Sec. 98, PM + PM = 08 ! + Pl)u* + {API + API)™ 2 , and here, since by (164) /3| = ^Pl/vi and by (150) m + /x 2 = 1, ei + p 1 = ^ M Mi Thus g— /9l 2 tu2— /32 2 V2 J — g~jSx a (u 2 //Xl+M2W 2 ) ^ The integral now can be reduced easily; the term in u x contributes nothing by symmetry, w x is handled in terms of polars as in reducing (1526), and (164) is used. The result is: where D^vix 4 3 7rnift 2 VM2Ch S d Sd = 2 tAPiJ* 0 J G(w,-d)(l — cos 0) sin 0 d$~^w*e~** lh ** dw (165a) or Sa - H s ) a - c “ •> “ 1 ’• "] x'tr* dx. (1656) 1 Sec. 109] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 193 The effect of /oi colliding with / s2 will clearly be represented by the' same expressions with C 2 A 1 replacing C 1 A 2 and v^ x or u x — ix\W x replacing t>i» = u x + mw x ; hence it will simply be equal to D iStfi* multiplied by CjA iau _ wig| ftiMi _ ni C1A2M2 /31a* 2 Wa by (162) and (164). The total value of D Svj* is thus times the value of D iS»i* or D'Zvix = We can eliminate Ci now by dividing this expression by the value of Lvix as found in the beginning, which gives D2vu 2«i» 8 nVM2 3 \Arr^i Sd — — g V mnviSa in terms of the mean speed v\ =' 2/(\/V/3i). It is the presence of the factor V/ia in this expression that prevents the occurrence of a great variation in the diffusion coefficient with composition, such as was ♦ predicted by Meyer’s free-path theory in the case of molecules of very unequal masses. If mi/m 2 is very large the heavy molecules sweep the others out of their way and tend to keep on going, thereby building up sizable amounts of Svi* in spite of their more sluggish motion. Finally, if we solve the last equation for and, after substituting for D2v lx from (163), insert the value so found for Y*v lx in (160a), we obtain as our approximate value of the coefficient of diffusion 8 y/ fxznviPlSd or, after substituting 1/fij = «|/4, n — n _ 3 tt vi 3tt h 32 y/jj^nSd 32 y/^nSd (166) since v\y / mi = v^y/ m 2 . Here mi — mi / (mi + m 2 ), M2 — ■ m t/ and n is the total number of molecules per unit volume. A formula equivalent to (166) was obtained by Langevin in 1905. According to our assumptions the formula is restricted to low densities and to small values of the composition gradient. By inserting the 194 KINETIC THEORY OF GASES [Chap. IV value of and the value of V\ from (66a) in Sec. 30 we can also write it in the more symmetrical form D u -i Jl i [»L±«! fcr T , _ 8\2n/SiL Tfi\Tri'i J r * t T ■ (I66 “ ) Here h = 1.381 X 10” 16 or R M = 83.15 X 10 6 in c.g.s. units and M h M 2 are the molecular weights. The new molecular constant Sd that appears in the formula for D and is defined in terms of the scattering coefficient G for any relative speed by eq. (165a) or (165b) obviously functions as an equivalent mutual cross section of these two kinds of molecules for diffusion. We have so defined it that it reduces in the case of classical spheres of diameters <r x and <r 2 to the usual cross section, S d = S = 4 [cf. (llOd) in Sec. 70]. 110. Self -diffusion. The case in which Mi = M 2 is of special interest. The formula for this case can be applied to the diffusion of a gas through itself. Such a phenomenon could be realized in the labora- tory if we could mark in some way part of the molecules of a homo- geneous gas and then observe the diffusion of the marked ones among the unmarked. Now according to present atomic theory such mark- ing is essentially impossible unless the physical nature of the molecules is changed in some way; accordingly, strict self-diffusion has in reality become a notion devoid of physical meaning. There are, however, several ways of modifying molecules without appreciably affecting their outer fields, for example, by bombarding them with neutrons and thereby altering the nucleus of an atom without changing the nuclear charge, and the formula for self-diffusion should then hold for the diffusion of such modified molecules among the normal ones. Ordinary gaseous ions behave, of course, like a different gas, because of their strong fields. If all molecules have the same mass, n = hence we can write for the coefficient of self-diffusion in a homogeneous gas, as a first approxi- mation, from (166), n = v __ 6 S vc r\ 16\/2 n Sd 5 Sd p (167 a) where P - nm and represents the density in grams, and the second of the two expressions given is got by comparing the first with the cor- Sec. Ill] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 195 responding approximate value of the viscosity as given in eq. (1566) in Sec. 101. For the special case of hard spheres S d /S vc = 1, both cross sections reducing to S 7r(cri + <r 2 ) •; in the case of repulsion according to the inverse fifth power of the distance Maxwell's results show that S d /S VC 0. 778. The forms of (167a) for these two types of molecules are,* respectively: Du = 1.200 — ; D n = 1.543 2. (1676, c) P P 111; The Corrected Diffusion Coefficient. According to the result of our approximate calculation as expressed in eq. (166) the coefficient of diffusion in a mixture of two gases should be entirely independent of the proportions of the mixture, for ui and n 2 do not occur separately in the formula but only the combination fti + n 2 — n . In this feature our new result goes to the opposite extreme from that of the simple theory as developed in See. 108 and agrees with the formulas obtained by Stefan and Maxwell. . When, however, we turn to the fully corrected theory as worked out by Chapmanf and by Enskog,J we find that in general there really should be a small variation of D u with composition. There is none, however, if the molecular masses are equal; and this fact facilitates the application of the formula to the phenomenon of self-diffusion, for it makes the coefficient the same whether we contemplate the diffusion of a large group or a small one among the rest. Furthermore, in the mathematically simple case of gases in which two unlike molecules repel as the inverse fifth power of their distance apart, which was solved exactly by Maxwell, Du again comes out independent of the composi- tion of the mixture; and in this latter case our values of Du and Du as stated in eqs. (166) and (167a) agree with Maxwell's formulas and hence are actually exact. For the general case it is convenient to write these equations in the slightly more general forms, Dn = (1 + Xu) Uf ^ kTj , (168a) *The coefficient is given as 1.504 in Jeans , “Dynamical Theory of Gases/’ but the formulas and numbers there given, and also those in Maxwell’s paper, “Scientific Papers” vol. II, p. 26, lead to the value 1.543. t Chapman, Phil Trans., 217A, 115 (1918). t Enskog, “Kinetisehe Theorie der Vorgange in mftssig verdtinnten Gasen,” Dissertation, Upsala. 1917. m KINETIC THEORY OP tiA'SES [Chap. IV Du == (1 + Xu) 3t V __ ^ rj 16 \^nS~d^ ? (1686) in the first of which Sd refers as before solely to collisions between unlike molecules. The result 'of the corrected theory as regards D n can then be stated : at once by saying that Xu is a small positive number, which vanishes for the inverse-fifth-power law of force and reaches a maximum for hard elastic spheres; this maximum value Chapman estimates at 0.017 within a thousandth or so. The quantity X i2 is likewise positive and is also, of course, zero for the inverse fifth power, but it can rise to a maximum value of 32/9x — 1 = 0.132 for hard spheres of extremely unequal mass, a limiting case in which, as it happens, the modified distribution function can actually be found by solving the Boltzmann differential equation for nf. In other cases than these two, X x2 varies somewhat with composition and usually in such a direction that the rate of diffusion increases as the lighter gas is made scarcer. The formulas exhibiting this variation are extremely complicated, however, and none of them will be written down here. The theoretical variation in most practical cases is small, and if a formula is needed for practical use, and a linear one will not answer, an empirical one of quadratic type such as D 12 = a n n\ + dunin* + will probably be found to meet all requirements. The coefficient C in (1686) is shown by Chapman to have a magni- tude lying between the value 1.200 for hard spheres, which just happens to agree to this number of places with the approximate value that we obtained above, and the value 1.543 for inverse-fifth-power repulsion. 112. Experiments on the Variation with Composition. The con- flict between Meyer’s theory, on the one hand, predicting large varia- tion in the diffusion coefficient with changes in the relative concentration of the mixed gases, and the Maxwell-Chapman-Enskog theory, on the other, predicting no change at all, stimulated a series of careful observa- tions at Halle designed to settle the question experimentally.* The method of experiment as finally perfected was to fill one half of a long uniform vertical tube with a pure gas and the other half with a mixture of this gas and another one, the denser of the two resulting masses of gas being put below the other, and then to open a stopcock and allow diffu- sion to go on for a known number of hours; the composition of that gas which had been pure at the start was then determined, either by chem- * Cf. Lonitjs, Ann. Physik , 29, 664 (1929). Sec. 113] VISCOSITY , THERMAL CONDUCTION, DIFFUSION 197 ical analysis or by weighing it in order to find its density. With this arrangement, when the stopcock is opened, the concentration at the middle of the tube promptly changes to a value halfway between the two initial concentrations and thereafter remains at this value; the observational result, interpreted with the help of the usual one- dimensional theory of diffusion, thus furnishes a value of D correspond- ing to this intermediate value of the concentration. These experiments demonstrated beyond a doubt that some varia- tion of D with composition does occur, but the variation is always small and is hard to observe unless the molecular masses are very unequal. The most interesting of the results obtained are shown in the table below; the observer is named in each case just under the names of the gases, and the fractional part which the heavier gas formed of the mixture in each case is given in the second column. The temperature was 15°C, the pressure approximately atmospheric. The values given under Di are the original ones, expressed in terms of meters and hours; under D are given corresponding values in terms of centimeters and seconds, obtained by dividing the original numbers by 0.36. In his last paper Chapman made theoretical calculations from his formulas to fit these experimental cases, adjusting the mean value of D arbitrarily in each case to fit the observed mean; values of the molecular diameters were calculated from the known viscosities. His theoretical values are given in the last column. The agreement is probably to be considered satisfactory and thus serves to confirm the theory. Diffusion Data Gases ni D Di obs. Dr theor. n\ H- n 2 Oa- 1"1 2 (Deutsch) 0.25 0.50 0.75 0.767 0.778 0.803 0.276 0.280 0.289 0.276 0.282 0.289 C0 2 -H 2 (Deutsch) 0.25 0.50 0,75 ' 0.592 0.606 0.633 0.213 0.218 0.228 0.212 0.222 0.226 0.273 0.678 0.244 0.248 Ar-He 0.315 0.694 0.250 0.250 (Lonius) 0.677 0.711 0.256 0.257 0.763 0.731 0.263 0.259 113. Diffusion at Various Pressures and Temperatures. The theoretical results indicate that at a given temperature the coefficient of 198 KINETIC THEORY OF GASES [Chap. IV diffusion should be inversely 'proportional to the total density and so proportional to the mean free path, which is as we should expect. This is shown clearly by our approximate formulas, in which the molecular density n occurs in the denominator and S d should depend only on the violence of the collisions but not on their frequency; and Chapman showed that the same conclusion should hold with precision. This property of the diffusion coefficient was verified in certain cases by Loschmidt in 1870. Accepting it as universally valid at sufficiently low densities, we can write D = F(T)/p f where p is the pressure and F(T) is a function of the absolute temperature T alone. At constant pressure , on the other hand, our formulas (166) and (167a) predict that D « since v oc T^ 2 and at constant pressure 1/n oc T, provided S d is independent of T ) as it would be for hard spheres. According to statements made in tables of physical data, the observed rate of variation is greater than this, D being roughly pro- portional to T n where n is around 1.75 for the more permanent gases but around 2 for the more condensable ones. The latter values of n are roughly greater by unity than the corresponding values for the viscosity rj of the same gases, so that for these gases D/rjT or pD/rj (p = density) must be nearly independent of the temperature at constant pressure. Such a relation is suggested by our approximate formulas; it follows, for instance, from eq. (167a), expressing the coefficient of self-diffusion, provided the ratio S VD /S d is independent of temperature, or at once from (167 b) or (167c) in the case of hard spheres or the inverse-fifth-power law of force. Further experiments to test the variation of pD/rj with temperature would be of interest. An increase of D more rapid than as T&, resulting from a decrease in S d with rising temperature, is just what we should expect in accordance with the considerations brought forward above in order to explain the rapid variation of the viscosity. For molecules repelling as the inverse sth power of the distance the method of similitude can be employed, as was done in dealing with the viscosity in Sec. 89. In that section modified motions were con- templated in which all lengths were changed in the ratio X and all times in the ratio L ±j («:) t = X 2 . It was then shown that the absolute temperature T is changed in the ratio (T 7 :) X 1 "*. 199 Sec. 114] VISCOSITY, THERMAL CONDUCTION, DIFFUSION Now D, being the ratio of molecular flow over unit area per second to a molecular density gradient, will be changed in the ratio (l/AV)/([l/\3]/X) or 3 - a m x 2 . But we saw that we could write D = pp ■ Here, the masses being unaltered, p is changed in the ratio 1/X 3 ; hence F{T) or pD must be changed in the ratio X«-*)/2 X s 3 8 2~~ y or like T n , where n = 3 5 20T=T)‘ Accordingly, D cc - T n , p i This is the same value of n that is given in eq. 128 in Sec. 89 for the variation of the viscosity with temperature for molecules repelling as the inverse sth power of the distance. Accordingly, for this type of molecule D « v /p, or pD/i? is constant, as was suggested by our general formulas. As an alternative, there is Sutherland’s theory in which the mole- cules are assumed to be hard elastic spheres surrounded by weak attractive fields. This theory we found to have considerable success in dealing with the viscosity. Sutherland showed that it gives for the coefficient of diffusion D _ oTW 1 + I (C12/T) in which a and Cn are constants. This equation he found to hold satisfactorily in certain cases. It must be recognized, however, that the diffusion coefficient has not yet been investigated over a sufficiently wide range of temperatures to make possible an adequate test of such formulas as these, and it seems pretty certain that when accurate data are obtained all formulas of such simple type will be found to fail for diffusion just as they did for the viscosity. 114. Numerical Values of the Diffusion Coefficient. Our formulas for the diffusion coefficient as expressed in eq. (166) or (168a) contain still another new molecular characteristic, the equivalent mutual 200 KINETIC THEORY OF GASES [Chap. IV collision cross section for diffusion denoted by S d and defined in terms of the scattering coefficient by (165a) or (1656). For this reason these formulas cannot be tested very exactly in terms of other known mag- nitudes but serve rather to enable us to calculate values of S d from diffusion data. In general is not quite the same as the quantity S vc which controls viscosity and heat conduction and which is defined in terms of the scattering coefficient by eq. (153) in Sec. 100. These two quantities, however, must be at least of the same order of magni- tude; in the case of hard spheres moving classically they are, of course, actually identical and equal to the ordinary mutual cross section. Instead of calculating values of S* itself from the formulas it is probably more convenient to calculate for reference what may be called the equivalent classical mean hard-spherical diameter for the two kinds of molecules in question, defined as a d = ( S d /r ) this is the mean diameter of hard spheres which, moving classically, would exhibit the same diffusion. If we had available for this purpose data on self- diffusion, such values of a d might then be compared directly with the values of cr calculated previously from viscosity data. Unfortunately, however, very few data exist on anything approximating to self- diffusion. An alternative idea would be to calculate values of an equivalent a vc from data on the viscosity of the diffusing mixtures themselves for comparison with their values of <r d . The significance of such a comparison would not be too clear, however, for the diffusion is controlled almost entirely by interaction between dissimilar molecules, whereas the viscosity is influenced also in part by interaction between similar molecules, and neither theory nor experiment has progressed to a point where these two kinds of influence upon the viscosity can be estimated separately from the data with much accuracy. In view of this situation we shall, for the purpose of testing our formulas, merely calculate S d and a d from Di% for several mixtures of gases by means of eq. (166a), in which n = no — and is known, omitting for this purpose the small and uncertain correction term X 12 which is added in (168a). The values of no, To, and Rm are given in eqs. (22), (19), and (20a). Values of c r d — ( S d /ir )** calculated in this way for a number of pairs of gases at 15°C are shown in the following table. For comparison with them, values of cr av - (<r x + <r 2 )/ 2 are also shown for the same pairs of gases in terms of the equivalent hard- sphere diameters for viscosity as given under “<r” in the table fol- lowing Sec. 86 above. The table contains also observed values of n mthe approximately valid formula, D = D 0 (T/To) n (p 0 /p), where p — pressure. The values placed in parentheses are less reliable. Sec. 115] VISCOSITY, THERMAL CONDUCTION , DIFFUSION 201 Diffusion Diameters Gases n <Td (unit, 1( j ^av )“ 8 cm.) Gases n <?d (unit, 1< | ^av )~ 8 cm.) H2-CH4 (1.75) 3 . 15 3.44 He-Ar 1.75 3.04 2.91 lls-HsO (1.75) 2.86 3.67 CH 4 -CO 2 (1.75) 3.97 4.36 Hj-Ne 1.71 3.02 2.66 H 2 O-air (1.75) 3.36 4.16 H 2 -C 2 H 6 (1.75) 3.62 4.02 H2O-CO2 (2.0) 4.06 4.60 HrO, 3.18 o 2 -n 2 1.80 3.40 3.68 H 2 -C 0 2 189 3.66 Air- CO 2 (2.0) 3.76 4.16 A glance at the table shows that there is a rough agreement between the values of Cd and <r av , probably as close an agreement as ought to be expected in view of the theoretical difference between the two effective cross sections. In most cases, however, c r av is somewhat the larger. Such a difference would be expected in view of the difference between the ways in which the angle of deflection 6 enters into the expressions for Sd and S <uc as given in eqs. (165a), (1656), and (153). The factor (1 — cos 6) in the integral causes Sd to depend more upon large-angle deflections than upon small ones, whereas the corresponding factor sin 2 6 in the case of S vc is symmetrical about d = v/2, and with any reasonable type of molecular field except the rigid-body type small deflections must predominate over large. Thus we should expect to find Sd < S v<1 . The physical reason for the difference can be said to be this, that momentum and energy can be transferred in a collision from one molecule to another, and the effect of deflections respectively greater and smaller than t/2 may therefore be much the same, whereas the inner nature of a molecule cannot be transferred, and consequently, while a small deflection will retard diffusion but little, a deflection near 180° actually reverses the diffusing motion. The general situation is accordingly very satisfactory for the theory. 116. Forced Diffusion. Up to this point wc have dealt with con- centration gradients as the sole cause of diffusion. Any influence, however, which affects the rate of flow of molecules across a surface might conceivably cause a differential flow that would constitute a phenomenon essentially of the same nature. A case that occurs very corpmonly in vacuum-tube work is that of the drifting motion caused by forces acting selectively upon certain molecules, such as the forces exerted by an electric field upon ions. This case is readily connected with the process of ordinary diffusion by the following argument and may be regarded as a sort of forced diffusion. 202 KINETIC THEORY OF GASES [Chap. I\l Let each molecule of a certain sort be acted upon by a steadv externa! force F, and let there be n y such molecules per unit volume wrth the distribution function *(*). Let the gas be uniform il com! position, and at rest and in equilibrium except for the forces F Hiff J he ? f the !: aX1 * ^ taken in the direction of F, the Boltzmann differential equation (87) in Sec. 51, applied to „„ becomes »i* 4- (wj/i) + — jl to V1 ' ^ mi dv lx Letus fio = = Aie-P^^+nvH-vu*) Tien fn tL*w “ dep “ dent of 1 **»»» temperature is uniform. ■Lhen m the last equation we can, as in that section, replace A bv A xn the collision term on the right, and by / 10 elsewhere; accordingly 7 ‘ dvi x ~ ~^fiiV lx nif w and, inserting ft = mi/2kT from (56) in Sec. 28, we have Vlx (^ ~ w) Aie ~^ = [l (ni/i.)] oon . This equation has exactly the same form as eq. (161) in Sec 109 which p dni HiF dx IcT' We may conclude that a uniform steady force F acting on each one of a group of molecules of density m per unit volume has the t “ of mSude aUSe dlffUS1 ° n ^ haS a d6nSity gradient of those molecules ( dnA _ _Wi F dx / eauiv kT’ (169a) tor e 5ratam SUal ““ 00 ““ 1 “ f ° r °" 6 molecule “ d T «» absolute The remainder of the theory of diffusion as we have developed it 11 now apply and we can utilize its results to obtain an expression for the mean rate of drift of the selected molecules. P “ f ° r Sec. 115] VISCOSITY, THERMAL CONDUCTION, DIFFUSION 203 One other circumstance must first receive attention, however. If other forces were acting in the opposite direction upon other molecules in such a way as to make the total force equal to zero, we could assume that dni/dx = 0. As it is, the total force niF per unit volume can only be balanced by a force arising from a pressure gradient dp/dx of such magnitude that (cf. the reasoning in Sec. 45 ) ; since the pressure equals nkT in terms of the total molecular density n, this requires a gradient — = dx kT dx from which we find, since the composition is assumed uniform, dni _ n\dn _ _1_ dp _ n \ F dx n dx nkT dx ~ ~n W The quantity P defined above thus becomes P = ElK. n x F n x n^ F nkT kT ~ ~ W where n, 2 = n - n x and represents the density of the remaining mole- cules. We find also that dn2 _ ^2 dn — 'ETE3 ^ _ p dx n dx ~ n kT ~ ~ ' The mathematical equations for both kinds of molecules thus take on the same form as in Sec. 109, where the two gradients were equal and opposite. Accordingly, we obtain the stream density of the molecules on which F is acting by substituting P for dn x /dx in (160a), which gives for it Ti = D n x ni F n W (1696) D being the ordinary coefficient of diffusion for molecules of the selected sort diffusing through the gas. The average velocity of drift u of the selected molecules under the action of the force F is then r i divided by their density n\ or u = Ui D 'nkT F. (170 a) The ratio u/F, representing the drift velocity with a unit force acting on each molecule, might be called the dynamical mobility U f of these 204 KINETIC THEORY OF CASES [Chap. IV molecules in the gas; it satisfies the relation, £ = a. kT. (170b) Uf n 2 In most practical cases, however, n i is relatively very small; then the last two equations take on the simpler approximate forms u ~ kT F ’ U f kT ' (170 c, d) Even ordinary diffusion can be interpreted as a kind of forced diffusion, the equivalent force per molecule being from (169a) 1 dni ni dx The total equivalent force on the n i molecules in unit volume is then ni F d = — kT^ = -“S where pi = nJcT and represents the partial CbX CtX pressure due to these molecules. Diffusing molecules can thus be regarded as being driven by their partial-pressure gradients, a view of the process that is often illuminating. 116. Thermal Diffusion. Another case of diffusion arising from causes other than a concentration gradient was predicted theoretically by Chapman (1916) and later revealed by experiment. He showed that a temperature gradient in a mixed gas might well give rise to a slight flow of one constituent relative to the gas as a whole, producing an effect that he called “thermal diffusion.”* Our approximate theory of heat conduction does not lead to this conclusion, since the approximate correction term that we introduced into the velocity distribution as given in eq. (1486) in Sec. 97 was such as to give rise to no net flow of molecules. The more complete analysis of heat conduction leads, however, to the contrary conclusion that collisions between different types of molecules in a mixed gas should usually give rise to a small differential diffusion. The effect vanishes for Maxwell's molecules repelling as r~ 6 , and for this reason was not discovered by him. It possesses a peculiar interest for theory as being one of the very few phenomena in gases which depend for their bare existence upon the particular form of the law of force; experiments upon thermal diffusion might be expected for this reason to throw a partic- ularly valuable light upon the nature of the molecular forces. The theoretical formulas obtained by Chapman are again too complicated to quote, so only a few illustrative figures will be men- * Chapman, Phil Trans., 217A, 115 (1918). Sec. 116] VISCOSITY , THERMAL CONDUCTION , DIFFUSION 205 tioned. Let k T denote the relative effectiveness of a temperature gradient in promoting diffusion as compared with a composition gradient, being defined as the ratio of the two resulting diffusive flows when (1 /T)(dT/dx) in the one case is numerically equal to (l/n)(dni/dx) in the other, n i being the density in molecules per unit volume of one constituent and n that of the whole gas. Then, accord- ing to Chapman, if the molecules were hard spheres, k T should not exceed 1 per cent or so in a mixture of two similar molecules like oxygen and nitrogen, but when the masses are very different, as in an argon- helium or oxygen-hydrogen mixture, it might range up to something like 0.13. For molecules having softer force-fields k T is less, and it vanishes of course for pure repulsion as r~ 5 . . Usually the direction is such that the heavier molecules tend to diffuse toward the colder region. The simplest case to study experimentally probably would be that of a steady flow of heat through a layer of gas; there thermal diffusion would result eventually in setting up a steady composition gradient parallel to the temperature gradient of such magnitude that the result- ing transport of molecules by ordinary diffusion in one direction would Just balance the transport by thermal diffusion in the other. Actual experiments of this sort were performed in collaboration with Chapman by Dootson* and the expected effect was shown to exist. Two bulbs connected by a tube containing a stopcock were filled with the mixture of gases to be studied, and then with the stopcock open the bulbs were held at different constant temperatures for a number of hours in order to allow the steady state to be set up; the stopcock was then closed and the gas in each bulb was analyzed. As an example of the results, a mixture of nearly equal parts by volume of hydrogen and carbon dioxide, after the bulbs had been held at 200 and 10°C, respectively, for 4 hours, showed a concentration of hydrogen in the hot bulb exceeding that in the cold one by 2.2 per cent of the total concentration; a mixture of hydrogen and sulfur dioxide in the ratio 3:2 gave a* difference in the same direction of 3.5 per cent. The theoretical values for these two cases as calculated from Chapman’s formulas for hard spherical molecules were 7.1 and 9.1 per cent, respectively. The difference between these theoretical numbers and the experimental values might easily be ascribed to the softness of the molecules and can probably be taken as a direct indication that some softness exists, but probably not so much as would result from an inverse-fifth-power repulsion. * Chapman and Dootson, Phil. Mag., 33 , 248 (1917). CHAPTER V THE EQUATION OF STATE For our next topic we may conveniently return once more to the consideration of gases in equilibrium; but we shall now drop the assumption that the gas is practically perfect. The principal effect of the intermolecular forces in causing departures from the behavior of perfect gases is manifested in the relation between pressure and volume, and to this effect the present chapter will be devoted. 117. The Equation of State. In the theory of a fluid the general relation between the pressure, the volume , and the temperature is of fundamental importance. An equation expressing this relation is called the equation of state of the fluid. For a perfect gas we have found for it the simple familiar form, pV = RT; and it has been remarked that according to experiment all gases follow this equation more and more closely as the pressure is decreased, so that it can be accepted as the universal equation of state for a gas in the limit of vanishing density. As the density is raised, on the other hand, departures from the perfect-gas law would be expected and actually occur; at extremely high densities all gases become, in fact, only slightly compressible and in general take on the properties that we commonly associate with liquids. In this chapter we shall survey the principal attempts that have been made to arrive at an equation of state that will hold for real gases at all densities, or at least over a considerable range. The problem is susceptible of attack along two fundamentally different lines. We may endeavor to refine our physical assumptions in regard to the properties of molecules in the hope of obtaining a theo- retical equation that will agree better with the facts; or, on the other hand, we may seek by trial and error to construct an empirical equation to fit the observations. Progress has actually been made along both lines; the earliest steps taken consisted mostly of modifications in the assumed properties of molecules, but during the last two decades the empirical method has received the greater share of attention. The ultimate theoretical solution should, of course, come out of wave mechanics, but as yet little has been accomplished in this direction. 118. The Equation of van der Waals. The most important of the earlier attempts at an improved equation of state was the proposal 206 Sec. 118] THE EQUATION OF STATE 207 made by van der Waals in 1873. * The following is a paraphrase of the argument by which he arrived at his new equation. In a real gas there must be cohesive forces acting between the molecules; in the liquid or solid state this is shown very obviously by their clinging together, and it is natural to assume that such forces act at least slightly, however far apart the molecules may be. Conse- quently, a molecule near the wall of the containing vessel must experi- ence a net average force due to the attraction of the other molecules tending to draw it away from the wall. The pressure of the gas on the wall will be diminished thereby, and, since the amount of the pull on the gas will be proportional to the number of molecules pulled and also to the number that pull on each one, the diminution of pressure should be proportional to the square of the density of the gas; this diminution can accordingly be written in the form, a/7 2 , where a is a constant depending on the exact law of attraction. On the other hand, the pressure will be affected also by the finite size of the molecules. Each one is compelled at least to remain outside of all the others, and the space available for it to move in is reduced in consequence; the impact rate of the molecules on each other must thereby be increased. Now each molecule should reduce the available space by a definite constant amount, hence the total reduction for a given number of them will have some definite con- stant value; let us denote its magnitude for a gram by 6. The effect of molecular size will then be to increase the pressure caused by molecular motion, which otherwise would have the value RT/V , in the ratio of the ‘whole volume 7 to the available volume, or in the ratio 7/(7 — b). Combining these two effects, we thus obtain for the net pressure upon the wall of the vessel p = v~^b ~ W (171a) This is the new equation of state, usually written in the form, (p + ^)(y-6) = RT, (171 b) and commonly known as “van der Waals’ equation.” It is one of the most famous equations in aJl kinetic theory. * Van der Waals, “ Essay on the Continuity of the Liquid and Solid States” (in Dutch; Leiden, 1873). Translated in Threlfall and Adair, “ Physical Memoirs ” (Taylor and Francis, 1890). 208 KINETIC THEORY OF GASES [Chap. V Perhaps the remark may be added that, like every important advance in physics, the equation had forerunners which went a long way toward accomplishing the same thing. Clausius, for example, introduced the term b, and Hirn wrote p + a for p. The advance made by van der Waals over his predecessors was twofold: first of all, he made the change from a to a/F 2 , and then he discussed the equation in comparison with experimental data and showed that in certain cases it fitted well. Both of these steps were important. We can probably say with safety, however, that, if van der Waals had not proposed the equation, some one else presently would have done so; it was, so to speak, in the air. In scientific work in general the individual scientist seems to determine when and in what form an advance shall come rather than to determine what advances shall ultimately be made. 119. The van der Waals Iso- thermals. The most interesting properties of van der Waals’ equa- tion lie in a region beyond the reach of present-day kinetic theory and will be mentioned here only briefly; a fuller discussion may be Fig. 49. — Van der Waals isothermals. Sought in books on heat Or on the properties of matter. * The general character of the isothermal curves on the p, F diagram as predicted by the equation is shown in Fig. 49. All isothermals corresponding to temperatures above a certain limit, which is called the critical temperature , slope downward toward increasing volume, and at high temperatures they approximate to the hyperbolas that are char- acteristic of a perfect gas. Below the critical temperature each theo- retical isothermal exhibits a maximum and a minimum, between which there is a rising segment (e.g., BC). The critical isothermal KK', corresponding to the critical temperature, is merely horizontal at one point, called the critical point (P in the figure). All isothermals rise to infinity as F approaches the value b. Now the p, F diagrams of all substances actually show a region of this general sort, with one important modification. Instead of ascend- ing segments on the isothermals, there is a polyphase region, outlined by the dotted curve in Fig. 49, within which the substance, when in equilibrium, is separated into at least two phases; one of the coexistent phases is a saturated vapor, the other or others are liquid or solid. * E.g., B. C. McEwen, “Properties of Matter,” 1923 , especially Chap. VI. Sec. 119] THE EQUATION OF STATE 209 This region touches the critical isothermal at the critical point. An isothermal of the homogeneous substance can be followed experimen- tally only a short distance into the polyphase region, as from A to A' (superheated liquid) or from D to D' (supercooled vapor), and the substance is then not stable. It is easy to see why no point on an ascending segment BC can be realized with the substance all in one phase ; for it would then be highly unstable toward small inequalities of density, any slight rarefaction tending to increase without limit. The actual isothermals for the substance in equilibrium pass, therefore, horizontally through the polyphase region, as illustrated by the dotted line AD. In many cases there exist other polyphase regions representing states in which two or more liquid or solid phases coexist, but with these we are not concerned. Particular interest attaches to the critical point. The corresponding temperature and volume, as given by van der Waals’ equation, can be found as follows. Differentiating (171a) with T kept constant, we have dp __ RT 2a ev ~~ (v — by + v s (172) Thus dp/dV = 0 when rt (f - by ( 173) Now the right-hand member of this equation is infinite both for V = b and for 7 = oo ; it has therefore a minimum when its derivative vanishes, i.e., for such a value of 7 that 372 27* ___ 7 2 (7 - 3b) __ A (7 - by (7 - by (7 - by or when 7 = 36. At this value of 7 we have dV /dp = 0, provided, according to (173), RT - 8a/276; with this value of RT but any other value of 7, the right-hand member of (173) is larger than the left and dp/dV < 0, so that the corresponding isothermal is horizontal at just one point. Also, for any larger value of T the left-hand member of (173) is always the smaller, and hence dp/d 7 cannot vanish any- where, the isothermal sloping downward toward the right throughout its course. The pair of values of 7 and T thus found refer, therefore, to the critical point. Calling them the critical volume V c and critical tempera - KINETIC THEORY OF GASES 210 [Chap. V ture T c , and adding the corresponding critical pressure p c , calculated from (171a), we have thus for the critical constants: V c = 3b, p.=»^g? RT C = ~- (174a, 6, c) Special interest attaches also to the critical ratio RT c /p c V c , which is a pure number and represents the ratio of the volume as given by the perfect-gas law at the critical pressure and temperature to the actual volume F c (or the ratio of the perfect-gas pressure at the critical volume and temperature to the actual pressure, or the ratio of the critical temperature to the perfect-gas temperature at the critical pressure and volume). The van der Waals value of this ratio is, from (174a, 6, c), !£=!«= 2.67. (175) Pc v c o 120. Quantitative Tests of van der Waals’ Equation. When we turn from its qualitative features to a quantitative comparison of van der Waals’ equation with the data for actual gases, we find very soon that it really does not fit the data very well. The easiest quantitative feature to test is the critical ratio RT c /p c Vc , for which the equation predicts a value of as stated in eq. (175). In the following table are shown experimental values of the critical con- stants for a number of gases and the corresponding experimental value of the critical ratio. It will be noted that the experimental values are uniformly larger than the theoretical value of 2.67, lying near 3.5 for the gases shown, except that it is above 4 for water, whose critical density Critical Data Substance tc, °C Pc atm d„ = 1/Fo, g/cc RTc/pcVc Hydrogen —240° 12 8 0.0310 3.27 Helium -267.9° 2.26 0.069 3,26 H 2 Q 374° 218 JO. 4 (?) 5.40 Neon -228° 25.9 J0.329(?) 0.484 4.45 3.42 Nitrogen -147° 33.5 0.311 3.43 Oxygen -119° 49.7 0.430 3.42 HC1 62° 86 0.42 3.48 Argon -122° 48.0 0.53 3.42 C0 2 31.1° 73.0 0.46 3.57 Ethyl ether, C*HioO 194° 35.6 0.262 3.82 U a " critical temperature, p c - critical pressure, V c — critical volume. Sec. 121] THE EQUATION OF STATE 211 is hard to determine accurately. Dieterici collected similar data for 23 other substances, all organic, and found values of the ratio mostly below 4 but ranging from 3.67 to 5. Probably, however, large as the discrepancy is, the fact that it is not even larger furnishes some ground for believing that there must be a considerable measure of truth in the reasoning leading up to van der Waals’ equation. Difficulties of a similar order are encountered when an attempt is made to fit van der Waals’ equation to a set of actual isothermals. It must be recognized, however, that in using the equation near the critical point we are taxing very heavily the general validity of its theoretical foundations. The equation might be expected to succeed better, although less spectacularly, at much lower densities. Such a test of its validity will be made later in connection with the dis- cussion of the second virial coefficient. First, however, let us see how the theoretical foundations themselves can be improved. 121. More Exact Theory of the Pressure in a Dense Gas. The argument by which we arrived at van der Waals’ equation above was distinctly sketchy and needs to be replaced by a precise analysis of the process by which the pressure is produced in a dense gas. Such an analysis will now be given. The changes that were made in our hypotheses concerning molecu- lar properties when we abandoned the assumption of a perfect gas amounted to allowing the molecules to exert appreciable forces upon each other while their centers of mass are still at considerable distances apart. To find the effect of such forces upon the pressure, let us con- sider as in Sec. 5 the flow of normal momentum across a small plane area drawn anywhere in the gas (cf. Fig. 50 below). In order to locate the molecules definitely with respect to this plane, let us define the position of any one to be the position of its center of mass, so that by definition a molecule “crosses” the plane at the instant when its center of mass crosses. The mean molecular density, which now means the density of centers of mass, is then simply the ordinary number of molecules per unit volume. Furthermore, according to classical statistical mechanics, the distribution of veloci- ties is quite unaffected by the presence of intermolecular forces (cf. Sec. 205). Accordingly, the entire elementary calculation made in Sec. 5 of the rate of convection of momentum across the plane by molecules that actually cross it still holds, and we have for the part of the pressure that is due to this cause, which we shall denote by pk and call the 'kinetic pressure , simply the perfect-gas value RT Vic - ~y> ( 176 ) 212 KINETIC THEORY OF GASES [Chap. Y in which V is the volume of a gram of the gas and R = Rm/M and denotes its gas constant calculated just as if it were perfect (22 at = universal gas constant, M = molecular weight). In addition to this convective flow of momentum, however, we now have an additional transfer through the agency of forces acting between molecules that lie momentarily on opposite sides of the plane, as at B and C in Fig. 50, irrespective of whether they them- selves ever actually cross it or not. This part of the pressure, meas- ured by the total force that acts across unit area, we shall call the dynamic pressure and denote by p & . A definite expression can be Fig. 50. — Illustrating the effect of molecular forces and finite size upon the pressure in a gas. Fig. 51. — Slater’s field for helium, to — potential energy; F = force; r = interatomic distance ; and <r = molecular diameter from viscosity. obtained for it only from a knowledge of the law of intermolecular force, but the following qualitative analysis possesses a certain interest. All that we know about molecules points toward the conclusion that the intermolecular force is usually of the nature of an attraction which rises as two molecules approach each other, reaches a maximum, and then quickly sinks and turns into a repulsion; the latter then increases rapidly as the molecules come close together. As an illustra- tion, the actual curve for the fotce F ) measured positively when repulsive, and its associated potential energy « = J r *F dr , for two helium atoms as obtained from an approximate treatment by wave mechanics* is shown in Fig. 51; the equations for the two curves are * Slater and Kirkwood, Phys. Rev., 37, 682 (1931). Sue. 121] THE EQUATION OF STATE 213 «- 7.7e" 2 <-£68 L (r/a 0 yj F = ~ = 0.354 e~ 2A3 »• X 10~ 10 erg, 0.0771 0/a 0 ) dyne, (177a) (1776) where a 0 = 5.29 X 10 -9 cm. (the Bohr radius) and r — distance between atomic centers. Conceivably the force might also oscillate more than once between attraction and repulsion, and in most cases it must depend a good deal upon the orientation of the molecules. In any case, we can at any moment classify all of the forces that are actually acting across the plane between various pairs of molecules into the two types of attraction and repulsion, and we can in thought imagine each type to contribute its proper component to the pressure. The component due to attractive forces may be called the cohesive pressure and we shall denote it by p c ; it is often called the “internal” or “intrinsic” pressure. The component due to repulsive forces will be called the repulsive pressure, denoted by p r . These two components together then make up what we have called the dynamic pressure p d ; and adding to this the kinetic part as given by (176), we have as the total pressure p - Pk + pa = -y- + Pd, Pd = Pc + Pr. (178 a, b ) It is interesting to note that the analysis up to this point would hold according to classical theory for a liquid or solid as well as for a gas. The kinetic pressure in a liquid or solid is, of course, enormous; tho dynamic pressure is almost as great and is, of course, negative, the algebraic sum of the two equaling the external pressure. Further quantitative progress along these lines cannot be made without additional information in regard to the molecular forces. Furthermore, it must be recognized that the introduction of wave mechanics blurs the sharpness of the distinctions that we have intro- duced here; even in a rarefied gas the distinction between repulsive and cohesive components cannot be so sharply drawn as in classical theory, and as the density increases, even the distinction between dynamic and kinetic pressures progressively loses physical significance until probably it scarcely retains any validity at all for the liquid or solid states. Moreover, wave mechanics furnishes an additional source of pressure in the “exchange” effects which are characteristic of the new theory and have no analogue in the old. Since, however, the latter effects are appreciable only for very light particles, or at 214 KINETIC THEORY OF GASES [Chap. V enormous densities, or at minutely low temperatures, and so are as yet of importance only in the electron theory of metals, and since little is really known that is quantitative in regard to molecular forces, and accurate calculations are in any case necessarily complicated, we shall follow the usual procedure of working out by classical theory a simpli- fied ideal case as a sort of model. The conceptions that are developed in doing this have at least some value as qualitative modes of thought in reasoning about the internal state of an actual gas. Problem. Calculate the kinetic and dynamic pressures in water at 15°C and under 1 atmosphere of pressure. Ans.: 1,312 and —1,311 atmosphere. (Estimates of the cohesive pressure by itself run to 10,000 or 20,000 atmospheres.) 122. Hard Attracting Spheres : The Repulsive Pressure. Let us suppose that the molecules are all of one kind and that they (а) are hai;d spheres of diameter a obeying classical mechanics, and (б) attract each other with a weak force that depends only upon, their distance apart and effectively vanishes at distances several times as great as a. Let us also restrict the density of the gas to be low enough so that the effect of these additional features upon the pressure is a small one; i.e., we seek only a first-order correction to the perfect- gas law. These are the assumptions that underlay the first quantita- tive calculation by van der Waals and others relative to his equation. Let us begin by calculating the repulsive pressure p r . With molecules of the type assumed, repulsive forces will act across the plan© introduced in the last section only when a molecule lies part way across it and is struck by another whose center lies on the other side. Thus p r arises here entirely from the finite size of the molecules. To calculate the total effect of such impacts, consider a particular molecule whose center is at some distance x less than c r from the plan© (e.g., A in Fig. 50), and around it draw its sphere of influence of radius <j, upon which the center of another molecule must lie at the moment of collision. Consider an element dS of that part oi the surface of the sphere of influence which projects across the plane. If the given molecule A has a component of velocity v L in the direction of the out- ward normal to dS , the element dS will sweep out in time dt, relative to any other molecule moving with corresponding component v L ', a volume (^i — vx)dS dt, independently of any transverse component of the motion of either molecule. Now the density of molecules with a component within dv± f of vx r is, by (62) in Sec. 28, n(/3/'\/^.)e~ l32v Y' i dv 1 / ; hence the chance that another molecule has its center in the volume swept out by dS and so gets struck is Sec. 122] THE EQUATION OF STATE 215 n(v ± . — vx') —z. e-P 2v ± ' 2 dv j.' <2$ ctt. V 7T If such an impact occurs, according to the laws of elastic collisions an amount m(y ± — v±') of momentum in the direction of the normal to dS is transferred to the second molecule, m being the mass of each one, and a component of momentum normal to the plane of magnitude m(v j_ — v j/) cos 6 is thereby transmitted across the plane, 6 being the angle between the normal to dS and the normal to the plane. On the other hand, the chance that a molecule lies within a volume element dr surrounding the position assumed for the first molecule and has also a component within dv± of v±, is n dv x dr. V A" The total normal momentum transmitted across the plane by such collisions in time dt is then the product of these three expressions integrated over all possible velocities and all positions of dS and dr or cos 6 dvj_ dv± dS dr. This expression divided by dt and by the area of the plane then equals the repulsive pressure. Instead of dividing the whole integral by the area, however, we may also simply confine the integration over dr to the space contained in a cylinder standing perpendicularly on a unit area of the plane; for each such cylindrical portion of space con- tributes an equal part to the whole. We can also imagine the integra- tion over the two variables y and z running parallel to the plane to be carried out at once, the result of this integration being simply unity since the integrand is independent of y and z; we have left then only an integration over x perpendicular to the plane. Accordingly, the repulsive pressure p r will be given by the last expression with dr replaced by dx, the limits for x being 0 and a since no molecule can lie farther than <r cm from the plane and still collide with another beyond it. Let us transform the integral by writing dS = <r 2 sin 6 dd dip in terms of polars with axis normal to the plane. Then, introducing obviously appropriate limits of integration, we have 216 KINETIC THEORY OF GASES [Chap. V p r = tt 2 m<r 2 er*rr T Jo Jo Jo sin 0 cos 0 dtf dp "* W i. r /* oo ’ I dax (a x - ax') 2 e-^’V+V 3 > dv x ’, in which a is the value of 6 at the circle along which the plane cuts the sphere of influence, and so depends upon the value of x, and the upper limit for v x ' is v x because faster molecules could not be overtaken by the first one. The double integral over v±, v x can, — however, be simplified. We can extend 1 the integration over the entire ax, v x - plane and then divide by 2, since the integrand is symmetric in v x and ax'; the original region of integration is that which is shaded in Fig. 52, but the integrand has the same value at any two points that are symmetrically situated relative to the line v x = a ,' Doing this, we find J- „ dv± /_ ~ dv x = I J ^ j* ” ( Ux 2 + ax' 2 )e-^(«x^x'») dv x dv x = dv x dv x ' Fig. 52. = x 2 e~P lxi dx j* e~i 2/S 4 the term in a ± ax' that arises from the expansion of (v x - ax') 2 gmng zero by symmetry. (Cf. table of integrals at the end of the The remaining integrations are then easily carried out with the help of the obvious relation, cos a = x/a, giving, since £* d<p = 2tt, f.'** "**•** - - |(l _ *■), J[' & _ ,, _ 1^ or, after inserting „„ - 1/F and 0- - 1/<2W) by (56) in Sec. 28, 2tt n Sec. 123 ] THE EQUATION OF STATE 217 For the repulsive pressure we can accordingly write RT 9 Vr = b b = ~7rnF<7 3 , (179a, b ) in which 7 is the volume of a gram. 123. Equation of State for Hard Attracting Spheres. On the other hand, the cohesive pressure, arising from the assumed attraction, is easily found merely by restating the brief argument of Sec. 118 in more precise terms. Because of the low density, the molecular dis- tribution will be very nearly the same as in a perfect gas, i.e., each molecular center is as likely to be in one position as in another and the various molecules are distributed independently. Accordingly, the chance of a molecule’s being in any element dr, say at B in Fig. 50, is simply ndr where n is the number of molecules per unit volume; the chance of another’s being in another element dr' at some point such as C across the plane is similarly n dr'; and the chance that both are in these positions simultaneously is n 2 dr dr'. The expectation of a resulting contribution to the cohesive pressure p 0 can thus be written n 2 F j_ dr dr' where F± denotes the component of force normal to the plane that one molecule exerts upon the other and so is some function of the positions of dr and dr'. Similar considerations apply to all elements of volume. Hence the total cohesive pressure is n 2 J* j* F± dr dr' or f n a> To = -a'n 2 = y2> where a' or a is a small constant. For the total pressure we have then finally, inserting our values of p e and p r in (178a, h) above, V = Vh + Pc + Pr = (l + y) - yj- (180) Here R is the gas constant for a gram computed as for a perfect gas (e.g., R = Rm/M ). This equation agrees for small values of b/V with that obtained from van der Waals’ equation, which can be expanded in powers of b as follows [cf. (171a) in Sec. 118]: b b ■ . . \ (a.. V y l 1 ^ y r v 2 ' / V 2 ( 181 ) 218 KINETIC THEORY OF GASES [Chap. V Our analysis thus furnishes a rigorous deduction of van der Waals’ equation for the type of molecules under consideration, but only for the case in which the effect of molecular size is relatively small. The question as to the validity of the equation at higher densities is left untouched. 124. The Value of b. In the last section we obtained a connection between our constant 6, which at low densities is the same as van der Waals' 6, and the size of the molecules, for the case in which the latter are hard spheres. The value given by (1796), 6 = f wriV cr z , is equal to four times the actual volume of the molecules in a gram of the gas. In our original rough analysis, on the other hand, 6 was introduced as representing a diminution in the space available for the molecules to move around in, and since each molecule excludes the centers of all others from its sphere of influence, we might perhaps have expected b to equal the total volume of all the spheres of influence, which would be eight times the sum of the actual molecular volumes, or twice as great as the value just found. The reduction from eight to four undoubtedly arises from the interpenetration of these spheres, of which an example is illustrated by molecules B and C in Fig. 50, where the spheres of influence are represented by dotted circles; but only a detailed calculation could show that the ratio of reduction is exactly H. It must not be forgotten that even the value obtained above for 6 has reference only to rarefied gases, i.e., it applies to the first-order correction to the perfect-gas law. If we assign this value to the constant 6 in van der Waals’ equation and then suppose the equation to hold even up to very high densities, a curious consequence results, since as V — > b the pressure becomes infinite; this ought to mean that when V — > 6 the molecules become tight-packed, but if they were, the volume of the gas ought to be less than twice rather than four times the actual volume of the molecules themselves. This considera- tion alone is sufficient to show that van der Waals' equation cannot be correct at all densities for an attracting-sphere gas. The equation might conceivably be found to hold closely for some actual gas even up to very high densities; but then we could infer that the molecules of that gas at least were not hard, weakly attracting spheres, and the empirical constant b appearing in the equation would then of necessity possess some other significance than that which we have found h efe. 125. Other Equations of State. Many attempts have been made to obtain an equation of state agreeing more closely with the behavior Sec. 125] THE EQUATION OF STATE 219 of actual gases. Besides efforts to improve the precision of calculations similar in type to that which we have just made, various other lines of approach have been tried, varying from strictly theoretical arguments to purely empirical procedures. Boltzmann, for example, carried the calculation for hard spheres a step farther by considering the effect of “ triple encounters,” or cases in which a third molecule lies close enough to have an influence upon the probability of a collision between two given molecules. In our calculation we assumed that the chance of finding a second molecule in an element of volume dr near another one (e.g., near A in Fig. 50) has the value n dr in which n is the number of molecules in unit volume. This is certainly the average number in dr during the whole time. We must recognize, however, that a second molecule can have its center in an element only when the latter is not over- lapped by the sphere of influence of some third molecule. Now, if dr is assumed to lie just outside the sphere of influence of a molecule A , the range of positions in which a third molecule might lie with its sphere of influence overlapping dr is easily seen to be restricted by the presence of A, and hence it is easier for a second molecule to get into such an element than into one that is out in the open. Thus in an element near the sphere of influence of a molecule the mean density of molecular centers is somewhat greater than n, the excess being itself proportional to n so long as n is small. (A similar enhancement of the mean density near the wall of the containing vessel actually serves to account for the entire increase in the pressure on the wall corresponding to p r , not merely for the second-order term which we are now discussing.) The resulting effect on the pressure can be calculated;* Boltzmann finds that it changes the parenthesis in (180) from ^1 + to (l+* + W \ V ^ F ' 8 V 2 ) The added term goes part way toward supplying the term y- in the expansion of van dor Waals’ equation; but the behavior of the equation at the critical point is found not to be improved. The calculation could be extended further so as to allow for encounters of four or more molecules, but this is scarcely worth while because the assumption of hard spheres is, after all, pretty wide of the truth. * Cf. also Uracil, references at end of Sec. 129, 220 KINETIC THEORY OF GASES [Chap, V In a more general way, both van der Waals himself and others have proposed to treat a and b in his equation as variables, either replacing one or both of them by some expression containing addi- tional unknown constants, or simply regarding them as functions of V and T . A better fit with experiment can, of course, be attained in this manner, but it must be noted that the original form of van der Waals' equation loses all significance if a and b are allowed to vary without restriction; for in this sense any relation whatever between p, 7, and T can be written in the van der Waals form (or, for that matter, in the simpler perfect-gas form pV = RT , by allowing R to stand for a suitable function of V and T\). Of the various other equations of state containing a limited number of disposable constants that have been proposed from time to time we shall mention only two. In 1899 Dieterici proposed the equation* V = RT V - b e **r in which a and b are new constants whereas R is, as usual, the perfect- gas constant. The equation rests theoretically on the assumption that the cohesive forces are on the whole equivalent to a force-field acting upon molecules in the surface layer of the gas in such a way as to tend to draw them back into the interior, with the result that the density is less at the surface in proportion to the Boltzmann factor, e ~o>/kT' wou i(i be natural, then, to complete the argument by saying that we should expect co to be proportional simply to the density of the gas or to 1/7; but such a statement would be open to serious question, since the force-field at the surface must owe its origin largely to those molecules which are in the rarefied surface layer itself, and accordingly its mode of variation with the density is not obvious after all. In any case, Dieterici himself adopted the assumption that a? cc 1/7 as a result of empirical trial, and his equation therefore rests on a half-empirical basis. It fits rather well in certain cases, and in particular it makes RT c /p c V c = J e 2 = 3.695, which lies right among the observed values, in contrast to the van der Waals value of 2.67 (cf. the table in Sec. 120). The most ambitious attempt at a closed equation of state is perhaps that of Beattie and Bridgman :f * Dieterici, Ann. Physik, 69, 685 (1899). f Beattie and Bridgman, Jour. Amer. Chem. Soc., 49, 1665 (1927); 60, 3133 and 3151 (1928); Zeits. Physik , 62, 95 (1930). Sec. 126] THE EQUATION OF STATE 221 _ RT f c Yt/ n bB 0 \ A 0 (, a\ P y 2 ^1 y^J^V + Bo yj y 2 ^1 yj- By suitably choosing the five disposable constants in this equation (R being the ideal gas constant), the authors show that the data can be fitted within 0.5 per cent over a wide range of pressures and tem- peratures, even near the critical point, for at least 14 gases including all the common ones. Some theoretical justification for the equation can be given, but its chief interest seems to lie in its possible utility, since, as the authors point out, its algebraic form facilitates its use in thermodynamical calculations. Problem. Show that the equation of Dieterici obeys the law of corresponding states, and gives at the critical point V c = 2 6, 1 e 2 = 3.695. VcVc 2 126. Series for pV; Virial Coefficients. The difficulties encoun- tered in seeking a satisfactory equation of state in closed form led Kammerlingh Onnes* in 1901 to turn to simple expansions in series, and this procedure has been widely adopted. The most usual form is an expansion of the product pV in powers of the density or reciprocal of the volume, thus, p v = A +- + ■!!+■■■ , (182a) or else, as preferred by many, in powers of p, pV = A B p p + C p p 2 + * * * . (1826) Forms such as pV = a( l + £ + £+---) are also frequently employed; and d is sometimes written for 1/F. It is quite common to take as the unit for V the volume under standard conditions, or sometimes the ideal or perfect-gas volume under standard conditions, which is, of course, slightly different; if the perfect- gas unit for V is employed A = 1, whereas if the actual volume is taken as the unit A differs slightly from unity. The coefficients A, B • • • or A, B p * * • are functions of the temperature which were called by Onnes first, second, etc., “virial *Kammeelingh Onnes, Commun. Leiden, 71 (1901); K. Akad. Amsterdam , Proc ., 4, 125 (1900). 222 KINETIC THEORY OF GASES [Chap. V coefficients.” In all equations we must have A = RT to make the perfect-gas law hold at zero density or pressure; and B, C ■ • ■ often written B v , C v , • • • are related to B p , C p • • • by the equations B — AB P = RTB P , C = A{B J + AC P ), etc., (182c) as is easily shown by substituting the value of p from (182a) in the right-hand member of (1826) and equating the coefficients of powers of 1/V in the result to those in the right-hand member of (182a). 127. The Second Virial Coefficient. Much interest has been taken lately in the experimental determination of the second virial coefficient S as a function of temperature, and the results are of considerable Temperature Centigrade, Deg. Fig. 63.— The second virial coefficient B. (p in atmospheres, V in terms of the standard volume.) theoretical interest. This is done by running isothermals at different temperatures, i.e., determining values of pV for a series of pressures at each temperature and then fitting a series to the observations. Equation (1826) often fits exceedingly well with only the three terms that we have written, and even C tends to be small. Experimental values of B for several gases are plotted against the absolute temperature T in Fig. 53, p being expressed in atmospheres and pV being made unity under standard conditions. The data were taken from papers listed below, values of B v being multiplied by 77273.1* to get values of B, and Holborn and Otto’s being mul- tiplied by 0.76 to convert from meters Hg to atmospheres, f * This is not absolutely correct when pV is made unity under standard condi- tions rather than at infinity, but the difference is negligible for our purpose. "f "Whitelaw, Physica, 1 , 749 (1934); Holboen and Otto, Zeits. Physik, 23 , 77 (1924), 33, 1 (1925), 38, 359 (1926). Sec. 128] THE EQUATION OF STATE 223 The sign of B tells us the initial direction of the variation of the product pV with density. If J3 < 0, < 0 at p = 0, and the first change in pV with rising density is a slight decrease. In all such cases, however, the subsequent terms of the series are such that pV passes through a minimum at a certain density and thereafter increases, finally attaining large values under pressures of the order of 10,000 or 20,000 atmospheres. All known gases exhibit this type of behavior at sufficiently low temperatures. As the temperature is raised, however, B eventually becomes positive in all cases shown in the figure, and probably this is true for all gases; at such temperatures, pV rises with increasing density from the start. The temperature at which B changes sign is called the Boyle tem- perature. It is more often called the Boyle point , but this term might better be applied in a more general way to any point on the volume- temperature diagram at which pV passes through a minimum along a given isothermal, so that in the immediate neighborhood of that point Boyle's law holds; the Boyle temperature is then that tempera- ture at which the Boyle point on the corresponding isothermal occurs at density zero and above which no Boyle points occur at all. Holborn and Otto give the following Boyle temperatures (centigrade) for six common gases; these values agree approximately with the tempera- tures at which B = 0 in Fig. 53. He h 2 Ne n 2 A o 2 Air -254° -167° -139° 50° 137° 150° 74° For most gases the Boyle temperature lies above 0°C. 128. The Second Virial Coefficient and van der Waals’ Equation. The behavior of B as a function of temperature furnishes a very con- venient test of van der Waals' equation at low densities, for comparison of (181) in Sec: 123 with (182a) above shows that according to van der Waals’ equation B = bRT — a. (183) The same value of B follows from eq. (180), representing the result given by the theory of hard attracting spheres. Thus, according to these equations of state, B should be a linear function of T , and the slope and intercept of the line representing B plotted against T should give us at once the values of the constants a and b. If B p = B/RT 224 KINETIC THEORY OF GASES [Chap. V is employed in place of B, we have B p b — and B p should there- Jtil fore approach a constant value equal to b at high temperatures. Now, the curves for B in Fig. 53 do approximate roughly to straight lines, showing again that the ideas of van der Waals must contain a good deal of truth; but they all show some curvature. Their convexity upward suggests the equivalent of a decrease of b with rising tempera- ture; the latter is what would be expected if the molecules are not hard spheres but interact by means of extended force-fields. Apparent values of a and b can be calculated from the curves by drawing tangent lines at some point and assuming these lines to obey eq. (183). With our choice of units the values so found are also the values in terms of any units of the dimensionless quantities a/p^Vl and b/V o, po and F 0 standing for the standard pressure and volume; these quantities give an immediate idea of the degree of departure from the perfect-gas law at 0°C and 1 atmosphere pressure, and the general van der Waals equation can be written in terms of them if desired, thus : Values of a/p 0 V% and of b/V 0 so derived from, tangents at 0°C drawn in Fig. 53 are given in the following table. For comparison, values of the same quantities are also shown calculated from the critical data by means of formulas (1746) and (174c) in Sec. 119, these two formulas being preferred because they do not contain V c \ such values are distinguished by a subscript c. Finally, we have added for comparison with b/Vo values of this quantity calculated by inserting in Van dee Waals Terms (Unit in all cases, 10~ 3 ) H 2 He Ne N 2 0 2 Ar h 2 o C0 2 C4H10O (ethyl ether) a/p 0 Vl b/V, ac/poV* 6./F, 0.34 0.95 0.48 1.18 —0.04 0.46 0.07 1.06 0.30 0.77 0.44 0.80 3.07 2.55 2.68 1.72 3.60 2.60 2.70 1.42 3.04 2.06 2.68 1.44 .10.9 1.36 (17.5) (9.9) 7.17 1.91 34.6 6.01 |7rno<r 3 1.16 0.58 0.98 2.97 2,65 2.71 5.48 5.44 Sec. 129] TEE EQUATION OF STATE 225 the hard-sphere expression for it, or f 7rn 0 cr 3 as given by eq. (1796) in Sec. 122, the values of a obtained from the viscosity at 15°C, which would be only slightly different at 0°C (cf. table in Sec. 86); no = molecules per cubic centimeter under standard conditions. A broad agreement between the values of a/poVl and b/Vo derived from these different sources is at once evident. Assuming the data sufficiently reliable, we have here another indication that there is a good deal of truth in van der Waals' ideas but that his equation fails in the finer details. The values of b/Vo derived from viscosity data (last column) agree on the whole distinctly better with those obtained from the virial coefficients (b/Vo) than do values calculated from the critical data ( b c /Vo) 7 as would be expected. The values from viscosity are, however, all 2 to 32 per cent larger than the virial-coefficient values, except in the case of CO 2 , for which precise data on B do not seem to exist; this uniform difference may well correspond to the dif- ference in the molecular processes involved in the production of viscous forces and of pressure, and it at least serves to indicate again that the molecules do not behave quite like hard spheres. 129. Theory of the Second Virial Coefficient, B. The value of B that follows from classical mechanics for any spherically symmetrical t^pe of molecular field is readily obtained in the form of an integral by making suitable modifications in the calculation given in Sec. 122. Let us write w(r) for the mutual potential energy of two molecules when their centers are a distance r apart; the force on each, taken positive as a repulsion, is then — a/ = —du/dr. Probably the value of co will be negative at moderate distances, but as r approaches zero it will become positive and rise with tremendous rapidity, somewhat after the fashion of the example plotted in Fig. 51. We must still assume the density to be very low, since B refers only to the first-order effect of increasing density, but we need make no restriction upon the numerical magnitude of co other than to assume that it vanishes at least faster than 1/r 3 as 00 , in order to secure convergence of certain integrals. Resuming then our usual procedure of calculating the pressure as the rate of transfer of momentum across a small plane of reference drawn in the gas, consider the force due to a molecule, such as A in Fig. 54, acting upon other molecules lying on the opposite side of the plane. The chance that, when A is in the position assumed, a given other molecule simultaneously has its center in any element dr' distant 03 r from the center of A is Ce w dr ' by (94a) in Sec. 55, provided all e kTdT r , integrated for all possible 226 KINETIC THEORY OF GASES [Chap. V positions of the second molecule in the whole volume V of the gas, must be 1. The integral itself in this expression, is practically equal to V, however, since the region in which differs appreciably from 0 is negligibly small as compared with V itself; hence we can write as a first approximation C = 1/V. Because of the assumed rarity of the gas we can also neglect all encounters except binary ones and so can as usual treat the occurrences of molecules in dr' as independent events. Accordingly, we can write for the probability that some other molecule has its center in dr' the product of Ce kT dr' by the total numbernF of molecules in the gas, or w ne kT dr'j n being the number of molecules in unit volume. In Sec. 122 we wrote simply ndr for this probability because there by hypothesis co was necessarily small. If, now, a second molecule does lie in dr', A will exert upon it a component of force normal to the plane of magnitude — a/ cos 0, 0 being the angle between the line joining the centers of the two molecules and the normal to the plane; and there are on the average n dz such molecules as A lying at a distance between z and z + dz from a given unit area of the plane. The total normal component of force exerted by the latter on all molecules beyond the plane is, therefore, — n 2 dz J («' cos 6)e~kT dr' integrated over the whole space beyond the plane. The integral of this expression from z = 0 to z = oo then gives the dynamic pressure p d . Putting dr ' = 27 rr 2 sin 6 dO dr with the polar axis perpendicular to the plane and introducing appropriate limits of integration (cf. Fig. 54), we thus find Vi = —2jm 2 J’ g " dz£ Ve “**>•* dr£° B ~' ' cos 0 sin 6 dO — ~^ wn2 f 0 r2(0 ' e kT drj^ cos 0 sin 0 ddj ' 009 6 dz or Vi — — f rn2 J 0 r 3 u'e~kf dr (184a) Sec. 129] THE EQUATION OF STATE 227 7T since cos 2 6 sin 6 dd = We can also write for this last integral in r in which we have inserted the 1 to secure convergence at infinity; and if we then integrate it by parts, the integrated term vanishes at both limits, reducing at infinity to r 3 co, and hence to zero according to our assumption as to the smallness of co there. We thus obtain the alternative expression : p d = 27 rn 2 kTj* o r 2 ( 1 — e~~kr^ dr. (184 b) Now, from (178a) in Sec. 121, we have pV = RT + p d V; and comparison of this with (182a) in Sec. 126, in which A — RT , shows that at low densities B = p d V 2 . Hence, inserting in this equation either of the values of p d just found and writing N for the number of molecules in the mass of gas whose volume is 7, which can be chosen arbitrarily, and R for the perfect-gas constant appropriate to this chosen mass, so that nV = N and Nk = R, we obtain finally: B = ^ 2 J o ”^ 3 ^ e ~^ dr = j Q r 2 (l - e"^) dr. (185) This value of B may be contrasted with that for hard, weakly attracting spheres as stated in eq. (183) in the last section. The expression obtained here is more general, not only because of substitu- tion of a general potential-energy function co for the hard-sphere assumption, but also because we have here employed the Boltzmann distribution formula in place of the more restrictive assumption that the distribution is sensibly uninfluenced by a>. From the last expression for B given in eq. (185) it is evident that positive values of co result in positive contributions to B, and negative to negative. We can say, therefore, that a positive value of B implies a predominance of repulsion between the molecules, whereas a negative value implies a predominance of attraction. By employing still more precise forms of the Boltzmann formula it is possible to take account of ternary and higher orders of encounters A /AT / ti' Till OP) Ob it IMN V m and tlmmby In obtain iniirnpimding v\po * am* fur tlm login r virial ruollmmnK (iunmnl fnimula* of l hi"* oil lm\* l«»o v**»rK* d nut liy Uim‘ 11 nnploymg it diirt t appto'n h on tlm l*a m ot f » tin nl no » Inn icH,* Iml ns yH tmt Mini'll M sl « hu^ Utou nmdo of flmm m tin di »* n mon of oxpmnmnhd n Mill** 1'nMutt A^mmug that m ( I ■ * for i - <» and t \ immII fur r '* <r, obtain llm van dor \\ iml » * \po - am fm /f mu giu m m t Kd with b » i Mr* 1 um usual ami n 2a \ 3 1 ^r*ft 130. Nature of Molecular Korean. Tlm formula* )ml *•! tlnitic ct run 1m utihmS fur ruU iilmt 111 MC W only if kimu tlm Inv^ ot iMHlffudar fin cn Upon dun lad nr quo’dmn I In* nmtlnit I lit **r v of tlm nlum hna lluuwn n gioal dual nf light. H inpjH urn f Iml umlt'i uhtr form* uri** frum novora! dintinot mihu'ih mul am n»ri« sjHmdifiglv *nu»d »n Hair Mpalial itint i ihiiliun ; of dm fnllmvmg Hjw« probably Minim at loanl, with hiibhi 1 mohmuloH U Uoliitivuly hugo olotlHHlafm form* tuning m» \ r* l«hmn lunimil mul llmiufmo rhrugt'd umlimulr* 2 KloHroMlulm forms wuyiugu* l/rMmtttiu n an inn nod n inulral molmulo jiului ivmm! by I la* hold nf Ilia am H. Klouhnstalm forma vaijiug mi 1,'r 1 bt fui ♦ n dlpnh* a nr nmlm ulr* puKM'H^mg poiiimunid nloolrio momiml*, *imh aa flior** nf am «!• Klmlrontulm fill ran varying a*« l/r 1 Ur hu an Hm Irm dq*o|i h and iumh ft Klrrtinalndr forma \nrying aa l/r* Imlwron (hr dipnto mmimut of ono inoiooulo and a mound tnolunuln po*tir «ing mm Maturat mnmnii lad {mlanv.ud by iltr fluid nf thu bud tt AUrartivr foirra ulhomluly proportional to l/r 9 dim to dan lortiun nf nm* innlmadn nndnr tliu ndlinaa n nf another liirli nrilhrr a not rlmigr nnra clipulr numund, arising fmm Urn ioMi|«la alnl mlrmrium bKwwn dm rlrrtroua and flm nm lm 7. Foirra of quantum origin rliirfly dim tn Urn M rxrliniiKr M r-fTrrl in Urn nlrdUuii almlln, falling off rapidly and roughly rx|anmutmlly with r. In rm*h of il aw vw* r daunton lla* di^tanm Iriwuat iiinlcrulnr rrnLrrH of num and llm pnhmliid mmrgy m vann an imur^u jmwrr of r one* Iran than that Htatrd for thn fnn n In nimw d In ft, at l*wt, U»? fotro vaiinn gmally with llm nrirntatnm of dm mnlmulu and i* ftmmnjmmud by a lorqim tunding In produrr rotation Foiwh of type 7, original lug Himfly in tlm oxrdmngo plmnomunon in tlm rlrrtion almllM, uir now Imliuvrd to amount, on tlm mm Immh 4 Uitfiiair, Ctimb Phi Sor Pro? SUh Wi U It lAim^a "KMtwUmt Morhiunpw," KI2D. il Pill 8m. 101 ] TBS EQUATION OF STATS 229 for the formation of ohemioal compounds through primary or Batlable valence and, on the other hand, for the observed Impenetrability of atoms. The impenetrability was always difficult to explain in terms of the e land c al conception of an atom composed solely of electric oharges; spherically symmetrical shells of oharge about a nucleus, for instance, should pass right through each other, the effect of suoh a shell on any element of oharge outside it being tho same os if the whole shell were concentrated at its center (and on on element of oharge inside it, nothing at ell), so that no specie! repulsion would develop when two shells oamo Into oontaat. Forces of type 6, on the other hand, are believed to account for the van der Waals attraction in most of the oommoner gases; the ohemioal combination by “ secondary " or nonsatlable valence is believed to be an enhanoed effect of the some sort. ' Permanent moments when preeont must also contribute, however, to the van der Wools attraction; if the dipoles wore oriontod at random their effect would vanish, as must tho moon eleotrlo Hold duo to any distribution that is olootrioally neutral as a whole, but tho Boltimnnn factor results In a Blight preponderance of molecular positions having less potential energy and so gives rise to an avorogo attractive effect. .The force-action between actual molecules may, of oourse, and no doubt oommonly does, represent a combination of several of the elementary types listed above, 181. B with tax Inverse-power Force, When the force is both spherically symmetrical and proportional to a simple power of tho intermolecular dlstanoo, the variation of B with temperature Is easily found. Suppose that the mutual potential energy o of two moleoulos is proportional to 1/r*. Then wo oan write w/hT “ o/(r*T), where o Is a constant, and, If we take r*T as a new variable a, so that r - and r*dr -> we find from the lost expression in eq. (186) u — ~ war 1- * f * J _1 (i - «"5) <fe>. • Jo Before drawing conclusions from this formula, however, we must first discuss the Integral In relation to the value of a. It diverges at the upper limit unless a > 8, so that a value of a equal to 8 or lass! corresponding to spherically symmetrical foroos decreasing at beet as rapidly as 1/r 4 , would lead to a pressure arising mostly from distant parts of the gas and therefore depending upon the shape of the ; con- taining veasol, Now, of tho forces listed in the lost section, types 8 and 4, arising from dipole moments, fall fa diminish with distanpq 230 uxHnr rut nu r at r«* i .*»/*< 1* mr v more laphiiy 1 Imti l/r‘, f till r till Hi** nlhor hand, fm random « »rt* iifntinn of tlt<* molooiilrn Ihoy HVoriigo l<i /t*M( Tin nr mih ■ if* > t tint*, hhuh from till* BolUmium fat Im, il H found It* ilnmtti !■ mti< It rapidly willt (IlslaUiT, KtilOOH <*f typo I * Inn f < t “im|iln * l> « 1 1 •> I n ( m k« on tin* oilier lialid, mo *»j tlmrinnlly '•vnmintrn »il. and I hoy mindly fall tiff only as l/r* (<o a'< I ,‘VI Sin li furor r * mu I umn l« i«n n imiii in a kiin. Amndinglv, if a mi*" 1 "on’ minim »«• I nf pi >i mo hind nf ion, lh<* pionMui* would umloiiltl mil v ilt point uj h * n tin «liu|« ,,} ih,. iimmh of (Im gm, which would llu*ii •amply Im* ngurdid >i » anting accouliuK to Dll* ihuuI laws nf t til upmi if . mill prn , * h irgo TIiih imml Im ti in* of an oloolinii gin, fur nmt jim-n, miiIv o pi, .urn in urinal cuwh im loo amall In la* nmiiMiind Id iih*>! « i , \ Immor, ioilH of oppimiln Migh timir with praotn ally otpnd iimuiistl < id i Imrgo par unit volunu*; (la* not oflont h llu’ii it ililfnnnlml him* mnl i Im cliflievilty in quoMlitm due* not uriw* Will'll OHO of UlOMO 1‘OMilllltillH la nnl, till* inlngtni Inn >i dr filiiln , 1 numerical valtm; anil (In* npmliim flmii tlmi It > T ♦ IW « > 3 llila ia a far wlnwor varinlnm of H with T limn m «dmwn in any of the aix HirvoH in Kig fill, with tin* nxitplmn ptrlmp- »•( tint for helium, whit’ll hIiowk It not fui fiom piopnrlhainl to T it < If. » . might happen if s won* voiy largo i''or fin* ulhoi iivi* gn «*•*, w«* umv nun holt*, thoioforn, llial I hi* mwumplmit that tin* wlnih'<ifwiwpnij*.,iti<,im| »t nil cliHlanrt'H lo iiMiiiglo pimorof (In* ih>liiini< mti*! In* wnl<* <>! tin* truth TIiih in waii'uly mu pi icing, nf mill ho, fm tin In i>|< * uinimiht, dly oxiuhih ropultunu ul miiiiii* ihtdiuier* Iml allisu fimi at ntln r» 132. Clamdcal Calculations of B, I 'or f>* Mini law-* *«f im«|i ,'iilnr foiro, which conihiiu* two rumple typo* am h a« tliovn lj»t* •! in A < I HU, llolniltitl oalculaliniiM nf It lm\i* I m*«*ii mailn in ti tinnil « Iiik>m al him han ic'H, Kci'moiii, in 11)12, fninnl It for iignl nphoriv nilrmtmg in 1 r**, ho hIho worked mil l In* eiiw* nf Mpltnrin mnlnining qiuidriqtxh >», nohnl« ing tho UMMnoialoil polarisation offoein, wlmh requin a ratlnr mm phoalod analyniM huouioti* a Minniinhmi mu**t l«* ninth’ mor varnuiR oiionlalimiH. A much holt or Itypnlltonl* in hy m tho two pmu r fun.* fur whn*h (mleulalimiH won* nmdo hy Jmmmrd ,lniu i ami hi* < nunrkorN * ,\«>,nni ing Hphorioul syimiiolry, thoy wiito (ho l.j ..no mnlnnlo on anolhor al a iIihIiuioi* r, mcnmitod pu,,itivolv in a ropnh'tnn, in tho form * Lknnaiid Jonkm nml CIiiok. Itw t S,* /*„»-. uj, an UWWi, J.nm™ lt„n fioo. Proe , 108, -III!) (1(12 h, It II I ntmit, ‘ Miw,li«nrttl Mrohmm#," r»ntl.n.lso, 102D, ahnplrr h,v I.knnaioi Jnso'*, p ‘ill Sec. 132] THE EQUATION OF STATE 231 X M |Oi) By adjusting the four positive constants X, n, n, in they were able to secure agreement with such accurate observations on B as have been made. The theoretical results wore, however, found to be extra- ordinarily insensitive to the values assumed for n and m; for simplicity they merely always set m — 5 and made n integral, or sometimes, for mathematical convenience, equal to The approximate best values thus found for n and the correspond- ing values of the force constants X and n are shown in the table below. To give a better idea of the spatial extent of the corresponding molec- ular fields, we givo also under the heading <r', as a sort of equivalent diameter, the distance of closest approach of two molecules which approach each other at speeds each equal to the root-mean-square speed at 16°0 and along coplanar lines inclined at 46° to the line j oining their centers. The distance ?'o at which the force itself vanishes, calculated from the equation X/rJS = m/>«' is also given; it is of course much larger than a'. Finally, values of the equivalent hard-sphere diameter <s m as obtained from viscosity data (cf. table in Sec. 86) are appended for comparison. n M X (c.g.H.) a (o.g.B.) <r' (unit 10~ 8 cm) ro (unit 10~ 8 cm) OVd (unit 10~ fl cm) Helium WA 11 5 2,35 X 10~ 11B 2.33 X 10~« 2.04 3.17 2.18 II, 5 ! 7.38 X 10- 80 1.G8 X 10-“ 2.52 3.91 2.74 Neon 11 5 4,38 X 10- 8 * 1,72 X 10““ 2.39 3.70 2.59 N a 9 6 1.58 X 10" 72 1.82 X 10““ 3.49 5.43 3.75 Argon UH 5 1.04 X 10” 111 1.13 X 10““ 3.25 4.21 3 64 The agreement between <r' and a vo is quite good enough to bo satisfactory. Of course a much better test would be actually to calculate the viscosity using the law of force in question. This was done by Lennard-Jones and Cook only in the case of hydrogen, whore the attractive field has relatively little effect upon the viscosity; excellent agreement was obtained, the values found for X being 7.19 X lO” 84 from viscosity and 7.38 X 10~ 83 from the virial coefficient. Revised calculations of this sort are described by Lennard-Jones in the second edition of R. II. Fowler's “Statistical Mechanics,” published in 1936. The above discussion serves, however, to give a general idea of what can be accomplished in this manner. 232 KINETIC THEORY OF GASES [Chap, V If such a field yields theoietical values both of B ancl of iho viscosity agieeing with expenment, we can pm haps oonoludo that it gives us some rough idea of the actual field The significance of the agicomont must not be ovci estimated, however, for two roasons. In the first place, the assumed law of force contains, after all, four adjustable constants, and any fomiula containing so many can bo made to fit a considerable range of experimental data unless the experimental values are much moie accurate than aro tho existing ones for B and the viscosity In the second place, and this is even moro important, classical mechanics is undoubtedly inadequate to givo anything bettei than rough qualitative results in dealing with forces between molecules that aie close together, accuiato results can bo obtained only by means of wave mechanics To this wo shall now turn. 133 Calculations of B by Wave Mechanics* In applying wavo mechanics to the interaction of molocules theio is a ohoioo between two diffeient starting points, and the significance of tho results is decidedly diffeient accoiding to the choice that is made. We may stait out from some arbitrary assumption as to tho law of molecular force, as m the classical calculations just described, and then try to fit the data by assigning suitable values to certain dis- posable constants If agreement with the data can bo secured, such agieement then constitutes evidence in favoi of tho assumed law of force, and the results will possess significance chiefly in proportion to the extent to which the same law can be mado to explain different phenomena On the othei hand, in wave mechanics it is also possible to approach the problem along much more fundamental linos The simple genoial principles of the theory are behoved to be adequate for tho deduction, by mathematical calculation alone, of all of the piopoities of atoms ancl molecules and so even of matter in bulk, Tho fundamental equations contain four univoisal constants, 1 o , Planck's h f tho speed of light, and the charge and mass of the election, to theso wo need add only tho atomic number ancl the nuclear mass of any particular atom in order to be able to deduce all of its piopeitios Unfortunately, howevoi, the piucly mathematical difficulties in the way of canying out this lattei piogiam aie piodigious, and in tho particulai field of moloculai properties little has as yot been accom- plished by this method In regaicl to tho equation of state, little has been done beyond a few calculations ol tho second virial coefficient B i and these calculations, besides employing very rough mothods of approximation, have been limited to helium Tho results obtained by Kirkwood,* however, possess xoal intoiest because of tho fact that * Kirkwood, Phyi u Zeils , 33, 39 (1032) Sec. 133] THE EQUATION OF STATE 233 they wore reached along the more fundamental of the two lines of approach. Kirkwood employs the approximate expression for the mutual potential energy of two helium atoms which is stated in eq. (177a) in Sec. 121 and also plotted in Fig. 51. This curve for <o drops so low at the deepest that there is room for one discrete quantum stato for the pair of.atoms, at an energy ei » —0.6 X 10” 10 erg. This means that a pair of helium atoms can form a sort of molecule, loose but stable, in a quantum state of energy e\\ the center of mass of the pair can then also be moving with any positive amount of kinetic energy. The only other states in which a pair of helium atoms can exist aro states in which they are moving with positivo energy of relative motion and will eventually separate completely, perhaps after first approach- ing and undergoing something like a collision. The method of approximate calculation which Kirkwood follows amounts to assuming that so long as the energy of the relative motion is positivo, classical theory holds nearly enough and the probability for spatial position in dr and velocity in die is Ce-wkT-mMkT dr dK [cf. (88) in See. 52]; whereas, on tho other hand, the probability of the occurrence of tho discrete state of negative energy ei is Ch z c hT [cf. (93a) in Sec. 54],* An indirect method of obtaining tho pressure must then bo employed, since our simple geometrical analysis of it fails when quantization has to be introduced. Tho resulting values of B/IiT for helium aro shown in tho follow- ing table in comparison with tho observed values, tho units being cubic centimeters per mole: T (abs.) 15° 20° 25° 35° 100° 200° 1 300° 350° B/RT (obs.) -14.00 MB 4.80 11.06 BE 11.60 B/RT (oalo.) -10.10 -6.14 H 4.44 10,80 11.37 m 10.82 In view of the rough approximation involved in tho method, tho agreement shown by these figures is surprisingly good. Even the slight drop at high temperatures in. tho values of the ratio B/RT, which according to the hard-sphero equation should approach a con- * When a Byatom ia imagined to bo cnpnblo of existing either in ono of a soriea of discrete quantum states or in a condition to which classical theory applios, tlio classical olomont of phase spaco dp dq is to bo replaced for each quantum stato by h n t n being tho number of coordinates needed to dosoribc the motion (cf. Soc, 233). 234 KINETIC THEORY OF OASES [Chap. V stant value, is given by the theoiy Such a theoretical success is particularly interesting as an example lllustiatmg the general remark made above, because in the entile calculation, including the obtaining of the formula for w, no special quantity is introduced lefomng spe- cifically to helium except its atomic number 2 (and tho fact that its nucleus is veiy much heavier than an election). An improved method has been woiked out by Kirkwood in which the use of classical methods as an appioximation is ontnely avoided,* but this method is hard to handle and numoncal losults liavo not yet been published 134. B for Mixed Gases. When several diffoiont kinds of mole- cules aie present, them will be interaction not only between tho molecules of a given type but also between those of one typo and those of anothei It is immediately obvious that at low pressuics the effects of these interactions will be simply additivo in tho dynamic piessure p A and also, tlierefoie, in the second vnial coefficient B Accordingly, when just two kinds of molecules arc present with respective densities wj, ns, tho expression for p A will consist of throe teims each of the foim of (1846) in See 129 but containing, respec- tively, a,, nm, and n\ as factois and « u , «i 2 , or in place of w, denoting the mutual potential oncigy of a molecule of typo i m the piescnce of one of type j. If wo wiite for convenience ni — y t n, ns = ytfi, whom n - m + n 2 , so that 71 and 72 represent tho frac- tional concentrations of the two kinds of gas 111 teims of molecules and 71 + 7a - 1, it is easily seen that the resulting oxpiossion for B can bo written, as a generalization of (186), B — (71-811 + 27172.B12 + 7IB22), where B t j - 2^/ttyJ o V(l - g w) dr> N denoting as usual the total numbei of molecules and R tho valuo of the gas constant for the whole mass of gas For a bmaiy mixtuio B should thus bo a quadratic function of tho molecular concentrations, Now for a mixture of oxygon and nitiogon the observed relationship seems, rather, to bo lineax in yi } y% } \ this ean happen, howevei, if = ii{B n + J3 22 ), which is very plausible in the case mentioned since the molecular fields of oxygon and nitrogen * Kirkwood, Phys Rev, t 44, 31 (1933); 45, 110 (193d), t IIolboun and Otto, ZciU, PhysiL, 10, 307 (1922) Sec. 136| THE EQUATION OF STATE 235 me probably quite similar. For hydrogen mixed with nitrogen* or lielium,t on the other hand, a quadratic curve fits the data very well. 'The original data on these latter mixtures had reference to B p = B/RT ; the values of B p <j that were found to secure the best fit with the data Were n 2 -n 2 NjrlT* Iia-TIa Ha-IIo Ho-IIo 20 °C —2.80 0.10 0.50 26°G .... 6.50 7.00 5,10 Tho unit ia 10 H in torina of nlmosphorcs mid the standard volume at 0°C, These results aro interesting as indicating that repulsion predomi- nates over attraction, as shown by the positive sign, not only between the molecules of hydrogen itself but also between tho molecules of hydrogen and thoso of nitrogen, and, furthermore, that the repulsive effect is a little greater between a molecule of hydrogen and one of helium than it is when both molecules aro of tho same kind. 136. The Virlal Theorem. In place of tho treatment of the pres- sure in a real gas that wo have given in Sec. 121, the argument can "be rearranged in a more abstract but very neat form known as the virial theorem of Clausius. Tho discussion of tho equation of state will be closed with a deduction of this theorem. Let x, y, z denote tho cartesian components of tho center of mass of a molecule, and lot us write down tho equation expressing Newton’s second law as applied to its ai-coordinato and then multiply this equation through by x. Wo thus obtain cBx -7* cBx \r m -yp “ ."kAj in which X is tho total .r-componont of force on tho molecule. The second of these equations can also be written Now these equations must hold at all times; hence tho average values of the two members of tho last must be equal. Let us take such avorages ovor a vory long time and denote thorn by a bar. Then the average of the first term in the last equation is simply tho total change in mx dx/dl divided by the total time, Now, dx/dt does * Vehschoym, Roy. Soo. Proc., Ill, 562 (1020). t Gimiy, Tannmh, and Masson, Roy. Soc. Proe., 122, 283 (1034). 236 KINETIC THEORY OF GASES [Chap. Y not increase without limit ; x may do so, but the theory of the Brownian motion (q.v.) shows that it increases only as the square root of the time. Hence the average of the first term will ultimately be zero. The average of the second term is minus one third the average of m [(§) +(i) + (j?) provided we assume the motion to be on the average isotropic, hence by (25b) in Sec. 15 it equals kT. Thus in the long run -kT = xX. A similar equation holds for the y- and ^-coordinates and the cor- responding force components, Y and Z. We can also sum up these equations for all N molecules in a given mass of gas and write as our result NkT - -2iX = -2yY = -ZYZ\ (186a) or, adding these three equations and again inserting \rrw 2 = kT , we have the double result that 3 NkT = :M = - 2(xX J ryY + zZ). (1865) The right-hand member of the last equation divided by 2 was called by Clausius the “virial” of the forces, and it is this latter form of statement that is usually called the virial theorem; it can be expressed in words by saying that the mean translatory kinetic energy of the molecules is equal to the virial of all the forces that act upon them. If we can evaluate the virial, the equation of state can be written down at once. The forces consist in part of forces exerted on the mole- cules by the walls of the containing vessel and the term representing their contribution to the virial introduces the pressure into the equa- tion. This term is most simply calculated if we give to the vessel a rectangular form with its faces perpendicular to the axis; then if X\ and x 2 are the values of x at the two faces perpendicular to the £-axis, x 2 being the greater, the contribution at the first face to ZxT can be written xiEX = x x S x p where S x is the area of this face and p is the macroscopic pressure on it; the contribution at the second face is similarly —x% S x p; and the sum of the two contributions is — (z* - xi )S x p = — Vp, where V is the volume of the vessel. Sec. 135 ] THE EQUATION OF STATE 237 If the gas is perfect, there are no other contributions to the virial, and (186a) then gives at once the perfect-gas equation, pV = NkT. Otherwise, wherever two molecules interact, the equal and opposite forces on the two act at points for which, in general, x has different values. The calculation of the resulting term in 2xX, in case the density is sufficiently low so that only binary encounters need be con- sidered, differs only in nonessentials from the calculation given in Sec. 129 above in obtaining an expression for B. If enough knowledge of the molecular motion were available so that 2xX could be calcu- lated at all densities, we could at once find the complete equation of state. The same can be said, however, of the methods employed above, so that the virial theorem does not seem to be of much real use in kinetic theory. Problem . Making the same physical assumptions as in Sec. 129, calculate the contribution to 2xX in (186a) that arises from inter- molecular forces and so obtain the expression for B given in eq. (185). CHAPTER VI ENERGY, ENTROPY, AND SPECIFIC HEATS The physical state of a given mass of gas that is in equilibrium and subjected to ordinary conditions (e.g., not subject to any force-field) is completely fixed when we assign values to any two of the three variables p, V, T (pressure, volume, temperature). These three variables are connected by the equation of state, which was the subject of discussion in the last chapter, and by solving this equation any one of the three can be obtained as a function of the other two. In addition to these three there are other quantities that are characteristic of a gas in equilibrium. Especially important are the energy and the entropy , and, connected with the derivatives of these, the specific heats . These might be called collectively thermal magni- tudes. They will form the subject of discussion in the present chapter. The laws of thermodynamics require certain connections to exist between the thermal quantities and the equation of state. This connection is such that if we know the equation of state and either of the two quantities, energy and entropy, we can calculate the other one; furthermore, it is sufficient to know one of these quantities merely for the gas in its perfect state of vanishing density. Our dis- cussion of the thermal quantities in terms of kinetic theory can accordingly be greatly restricted; it can be limited to the problem of the energy of a perfect gas. In preparation for this discussion, however, it is advantageous first to survey the conclusions that are furnished by thermodynamics. Of course, the general validity of the laws of thermodynamics is also itself one of those properties of gases which it is the object of kinetic theory to explain as a consequence of the fundamental properties of the molecules; such an explanation of the second law of thermody- namics we shall, in fact, undertake in the chapter on Statistical Mechanics. The explanation there obtained is so abstract, however, and our confidence in the validity of these laws is so great, that it is best at this point simply to assume their truth and to direct our prin- cipal efforts toward developing a theory that shall supply the other half of the information which thermodynamics cannot furnish. The preliminary thermodynamic investigation of the relations between thermal quantities and the equation of state is necessarily 238 Sec. 136] ENERGY , ENTROPY , AND SPECIFIC HEATS 239 rather mathematical and abstract. Any reader who is interested primarily in the contributions of kinetic theory to the subject and prefers to take the thermodynamics on faith can without difficulty omit the next six sections entirely and pass at once to Sec. 142. INFORMATION OBTAINABLE FROM THERMODYNAMICS 136. Some Definitions and Basic Principles. Let U denote the internal or intrinsic energy of a gram of gas, i.e., that part of its energy which is neither kinetic nor due to an external force-field. For simplicity it will be assumed that energy of these latter sorts is entirely absent, and hereafter for brevity we shall refer to U simply as the energy (or specific energy, since we are dealing with a gram). In books on physical chemistry the mass is often taken to be a gram molecule, but the advantage in physics seems to lie in the direction of the gram as the unit. In any case, all of the formulas in this chapter will hold for a gram molecule, provided all quantities, even the specific heats, are understood to refer to that unit of mass. Then U can change only through exchange of energy between the gas and its surroundings; and this exchange may occur in either of two ways which from the thermodynamic standpoint are fundamen- tally different. Energy may pass into the gas by thermal conduction, or by some process equivalent to this so far as its effects on the gas are concerned (such as the absorption of infrared radiation) ; energy so passing into a body is called heat On the other hand, the gas may lose energy by doing mechanical work. These three quantities are related by a simple formula expressive of the conservation of energy. During some slight change let the gas absorb heat dQ, undergo a change of energy dU, and do work dW. Then dQ = dU + dW. Under ordinary conditions, however, a gas does work only by expanding while exerting pressure. Confining ourselves hereafter to such cases, we have dW = p dV , V being the volume (or specific volume) and p the pressure; thus dQ = dU + p dV. (187) To this equation, expressing the “ first law” of thermodynamics, the second law then adds the proposition, already noted in Sec. 13, that if the heat dQ is imparted reversibly, we can write dQ = T dS , (188) where T and S are two other definite functions of the state of the gas, i.e., functions of whatever pair of independent variables are being 240 KINETIC THEORY OF GASES [Chap. VI employed to specify its state. From (188) and (187) we have then also TdS = dU + pdV. (189) This equation was, in effect, employed in Sec. 13 in setting up the thermodynamic temperature scale, and in accordance with the pro- cedure followed there, we can interpret T to stand for the absolute temperature, the new function S then representing the entropy. Connections between U and S, on the one hand, and the equation of state on the other, are now arrived at by seeking the mathematical consequences of the equations just written down, especially when taken in conjunction with the fact that both U and S are, by, assump- tion, definite differentiable functions of the independent variables. Equation (189) shows that if the equation of state is known, so that, say, p is known as a function of V and T , then 8 can be found from [/, or, alternatively, U can be found from S, by a process of simple integration; for example, S = /¥ + (189a) It is sufficient, therefore, to discuss in detail only one of these two quantities. Experimentally, S is the more important one of the two, but U is more interesting from the point of view of kinetic theory; accordingly we shall select U for further consideration here. . 137. Differential Equations for the Energy IT. Taking as inde- pendent variables V and T, we can write dU-dU(V,T),(g) r dV+(^) r dT. Then (189) can be written In writing partial derivatives here we have indicated in each case by a subscript the variable that is being kept constant. Now this expression for dS is of the form dS = H dV + K dT; if we make dT = 0, dS/dV = (dS/d V) T by definition of the partial derivative, while if we make dV = 0, dS/dT = (dS/dT) v ; hence 241 SBC. 1371 ENERGY, ENTROPY, AND SPECIFIC HEATS But, since S is a definite function of V and T (and if we may assume its second derivatives to be continuous), = d * S _ d fdS\ dT\dVj T dTdV dVdT~dV\dTj r ' Hence In the case before us this means that __1 dU 1 d 2 U d p 1 d'XJ T 2 dV “ l " T dT dV + QT T ~ TdV df whence o** As an alternative, we might employ p and T as independent varia bles. We need then only to replace dV in (189) by and dU by a similar expression, so that, divided through by T, (189) takes the form dS I (?E\ p T\dpjr + T\dpJ T . +p( dv \ ' ap ^lT\dTj p ^ T \dT/ P Then, by similar reasoning, we obtain dT. (1906) Both (190a) and (1906) obviously rest for their validity in part upon the second law of thermodynamics. When the relation between p, V, and T is known, either (190a) or (1906) constitutes a partial differential equation for U. The two equations are mathematically equivalent, of course, for from (190a) one can pass back to the conclusion that ( dH/dT) v — (dK/dV) T with H and K defined as above, and then by a simple mathematical theorem it follows that H dV + K dT is a perfect differential, or the differential of some definite function of the independent variables, for which we can write dS; one then finds easily that T dS = dU + p dV, as in (189), 242 KINETIC THEORY OF GASES [Chap. Vl and from this we can then deduce (1906) as before. In a similar way we can pass in the reverse direction from (1906) to (190a). From (189), which is reached on the way in either case, one can also pass by the conservation of energy to (187) and (188), which latter expresses the second law. It is therefore evident that any form of the function U that satisfies one of the differential equations just obtained will automatically be in complete harmony with thermo- dynamic requirements. The solution of any partial differential equation contains, however, a large degree of arbitrariness. Additional information is necessary, therefore, for the complete determination of U . The same situation exists as regards the entropy. Problems. 1. Show that thermodynamics requires similarly that the entropy S satisfy the following differential equations (constituting two of what are called the “ thermodynamic relations”): Of), -(H)/ < i9i “- 6 > 2. Show that if U is a function of T alone, then p/T is a function of V only; and, conversely, if the equation of state has the form p = Tf(V ), then U is a function of T only and the heat of free expansion vanishes. 138. Experimental Measurement of Energy and Entropy; the Specific Heats. According to (187) and (188), the energy U and the entropy 8 of a gas can be found from the integrals U = JdQ- jpdV, S = in which T is the absolute temperature and dQ is the heat absorbed by the gas during a small reversible change in its state. Since, how- ever, an integral necessarily contains an arbitrary constant of integra- tion, values of U and S must be assigned arbitrarily to some convenient base state of the gas, e.g., the state at some point Aq on the pV diagram (cf . Fig. 55) ; in other words, it is only changes of U and S that possess physical significance. * We might then determine U and S in any other state A by carrying the gas along any reversible path from A a to A, such as AoBAj and observing the successive elements of heat dQ and of work p dV and the values of T ; from these we could then calculate the * If we take the view suggested by relativistic and atomic phenomena that mass and weight are associated with all energy in a definite ratio, then U itself is physically definite. Such considerations, however, lie entirely outside the range of kinetic theory. Sec. 138 ] ENERGY, ENTROPY, AND SPECIFIC HEATS 243 increases in U and in S. In practice, however, the thermal observa- tions are usually interpreted as yielding values of the specific heats, and the other necessary data are taken from observations on the equa- tion of state. The specific heats are very simply related to U and S. When the gas is kept at constant volume, so that it does no external work, dQ = dU = T dS (the process being assumed reversible). Hence, since we are dealing with 1 g of gas, we can write for its specific heat at constant volume in mechanical units Cv (§\ - (frX - (192o) If, on the other hand, the heating is done at constant pressure, the gas does external work and dQ - dU + p dV — T dS ; hence the specific heat at constant pressure in mechanical units is given by + *(!?)» Cp ~ \dT/ p dT/ P T (dS\ dTj p ’ (1926) in which the value of (dV/dT) p can be found from the equation of state. Here for Q we have written a total derivative because Q is not a definite function of p and T as independent variables, but does become a function of one of them when the other is held fixed. These formulas can be utilized in calculating U and S from observa- tional data in the following way. To calculate the gain in energy of a gram of the gas as it passes from the base point A a where its pressure is po and temperature To, up to any other point A where the pressure is p and the temperature T, let us first suppose it to be heated (or cooled, as the case may be) at constant pressure p 0 along the path A q C to temperature T (Fig. 55). During this process the gain in energy can be written, according to (1926), dT. The last integral here represents simply the change in volume, V(p 0 , T) — F(po, To). Then let us compress (or expand) the gas isothermally from C to its final state A ; using (190a), we can write for the increase in U along this path 244 KINETIC THEORY OF OASES [Chap. VI If we Ww denote by U(po, To) the assumed value of U at the base point At>, We thus ‘obtain finally for TJ at A U(p, T) = XJ(p Q , 'To) + PoW'ipo, To) - V(po, T)] r T fV(.p,T)/ a \ + L^™ dT + T, Lnml dV - (193o) Values of V(p, T) and of (B/dT)(p/T)v, the latter all being taken at the temperature T, are furnished by the equation of state. Of course, other paths of integration may also be followed. Problem. Show in a similar way that S(p, T) = S(p 0> To) + (cp)p=po dT - d V> ( 193fe ) the values of (dV/dT)^ being taken at the temperature T. 139. Specific-heat Relations. From the equations obtained in the last section it is clear that in order to be able to ascertain the energy and entropy of a gas, all that we need in addition to the equation of state is a knowledge of the specific heats. There exist, in turn, a number of relations between the latter and certain mechanical quan- tities which serve to simplify the problem still further. In the first place, we may note the familiar relationship that Cp _ (dp/d F) a Cv (dp/dV) t (194) or the ratio of the specific heats y equals the ratio of the adiabatic and the isothermal elasticities.* The proof of this equation requires only a straightforward application of the conservation of energy, but we shall refer for it to other books. t In the second place, an important expression for the difference of the specific heats can be deduced by means of thermodynamical rea- soning. From (192a, b) * As usually defined, these elasticities are, of course, — V(dp/dV)s, — V(dp/dV)r- t Cf. Poynting and Thomson, “Heat,” 9th ed., 1928, p. 288; also, E. Edsicr, “Heat,” 1936, p. 367, where the proof is indirect. Sec. 140] ENERGY, ENTROPY, AND SPECIFIC HEATS Hence 245 and accordingly / __ /d$\ \( f dp\ \dfjy ~ \dTj p + \&p) 1 \dT)y JdS\ fdp\ Cp - Cv = - T[ — ) I ~ ) > \dp/ r\dT / v or, by (1916), which latter rests upon the second law, (195) the second and perhaps more useful form here results from the mathe- matical relations* (dV\ (dp\ = f§V\ (dV\ = 1 \dpJ 2 \dTjv Vd? 7 // \ d P / T ldp/dV)v The values of the derivatives in (195) are all obtainable from the equation of state. We thus reach the important conclusion that separate observa- tions of both specific heats are unnecessary if the equation of state is known. As a matter of fact, (194) and (195) together could be employed to calculate the specific heats themselves from mechanical data alone, 7 being known from the velocity of sound in the gas and the value of the right-hand member of (195) from the equation of state. Certain qualitative information of a nonmechanical sort would still be required, however, for we should need to know what constitutes an adiabatic compression, and this is not a purely mechanical conception. In practice, furthermore, the mechanical data are scarcely complete enough at present to make it worth while to substitute them for direct observations upon one of the specific heats. 140. Variation of the Specific Heats. The experimental or theo- retical study of one specific heat which thus remains to be made can be reduced further by means of yet a third connection with the equa- * These can be obtained by writing dV dT , in which ( dV/dp)f (dp/0V)T « 1 and hence the last term equals zero; dividing out dT we have then the equation stated, 246 KINETIC THEORY OF CASES [Chap. VI tion of state. For by differentiating the first and last members in (192a) and (1926) with T constant, and then substituting for d 2 S/dV dT and d 2 S/dp dT from (191a, 6), we find fal-i&l- &),--<$,• <“**> In view of these equations it is clear that we need add to the equa- tion of state only a knowledge of c v as a function of temperature at a single volume , or of c p at a single pressure, in order to be in a position to calculate all values of the specific heats, and then from these the energy and the entropy. It is sufficient, for instance, to determine in some way either c v or c v for the perfect-gas limiting case of vanishing density. To a theoretical study of the latter problem we shall accordingly devote the major part of this chapter, tarrying only to note the special forms which some of the preceding equations take in certain simple cases. 141. Thermodynamics of Perfect and van der Waals Gases. If a gas obeys the perfect-gas law, pV = RT, we have at once dp\ R __ p_ (dV\ _R_V dTJy V T’ \dT/ p ~ p~T’ and hence, by (190a) in Sec. 137, The energy U is, therefore, a function of the temperature alone. The reasoning here constitutes the inverse of that in Sec. 13, where we started from the assumption that, for a perfect gas, U is a function of the temperature only and then proceeded to show that the equation of state must have the form, pV — RT. If we know the function U(T), we can evaluate both integrals in eq. (189a) in Sec. 136, the second = R log V ; we find thus for the entropy of a gram of a perfect gas, writing dU = c v dT from (192a), S = j c v ~ + R log V = j c v —jr + R log (RT) — R log p . Furthermore, it follows from (195) that ' Cp “ Cy =: R, giving idV = rJ dV V ( 197 ) Ski'. 1411 ENERGY, EhTROPY, AND SPECIFIC HEATS 247 ho that for a. perfect gas this difference is a constant, representing a constant work difference between the two specific heats equal to the gas const ant R. From this equation we have, in terms of the specific- heat ratio, 7 = c„/cv, cv = y f <V = ~l (198a, b) Differentiating further, we find also that and hence by (196a, b) Thus, for a perfect gas both specific heats and likewise their ratio 7 are, like f, functions of the temperature alone. The actual forms of tint functions U(T), Cv(T), or r„(T) are not fixed by any general principle but depend upon the special properties of the molecules. Ah a special case, if wo know, or are willing to assume, that o is independent of temperature and so is completely constant, we can at once integrate (192a) to find U and so obtain U == CvT + const..; (199) then, the above equation for 8 takes the form 8 ■» <v log 7' + li log V + const., ( 200 a) or 8 «* c p log T — R log p + const., (2006) since V » ItT/p and by (197) c* + J2 ** <y The two constants of inf.egrat.iou differ, if S is the same, by R log It, One of these formulas is often written as the general formula for the entropy of a perfect gas, but strictly speaking the constancy of Cv hardly forms part of the perfect-gas idea; there is no reason why cv should become indepfrulnd of temjH’raturc as the molecules of the gas get farther and farther apart and the gas therefore takes on perfect-gas properties. Kxperimental results, in so far as they exist., seem to show that actual gases approximate to the perfect gas in their thermal behavior under the same circumstances and to about the same degree as they do in regard to their equation of state. The specific heats and 7 all 248 [Chap. VI KINETIC THEORY OF GASES increase, however, when the density becomes considerable; they may ultimately become much larger than at low densities, especially near the critical point. Problem. Obtain the following equations for the van der Waals gas, valid .also at low densities for a gas of weakly attracting hard spheres, and compare them with the equations for the perfect gas: (w\ " w v - v ° m ~ r S - f^ + Klog(V-b). Thus for small values of a/RVT and b/v c P — c v = R + -yf' @02) SPECIFIC HEAT OF THE PERFECT GAS We shall now complete our study of the energy and related mag- nitudes of a gas by taking up the fundamental problem of its energy and specific heat at vanishing density, when it becomes perfect. It is instructive to consider first the conclusions that are indicated by classical theory, which contain a great deal of truth, and then to correct and amplify these results by introducing wave mechanics. 142. Molecular Energy. The energy of a gas, regarded as the energy of its molecules, can be divided more or less definitely into a number of different parts, as follows : a. Translatory kinetic energy. b. Rotational kinetic energy. c. Energy of vibration of the atoms relative to the center of mass of the whole molecule, partly kinetic and partly potential in nature (at least in classical theory). d. Mutual potential energy of the molecules as wholes. e. Internal atomic energy. Even in classical theory the distinction between 'these different ( types cannot always be drawn sharply, but it is sufficiently definite to make the classification helpful. At very low densities the mutual energy, type (d), becomes negligi- ble, and hence it will not further concern us here; its effect is, in fact, covered by the various connections with the equation of state that 249 Sec. 143] ENERGY , ENTROPY , AiV'D SPECIFIC HEATS -we have deduced from thermodynamics. The translatory kinetic energy of the molecules moving as wholes is likewise quite distinct a-t low densities from the other forms. The remaining types, which we have been lumping together and calling internal molecular energy, are much more closely interrelated; both classical and quantum theory Indicate, however, that they can be distinguished clearly enough so long as they are of quite different orders of magnitude, and such is the case at least when the temperature is sufficiently low. Each of these types of energy is a definite function of the coordi- nates and momenta, and if the form of this function is known, the average value of this part of the energy can be found by the methods of statistical mechanics. For one type of energy classical theory furnishes an extremely simple general rule: According to the famous principle of the equipartition of energy , if any part of the energy of a system is simply proportional to the square of a coordinate or of a component of velocity or momentum, then, when the system is in ■thermal equilibrium at temperature 1\ the mean value of this part of the energy is \ kT, k being the gas constant for one molecule (cf. Sec. 206). Now the translatory energy of a molecule is the sum of three such 'terms, corresponding to its three degrees of translational freedom; it can be written, for instance, (p* + pi + p£)/2m in terms of the cartesian components of momentum, p Xy p V9 p z . The mean translatory energy of a molecule should accordingly be f kT, in agreement with ■the usual elementary result. Multiplying this value by the number of molecules, we have therefore for the translatory energy of a gram U t = 4 RT, (203) JR being as usual the gas constant for a gram. 143, The Classical Theory of Specific Heat. We can conceive of molecules that possess no other form of energy than translational. This would be true, for instance, if they were simple mass points Incapable of rotation, or if they were smooth spheres initially devoid of rotation and having the center of gravity at the geometrical center, bo that their state of rotation could never be changed by any molecular Impact. For a gas composed of such molecules the total energy would Toe U = Ut] the specific heat at constant volume would accordingly "be, by (203), cv = ( dU/dT) v = f R, and, since for a perfect gas — cv = It [cf. (197)], the specific heat at constant volume would be c p = I R, and for the ratio of the two we should have c p /c v = If the gas constant Rm for a gram molecule or mole is substituted for JR we have § Rm as the specific heat in terms of moles; substituting 250 KINETIC THEORY OF GASES [Chap. VI R m = 83.15 X 10 6 and dividing by 4.186 X 10 7 , we find for this 2.98 cal per mole. This result is often cited as a specific heat of about 3 cal per mole for a monatomic gas. Since any internal energy that the molecules may possess is almost certain to increase rather than to decrease with a rise of temperature, its presence should increase not only U but also the specific heats. Accordingly, we are led to expect all real gases at low pressures to satisfy the following inequalities: U £ | RT, c v ^ | R, c p t | R, y g (203a, b, c, d) The last two of these is based upon the assumption that c p — c v = R. The next simplest type, after the simple mass point, would be a dumbbell , or two atoms rigidly united into a molecule possessing a single axis of symmetry. Such a molecule would be incapable of any change in rotation about this axis; it would have, therefore, two degrees of rotational freedom, corresponding to independent rotations about two axes perpendicular to each other as well as to the axis of symmetry. The principle of equipartition also asserts that the mean kinetic energy associated with each of these degrees of rotational freedom would have an average value of % kT (cf. Sec. 206). Accordingly, the molecules in a gram would possess, in addition to U t , rotational energy of magnitude Ur = RT and we should have U = U t + U r = f RT, c v = -f R, c p = $ JS, 7 = b , If, on the other hand, the molecule were rigid but possessed no axis of symmetry, as would almost certainly be the case if it were composed of three or more atoms not lying on the same line, all three degrees of rotational freedom would take up their share of kinetic energy and we should have U r = | RT; hence U = ZRT and c v = 3 R, c v = 4#, y = %. Another possibility is that the atoms may vibrate relative to each other. If these vibrations are of rather small amplitude, they should be very nearly harmonic; the expression for the corresponding potential energy would then be of the squared form to which the principle of equipartition applies, e.g., | aq 2 , where q is a coordinate and a a constant, and so would likewise average \ kT, The total mean energy, kinetic and potential, associated with each mode of vibration would then be kT , and the total amount of this kind of energy in a gram would Sue. 144] ENERGY, ENTROPY, AND SPECIFIC HEATS 251 be RT. A gas composed of dumbbell molecules in which the atoms can vibrate along the line joining their centers would thus have energy U = U t + U r + RT — RT, so that cv = % R, c P — % R, y = A general formula can easily be worked out for an asymmetric molecule containing any number r of atoms which are capable of rela- tive vibration in simple harmonic motion. We require 3r coordinates to specify the positions of the atoms, but three combinations of these represent the position of the center of mass of the whole molecule and three more are accounted for by the three possible independent rota- tions; hence there are only 3r — 6 independent modes of vibration. Thus there will be in a gram U v — (3 r — 6) A 7’ ergs of vibrational energy, U T = | RT of rotational and U t = -f RT of translational, or a total of U = 3(t — 1)RT ergs: consequently c v = 3(r — 1)12, c p = (3r — 2 )R, 7 = ^ _ yj - The specific heats in an actual case might be smaller than these values if some of the atoms were rigidly bound together; or, if some of the modes of vibration were anharmonic, the specific heats might even exceed the values stated, but probably not by a great deal. The classical results thus worked out are summarized in the follow- ing table, the column headed U/U t giving in each case the ratio of the total energy to the translatory energy alone : Molecular type u/u, cv/R Cj)/ R 7 Spherically symmetrical 1 % 2 % 2(r - 1) X X 3 % 3(r - 1) H Yi 4 94 3r - 2 % = 1-667 % = 1.400 % = 1.333 y, = 1.286 1+ 1 L + 3(r - 1) Dumbbells Rigid nonsymmetrical Diatomic, vibrating S.H t atoms vibrating S.H. (nonsym- metric, r >2) 144. Comparison with Actual Specific Heats. For comparison with the theoretical formulas the principal thermal data for the com- moner gases and a few organic ones under ordinary conditions are col- lected in the annexed table on p. 252. Under the heading y is given the ratio of the two specific heats, mostly determined from the velocity of sound, and under the head- ing c' P is given the specific heat at constant volume stated in terms of calories per gram for convenience in comparing with other specific heats; c' P must, therefore, be multiplied by the mechanical equai valent of the calorie, J = 4.186 X 10 7 ergs, to obtain the quantity c„ that occurs above in our formulas. In the fourth column are shown val- 252 KINETIC THEORY OF OASES [Chap. VI Specific Heats Gas 7 r c v Cp — — Cy R cv R Helium 1.659 (18°) 1.64 (19°) 1 . 66 s 1.252 (18°) 1.001 1.519 Neon Argon 0.125 1.008 1.509 Krypton 1.68 (19°) 1.66 (19°) 1.410 Xenon h 2 3.39s .9995 2.438 HC 1 1.40 (20°) 1.404 0.194 1.02 2,54 n 2 0.247s 1.005 2.448 CO 1.404 0.248 1.005 2.488 0 2 1.401 0.218o 1.004 2.504 NO 1.400 0. 233i 1.005 2.512 Cl 2 1.36 0.115 1.09 3.02 H 2 0 1.32 (100°) 1.32 (18°) 1.304 0.48 (100°) 0.253 1.06 3.3 H 2 S 1.05 3.29 co 2 0.199 1.027 3.38 so 2 1.29 0.152 1.10 3.79 NHa 1.31 0.524 1.06 3.42 c 2 n 2 1.25s 0.205 1.09 4.27 CH 4 (methane) 1.31 0.529 1.01 3.25 C 2 H 4 (ethylene) 1.25s 0.360 1.03 4.04 C 2 He (ethane) 1.22 0.386 1.05 4.78 C 2 HoO (ethyl alcohol) 1 . 13 (90°) 1.08 (35°) 0.454 1.21(?) 1.23 9.3(?) 15.4 C 4 H 10 O (ethyl ether) (100°-223°) 0.445 (35°) The data are for 1 atmosphere pressure and, unless otherwise stated, 15°C. cp and Cp are in mechanical units, cp in calories per gram. {q — (*y\ (' y — * J ues of the ratio — - calculated as or mostly as (y — 1 )C P X 10 7 /yR M) Rm being the molar gas constant or 8.315 joules per degree and C p , the heat capacity of a gram molecule in joules per degree. The fifth column contains values of c v /R, Cv being calculated as c' p J /y. The data were taken largely from the International Critical Tables, where the values given are mostly those of C p , from which c r p was found by dividing by 4.186 and by the molecular weight. A glance at the table shows at least that inequalities (2036) and (203d) are always satisfied. It is really sufficient to discuss only one of the two quantities c v and y, since, as we have seen, the other is determined in terms of it by means of relations obtained from thermo- dynamics, but as a matter of interest we shall discuss both. We note that for five gases y lies close to the theoretical value of % that was found in the last section for mass points; and for two of these Sec. 144 ] ENERGY, ENTROPY, AND SPECIFIC THE ATS 1253 cv/R is known and lies close to the theoretical wdke of 1.5. These are the rare gases, whose molecules are believed to consist of single atoms. When these gases were first discovered, to be sure, their monatomic (character was actually inferred from observations on y and the (interpretation lin ithe light =of kinetic theory, for the reason that these (gases scarcely (enter into chemical combination, but modern atomic (theory has now lent strong support i.te ..the conclusion that they have (monatomic molecules. There follows next a group from 'B^tte lNC) for which y is close to 1.4 and Cv/R fairly close to 2.5, which are the .theoretical values for rigid dumbbell molecules. Since there are abundant chemical reasons for believing these gases to be diatomic, classical theory was able to explain their values of y and Cv very nicely by supposing the two atoms to be bound rigidly together in the molecule and hence incapable of vibration. The next gas in the table, however, Cl 2 , is also certainly diatomic, yet it has 7 = 1.36 and c v /R = 3.02. These values do not correspond exactly to any simple classical type; if the explanation of the departure lay in a classical vibration of the atoms, which is the most attractive supposition, we ought to have y — 1.286 and c v /R — 3.5 (cf. table at end of preceding section), whereas if the dumbbell were rigid but some- how asymmetric we should have y = 1.333 and cv/R = 3.0. The latter number agrees well with the experimental value, but the value of 7 definitely does not ; such an assumption is, moreover, very improbable. Thus in this case classical theory fails. All of the gases in the table with more than two atoms in the mole- cule have 7 less and c v /R greater, respectively, than the theoretical values % and 3 for rigid asymmetric molecules. This much is satis- factory, since their molecules all contain at least three atoms; and we note also that cv/R never exceeds the value 3(r — 1), which is the maximum that could be accounted for by allowing all of the r atoms in a molecule to execute classical simple harmonic motions. No definite progress can be made, however, using classical ideas, toward accounting for the data in detail. It is interesting to note that the classical results were supposed to be applicable to liquid and solid phases as well as to gases. In the case of pure elementary substances the molecule was commonly thought to be monatomic in condensed phases, the individual atoms moving inde- pendently. If we then assume that in a solid the atoms of such a substance vibrate approximately in simple harmonic motion, we arrive at the conclusion that the heat content should be just twice that of a monatomic gas, the energy averaging half kinetic and half potential, and the specific heat should accordingly be 31?. The heat capacity of a 254 KINETIC THEORY OF GASES [Chap. VI mole would then be 3 Rm or 3 X 83.15 X 10 6 /4.186 x 10 7 = 5.96 or about 6 cal per mole. Now, it is a fact that for most solid elements at ordinary tempera- tures the specific heat is not far from 6 cal per mole. There are a few notable exceptions, however, such as diamond. Furthermore, later work has shown that the specific heat of all substances drops eventually if the temperature is lowered sufficiently, apparently tending toward 0 at T = 0. The first satisfactory explanation of these facts was given by Debye in 1912, but it lies outside the scope of this book.* 145. The Specific-heat Difference. The values of (c p — c v )/R are in quite a different status from those of y or c V) since the value of c p — c v is fixed by eq. (195) above in terms of quantities derivable from the equation of state. Departures of ( c p — c v )/R from the perfect-gas value of unity thus serve as an indication in thermal data of a departure from the behavior of a perfect gas, and some interest attaches to them for this reason. For the first eight gases for which values are given, with the excep- tion of HC1, and for methane, ( c p — c v )/R is actually within 1 per cent of unity; these are all gases whose critical temperature lies below 0°C (even that of methane being — 82.5°C). The remaining gases in the table all have critical points above 0°C and would be expected to show greater departures from the perfect gas. We found above that a gas composed of hard weakly attracting spheres, or a van der Waals gas, should obey eq. (202) in Sec. 141; since a is, according to its theoretical origin, necessarily positive, for such a gas ( c p — cv)/R should exceed unity. This is uniformly the case for actual gases except for hydrogen, where the slight defect in the ratio may easily represent experimental error. 146. The Problem of the Internal Energy. It is obvious from our discussion that classical theory was unable to deal in any satisfactory way with the internal energy of the molecules. The worst feature of the situation was that classical statistics indicated a mean value of \ kT for the kinetic energy associated with each degree of freedom, and the nature of atomic spectra pointed very clearly to the existence of many internal modes of vibration even within a single atom; it was necessary to assume that for some unknown reason these modes did not contribute appreciably to the specific heat. The difficulty was not quite so serious in the present connection as in the theory of radiation, where the degrees of freedom of the electromagnetic field were actually infinite in number, and where the * Cf. F. K. Richtmyek, “Introduction to Modern Physics,” 1934, p. 280. Sec. 147] ENERGY, ENTROPY, AND SPECIFIC HEATS 255 difficulty had in 1900 led to Planck’s invention of the quantum theory; but as time went on, matters became worse instead of better. The success of the Rutherford atom in 1912 was the last straw; for the concentration of the positive charge into a minute nucleus opened up a deep hole in which, according to the classical Boltzmann formula, the electrons would be completely swallowed up. The poten- ' tial energy of an electron of charge e at a distance r from a nucleus of positive charge — Ze being <o = — Ze 2 /r, the probability, according to the Boltzmann formula (92a) in Sec. 55, that the electron is in an Ze* element dr of space is C mq e rkT dr, and the chance that it lies at a dis- tance between r and r + dr from the nucleus is, therefore, Ze * F r dv = 4 tt r 2 C mq e rkT dr; the total chance of its lying within a distance n of the nucleus is then fri C r I — C 00 Ze ^ P r dr = 4 tt C mq r 2 e rkT dr = 4 7 rC mq e % Jo Jo J l/n M 4 This is infinite unless C mq = 0, in which case the electron would have to be right on top of the nucleus. Any departure from the Coulomb law that could reasonably be assumed did not help matters much. The first step toward a solution of such difficulties was the partial substitution of quantum for classical ideas in the atomic theory pro- posed by Bohr in 1913. Accordingly we shall turn now to the treat- ment of the internal energy that is furnished by modern wave mechanics. 147. Quantum Theory of the Specific Heat. The different parts into which the molecular energy can be divided, as described above in Sec. 142, fare differently in wave mechanics. As regards the tranala - tory kinetic energy, it can be shown (cf. Chap. X) that in practically all cases the classical expression is correct within the limits set by experimental error, and in those special cases in which perceptible deviations of quantum origin can occur, the effects of molecular forces are sufficiently great to mask the quantum effects. Furthermore, in the limiting case of indefinitely low density the quantum effects disappear entirely. Hence we can write with complete accuracy for the translatory energy of a gram of indefinitely rare and therefore perfect gas U t = -I RT } as in eq. (203) above. 256 KINETIC THEORY OF GASES [Chap. VI The internal energy , on the other hand, requires radically different handling, as was explained in Sec. 54. If for simplicity we suppose the various fundamental quantum states of a molecule to be numbered off in a single series, then for statistical purposes we may imagine the molecule to spend a fraction Pj of its time in each quantum state of energy where Pj has the value given in eq. (936), viz., Pi = e kT /Z, Z = kT , (204a, b) 3 the sum in Z extending over all fundamental quantum states. The mean internal energy of such a molecule will thus be 6 = 2 > 3 - p ,- = e ~^! z ( 205 ) 3 3 and the internal energy of a gram of gas containing N such molecules will be Ui = N^i e~^/Z = RT 2 -A log Z (206) J where R = Nk or the gas constant for a gram. The quantity Z( T) which thus plays an important role in the theory was called by Planck the state sum (in German, “Zustandgsumme”); it has also been called the “partition function.” We could equally well include in it by definition the factor N, so that it would have reference to a gram. If we know the energies e,- of the molecular quantum states, we can calculate Z and Ui as functions of the absolute temperature T; the specific heat at constant volume can then be calculated as c v = A (JJ t + Ui). (207) The quantum states for a given molecule ordinarily fall naturally into a number of distinct groups corresponding to the fact that the energy can be separated approximately into rotational energy of the whole molecule, vibrational energy of the atomic nuclei, and electronic energy, and for this reason it is customary to number them by means of not one but three quantum numbers; the latter are often denoted by n, v, J } being assigned so that changes of J imply chiefly changes in the rotational motion, of v , in the relative motions of the nuclei, and of n , in the electronic configuration. The change in energy involved in a jump of one of the quantum numbers from its lowest value to the next higher, which is the most important jump from our standpoint, is of a different order of magnitude in the three cases; expressed in Sec. 148] ENERGY, ENTROPY , AND SPECIFIC HEATS 257 terms of kTu, the value of kT at 15°C or two thirds of the mean kinetic energy of a molecule at that temperature, the first step in rotation or in J requires about 0.15 to 0.3 kT i5 ergs, whereas the first step in vibra- tion or in v requires 1 to 10 kT i 5 , and the first step in electronic excita- tion or in n usually requires at least 100 kT n . A further multiplicity due to the nuclei must often be allowed for, either by the use of additional quantum numbers or by the intro- duction of suitable statistical weights or multiplicity factors into formulas such as (205) or (206). (Cf. Sec. 54.) This is illustrated in the discussion of hydrogen below. 148. Variation of Specific Heat with Temperature. In the light of the facts just stated the general course of the specific heat as a func- tion of temperature can at once be predicted. If we first go to extremely low temperatures, the probability Pj of the state in which the internal energy €,* has its lowest value is very much larger than is Pj for any other state, and the molecules remain, therefore, nearly all of the time in this lowest state; all terms in the series for Uj in (206) are then extremely small in comparison with the first term, in which tj = ei, and the series for Z in (2045) likewise reduces to its first term alone. Thus Ui = Ne i and is independent of temperature, and the specific heat reduces to dU t /dT = 322/2, just as for a gas of mass points. If we then gradually raise the temperature, in the case of polyatomic molecules the higher rotational states eventually begin to be occupied; and a little consideration shows that when kT comes to exceed the sum of the first few energy steps between these states, there will be’ an approximation to the classical value of the rotational energy. This condition can often be met before the higher vibrational states begin to occur with appreciable frequency; in such a case there may be a con- siderable range of temperature over which c v has its classical value, including the part that represents rotational energy but nothing more. When the temperature is raised sufficiently, however, vibrational energy will begin to occur in appreciable amounts, and c v will then increase further. Electronic excitation, on the other hand, can occur in appreciable amount only at temperatures of the order of a hundred times normal, or above, say, 20,000°. It should be noted that in the electronic energy there is included all energy of rotation of monatomic molecules, and also, in the case of molecules with collinear nuclei, the energy of rotation about the axis through the nuclei. The general shape thus predicted for the specific-heat curve of a perfect gas with polyatomic molecules is shown in Fig. 56. In the case of the more permanent gases there is evidence to show that the curve actually has such a form, the gas at ordinary temperatures being on 258 KINETIC THEORY OF GASES '[Chap. VI the flat part of the curve between A and B ; only in hydrogen, however, does the drop at A occur at easily accessible temperatures. The success of the classical theory in dealing with these gases is thus accounted for. Most gases are at B or still farther to the right. In order to construct a quantitative theory, we might now adopt some simple model for the molecule and try to adjust its assumed moments of inertia and vibrational properties in such a way as to make the theoretical values of cy at different temperatures fit the experimental data. Several attempts to do this were made in the case of simple molecules such as hydrogen, but for various reasons complete success was never achieved. A much better procedure, in general, is probably to make use of the rich material concerning molecular energy levels that is furnished by the study of band spectra and to leave the theoret- ical interpretation of the levels themselves as a problem for the theoretical spectroscopist. The la- bor involved even in this procedure is considerable, and it is increased by a peculiar complication due to nuclear spin; but several cases have Fig. 56. Specific ^heat of a polyatomic WO rked Out. As an example the famous case of hydrogen will be discussed in detail presently; but first it will be worth while to consider briefly an ideal case that can easily be treated completely. 149. The Case of Harmonic Oscillators. As a special case, suppose the molecules can vibrate harmonically in some way with a definite frequency v . Then there will be some coordinate q that can vary sinusoidally with the time and can be written q = a sin 2wv(t + 6), and there will be a corresponding term in the energy of the form %(aq 2 + 0q 2 ) where q = dq/dt and a and are constants. In such a case wave mechanics predicts a series of quantum states whose energies are of the form (n + M)hv, n being a positive integer or zero. The state sum for such an oscillator is easily calculated. We have only to establish first the mathematical formulas e x e x = i- o (l y-\ dx^/ e -jx — (1 - e~~ x ) 2 Putting x = hv/kT in these formulas and e,- = (j + }4)hv in (2045) and (205), we obtain at once for the state sum Z and the mean energy Sec. 149] ENERGY , ENTROPY , AND SPECIFIC HEATS Z per oscillator hv 00 jhv Z = e = hv 2kT hv ™ i nv " j/n> 6 = Jive 2 kT '^j e kT /Z~\~ 2 ^ ve 2kT "^ e kT /Z hv i = o jhv hv plf 259 (208a) hv jhv or y-o J“0 hv e kT — 1 +!*’• (2086) IV molecules in a gram, each containing such an oscillator, would then contribute IVe to £/*, and to the specific heat Cv the amount Cw d dT m R h 2 v 2 ik 2 T 2 / siuh 2 hv 2 kf (209) in terms of R — Nk and sinh x = (e x — e~ x )/2. In Fig. 57 are plotted l/hv and c Vv /R against the temperature T. For large T or, more exactly, for small hv/lcT, these quantities approach the classical values, kT and 1; for, if we expand the exponential that occurs in (2086) or (209), we obtain the series : i = kT(l + 1 h 2 v 2 Cw = r(i 12 k 2 T 2 1_ h 2 v 2 12 WT 2 ' > )• u u ■ ■ IE! ■ sa u u ■ B m ■ ■ ■ m & 21 11 p m u i ■ ■■ H m m ■ ■ ■ 8 ■ u ■ ■ m i i ■ i ■ ■ 0.2 0.4 F hi/ —The harmonic e = mean energy, 1.0 1.2 1.4 oscillator. For small T, on the other hand, Cw sinks exponentially to 0. It might have been anticipated that at some Fig - 57 intermediate temperature it would rise above the classical value in order to make up the deficit in energy that should exist at low temperatures, but no such rise occurs; according to present theory, there is really not a deficit but an excess of energy, due to the zero-point energy or the constant term \hv in Z. The physical significance of this term is not too clear, however; if it were missing, as in the older quantum theory, the energy at high tempera- tures would simply remain permanently below the classical value by the amount i hv. 260 TIN’S TIC THEORY OF GASES [Chap. "Vt To give some idea of numerical magnitudes, we may note that at 15°C the specific heat cw would be 1 per cent below R for a frequency v = 0.346 = icTu/h = 2.08 X 10 12 , corresponding to a radiant wave length '3 X 10 10 /2.08 X 10 12 cm or 0.14 mm. As the frequency rises, the difference between cw and R increases.* For the normal oxygen •molecule the second vibrational level lies above the first at a height -•equivalent to a frequency of 1,556 cm" 1 or v = 4.67 X 10 13 . If we 'treat this as a simple harmonic mode of vibration, which is certainly Jjustifiible at ordinary temperatures, we find from (209) that it corn- tributes to the specific heat'Cy, = 0:O26 jR. This is just appreciable. (; Chlorine, on the other hand, has a first vibrational frequency of about 560 cm" 1 , for which at 15°C, c Vv = 0.542. This is just about right to 'accouht for the observed excess of 0.52 R above the dumbbell value, 'Cv = 2.5i?.* In hydrogen and nitrogen the vibrational frequencies lie too high to affect the specific heat at ordinary temperatures. The hydrogen molecule, on the other hand, has such a small moment of inertia that the drop in its rotational energy occurs at a moderately low temperature. The behavior of this gas will accordingly be discussed in detail. 150. Hydrogen. The hydrogen molecule consists of two electrons and two nuclei or protons so tightly bound by the force-actions between the various particles that, as already remarked, at ordinary temperatures the vibrational and electronic energy practically never vary. Included in the electronic energy is also the equivalent of rotational energy of the electrons about the line joining the nuclei. There remains, therefore, as internal energy that does vary under ordinary conditions, only energy of rotation of the molecule as a practically rigid dumbbell about an axis perpendicular to the nuclear line. The quantum states for such a rotation are shown in wave mechanics to have the energies h 2 JV + 1), (210) where J is a positive integer or zero, h is Planck's constant, and I is the moment of inertia of the molecule about an axis perpendicular to the nuclear line.f In the case of hydrogen the moment of inertia arises almost entirely from the large masses of the nuclei, the electrons being 1,821 times lighter and so negligible; and its value is known from the spacing in * Cf. also Tratjtz and Ader, Zeits. Physik , 89, 15 (1934). t Cf. Condon and Morse, “Quantum Mechanics,” 1929, p. 69. Sbc. 150] ENERGY, ENTROPY, AND SPECIFIC HEATS 261 band spectra to be 7 = 4.67 X 10~ 41 c.g.s. units.* The factor h 2 /8w 2 I in (210) has thus the value, (6.62 X 10- 27 ) 2 8tt 2 7 1.19 X 10~ 14 erg = 0.30 kT u , so that only about the first three states (J = 0, 1, 2) would be well represented at ordinary temperatures; hence appreciable deviation of the specific heat ought to set in upon a moderate lowering of the temperature. The first attempts at a quantitative theory of the specific heat of hydrogen failed, however, because two important principles of quantum theory were unknown until about 1927. In the first place, nuclei as well as electrons exhibit the phenomenon called spin, or something equivalent to it. One aspect of the spin is an internal angular momen- tum, whose total value in the case of a simple particle like a proton or electron never changes but whose component along any chosen axis when quantized can take on only one of two possible values, cither Yl or — times h/2r. For two protons we should accordingly expect four times as many independent quantum states as there would be without spin; in the absence of interaction between the protons, the corresponding wave functions would be obtained by taking each allowed function in terms of the spatial coordinates and assigning the spin Y or — Yi in succession to each proton separately. But then a reduction in the number of quantum states occurs in consequence of the second of the two new principles, the Fermi- Dirac-Pauli exclusion principle, which applies to any set of identically similar particles. When we have obtained any wave function for such a set, we can always form another one corresponding to the same energy merely by interchanging in the given function the coordinates of any two particles. The exclusion principle now asserts that, for some reason as yet unknown, only those quantum states occur in nature for which the new function thus obtained is merely the old one changed in sign, and so does not represent a new quantum state. Functions having this property are said to be antisymmetrical in the coordinates of the particles. As a consequence of the exclusion principle, in the case of the dumb- bell rotator, rotational quantum states with even J would not occur at all if there were no spin, for the wave-functions of these states without spin are symmetrical in the particles, only those with odd J being Horn, Zeits. Phys., 44, 834 (1927), 262 KINETIC THEORY OF GASES [Chap. VI antisymmetrical.* When spins are introduced, however, it turns out that the functions for even J can be made antisymmctrical in the spins and so can be used. Those for odd J, on the other hand, can be made symmetrical in the spins and so antisymmetrical on the whole; and it happens that this can be done in three different independent ways, in two ways with the spins parallel and in one way with them anti- parallel, so that we obtain three different functions of this type for each odd value of J, as against only one for each even value. Thus there are two distinct types of hydrogen molecules. One type, which is called parahydrogen } has wave-functions antisymmetric in the nuclear spins and rotates always with an even value of J, includ- ing the state of no rotation at all with J = 0 ; the other type, called ortho- hydrogen, has wave-functions symmetric in the nuclear spins and rotates with an odd value of J. The para molecules have 2J + 1 quantum states for each value of J, corresponding to 2 J + 1 different possible quantized values of the component of angular momentum about any chosen axis, or, the statistical weight or multiplicity of the multiple state J is 2 J + 1 ; but the ortho molecules have three times as many states or a multiplicity of 3(2 J + 1). 161. Para-, Ortho-, and Equilibrium Hydrogen. The first treat- ment of the specific heat of hydrogen in which allowance was made for these new features of quantum theory was that of Huncl,f but his results did not fit the facts. Hund assumed that individual molecules would pass freely back and forth between the two types. Dennison then pointed out that,J since the process of conversion from one type to the other ought theoretically to be an extremely slow one, the proportion of the two types would probably not change appreciably during the time in which the gas changes temperature in an ordinary measurement of specific heat. Hydrogen should therefore behave like a mixture of two gases which can transform into each other at a slow rate and so will come to a definite equilibrium of relative concentration when the gas is allowed to stand long enough, but which will not remain in equilibrium when a change of temperature is made quickly. In ordinary experiments on specific heat we are dealing, therefore, with a mixture that is practically fixed in composition. The equilibrium * Polar angles 0 , <p can be used to describe the rotation, with the axis drawn from one particle to the other; then the wave-functions for even J are of such forms as 1, 3 cos 2 0-1, e itp sin 6 cos 0 [i.e., e im<p Pl n (cos 0) with even l], and such functions retain their value if we substitute in them <p + t for <p and t — 0 for 0 to represent an interchange of the cartesian coordinates of the particles, whereas the functions for odd J or odd l , such as cos 0, e H(p cos 0, etc., change sign. f Hund, Zeits. Physik , 42, 93 (1927). x Dennison, Roy. Soc , Proc., 115, 483 (1927). Sec. 151] ENERGY, ENTROPY , AND SPECIFIC HEATS 263 composition itself will, however, vary with the temperature at which the equilibrium is established, because of the differences in the Boltz- mann factors for even and odd J. This extraordinary theoretical conclusion has been confirmed by experiment. A good description of the relevant facts is contained in a recent book by Farkas.* At 20°K the molecules tend to crowd into the lowest state of no rotation, and in consequence equilibrium hydrogen at this temperature, according to eq. (212) below, is 99.8 per cent pure parahydrogen. It has been found that the process of con- version from the ordinary mixture into this form, which would take three years to go only halfway under standard conditions, can be catalyzed by adsorbing the gas on charcoal, so that at 20°K the process goes practically to completion in at most a few hours. If the tem- perature is then raised, the gas remains in its new form for a long time. Thus practically pure parahydrogen can be prepared and experimented upon; by comparison of its properties with the ordinary mixture, the properties of pure orthohydrogen can then be inferred indirectly. In most respects the two forms differ very little, but in specific heat and in related properties, such as thermal conductivity, as also in their band spectra, they differ decidedly. An exact expression for the composition of equilibrium hydrogen at any temperature can easily be obtained from the Boltzmann proba- bilities. The relative numbers of molecules in the various quantum states are given by (204a) in Sec. 147 with ey replaced by ej as given in (210) above or «/ = J(J + 1) xkT, x = ~ = ~ (211a, b ) For convenience we may, as explained in Sec. 54, group together states having the same energy and hence simply write for the probability of any multiple J state Pj ej (2 J + l)e w Z for even J and „ _ 3(2 J+ l)e w Pj z for odd J. The fractional part of the equilibrium gas that is in the para form will then be SPj- summed for even J beginning with 0, divided by 2P/ summed for all J, or * Farkas, “Orthohydrogen, Parahydrogen and Heavy Hydrogen,” 1935. 264 KINETIC THEORY OF GASES [Chap. VI 1 + 5e~ 6 * + 9<r 20 * + • • • 1 + 3 X 3e” 2 * + 5e” 6 * + 3 X 7e" 12 " + 9e" 20a: + • * •* K } This formula gives for the per cent of parahydrogen in the equilibrium mixture: 99.8 per cent at 20°K, 88.6 per cent at 40°, 38.5 per cent at 100°, 25.7 per cent at 210°, 25.13 per cent at 273°K. Thus at room temperature the ratio of para to ortho has practically its limiting value of 1:3; this latter conclusion is confirmed by the distribution of the intensity in the band spectra of ordinary hydrogen. 152. Specific Heat of Hydrogen. In a similar way, by inserting c,/ from (211a, 6 ) for ej in (206) and (2046) in Sec. 147, we obtain the following expression for the rotational energy of a gram of equilibrium hydrogen containing N molecules: U„ = RT‘±lo g Z.--gg±lo s Z., Z e = 1 + 3 X 3e~ 2x + 5er 6 * + 3 X 7er 12x + 9e~ 2Gx + • • • , in which x is given by (2116) whereas R = Nk and represents the gas constant. If the temperature were now changed so slowly that the hydrogen remained continuously in equilibrium, its rotational specific heat at constant volume would be dUje r> 2 d 2 . „ Cj ° ~ ~dT~ ~ Rx dtf l0g Ze ‘ Corresponding expressions for para- and orthohydrogen arc obtained by including only terms for even or odd values of /, respec- tively, thus: { n Cjp = Rx 2 ~ log Z 9 , Z p = 1 + 5e~ 6x + 9<r™* + • * * , c Jo = Rx 2 ^ 2 log Z 0 , Z 0 = 3c~ 2 * + 7e~ 12x + • • • . Finally, if a mixture containing fractional parts y p and y 0 of para- ancl orthohydrogen, respectively, is changed in temperature with ordinary rapidity so that its composition has no time to change appreciably, its apparent rotational specific heat will be Cvr = y pCjp + y oCjo- In all cases the total specific heat at constant volume is obtained by adding to these values the translational term f R. The series given above do not represent any ordinary functions, but fortunately they converge rapidly at room or lower temperatures. Sec. 153] ENERGY, ENTROPY, AND SPECIFIC HEATS 265 In Fig. 58 are shown data obtained by several observers on hydro- gen subjected to various preliminary modes of treatment, and also the theoretical curve for that mixture which gave the best fit with the data, the assumed percentage of parahydrogen being given near Fig. 58.— Rotational specific heat c Vr of hydrogen. R — 0.985 cal per g. each curve.* The ordinates represent in terms of R the rotational specific heat at constant volume, obtained by subtracting f R from the total specific heat c v . The uppermost curve is almost that for pure parahydrogen, and it shows an interesting maximum, well above the classical value, at about 160°K. The curve marked 25 per cent is for the ordinary mixture. Theoretical curves for pure parahydrogen and orthohydrogen are also drawn; and in Fig. 59 is shown the total specific heat c v for these two forms and for ordinary hydrogen. From these curves it appears that the modern theory is completely successful in accounting for the specific heat of hydrogen at ordinary or low temperatures, 163. Specific Heats of Mixed Gases. The heat capacity at con- stant volume of a rarefied or perfect gas composed of different kinds of molecules will be the sum of the heat capacities of the separate kinds; * Of. Eitcken and Hiller, Zeits. phys. Chemie , 4(B), 142 (1929); Clusius and Hiller, ibid ., 158. Fig. 59.- Abs.Temp.Deoj.T -Total specific heat hydrogen. of 266 KINETIC THEORY OF GASES [Chap. Vl for at low densities interaction can be neglected and the energies of the molecules are simply additive. Accor dingly, if molecules of kind v form a fraction y p by weight of the whole and have specific heat cy„, the specific heat of the mixture at constant volume will be Cv = ^yyCw . V Since according to (20c) in Sec. 14 the gas constant obeys the same “law of mixtures,” so that R = ^7 y R V) and by (197) in Sec. 141 v Cp v = Cw + R, and for the heats of the mixture c p = cv + R, the same law will hold for the total specific heat at constant pressure: v Few data exist by which these conclusions can be tested, but their truth is hardly open to doubt. As the density of the gas is increased, however, departures are to be expected because of the mutual energy of the molecules, and in the case of c v also because of departures from the perfect-gas law. CHAPTER YII FLUCTUATIONS In the preceding chapters we have dealt with gases in mass, con- fining our attention to physical phenomena on such a large scale that the gas behaves like a continuous medium. In developing a molecular theory of such phenomena we continually averaged molecular quanti- ties until we smoothed out all irregularities due to the particular behavior of individual molecules. In marked contrast with such phenomena there exist others in which molecular irregularities them- selves give rise to observable effects; the most famous case of this sort is the irregular dancing about or “ Brownian motion” of small particles suspended in a fluid. This chapter will be devoted to the discussion of such phenomena, which are often grouped together under the collective name of fluctuations . These phenomena possess great theoretical interest as constituting direct and striking manifestations of the molecular structure of matter; they are likewise increasing in importance as a nuisance for the observing physicist. Most of the cases of practical importance do not involve gases, to be sure, but the appropriate methods of treatment and the nature of the phenomena have so largely the special character peculiar to kinetic theory that it seems natural to step a little outside of our principal range of subject matter at this point. The methods that have been developed for the theoretical treat- ment of phenomena of this sort fall into two rather distinct classes. In one type of method the attempt is made to obtain results as conse- quences of the assumed properties of the molecules themselves; in the other type, only broad features of the molecular motion are made use of and a connection is sought with some mass property of the gas, such as its viscosity or its coefficient of diffusion. The latter method is the safer and more widely useful one, but the former, more funda- mental method does lead directly to a few observable results, and it also assists greatly in forming a lively picture of the chaos in the molecular world; so we shall include one or two examples illustrating this method as well. The fluctuations themselves can also be divided roughly into two classes, phenomena of dispersion and fluctuations about an average. We begin with the former.* * An extensive study of fluctuations is contained in R. Ftirth’s “ Schwankungs- erscheinungen in der Physik,” 1920. Cf. also the excellent book by T. C. Fry, 267 268 KINETIC THEORY OF GASES [Chap. VII N = 3 PHENOMENA OF DISPERSION 154 , The Simple Random Walk. The essence of all problems of molecular dispersion is contained in the simple one-dimensional problem sometimes called that of the “random walk.” A vivid and completely typical form of this problem is the following. A man takes steps of equal length either forward or backward at random. Where will he probably be after taking N steps? To solve this problem, we note first that under the conditions stated each individual step is equally likely to be taken forward or backward quite independently of the directions of the others. All possible sequences of steps, each taken in a definite direction, are thus equally probable; and the probability of any given sequence is for the probability that the first step is taken in an assigned direction is l A, similarly for the second independently of the direction of the first, and so on through the N steps. Now the only way in which the man can arrive, _ j n the end, j ust m steps away from his starting point Fiq. 60. — The various in the positive direction, [is by his taking on the groupings Of three steps. whole positive and JV 2 negative steps, where N i and Ni have such magnitudes that N i — Nt — m and N i + N't — N , it follows that Ni = f + f >Nt - f _ - J ■ Clearly m can only be even if JV is even, and odd if N is odd. But only certain sets of steps can result in any particular value of m (cf . Fig. 60) ; the number of such sets is obviously the number of combinations of N things taken either Ni at a time or Nt at a time, and so is equal to Nl/(Ni\Nt\). Multiplying this number by Q4) N , the probability that any particulai set occurs, we have, therefore, as the probability for the occurrence of any particular value of m P ( 213 ) This result is easily seen to hold for negative values of m as well. We may note in passing that if the probability of a positive step were p instead of A, and that of a negative one, therefore., (] 1 p, “Probability and its Engineering Uses,” 1928, and R. A. Fisher’s Statistical Methods for Research Workers.” 260 Sior. 15*1| FLUCTUATIONS the same expression would bo obtained except that (}4) N would be X ^ m N m replaced hy p ~ 2 </'- 2 . In kinet ie theory we shall ho concerned duchy with very largos valuer <>f ,V. In any sueh ease (213) can bo replaced by an approxi- mate expression t hat is easier to handle by moans of Stirling’s formula, which reads: n I (2th) w within 9/n per cent for integral n > 0 (the numerical error increasing, how ever, with increasing n), or leg n ! = (n + 1 o ) log n — n + 1 «) log (2tt) with an error less than 0,09/n.. Using the latter form, we can write, approximately, log i\ iV + (2 - ? + n ( 2 - - 1 "■« ^ ,t>K 2 - Now let us use the series, log (1 + x) »* X , so that log (U“) *■) log JV - log 2 m no N 2N' i TIkui, dropping powers of 1/iV beyond the fimt,, wo obtain Pur AT 4» hver 2 — ** loir f2*rt — 2 t | '>»/* log /’ w * - ;j log N + log 2 - ^ log (2 tt) - * 2 ^ and approximately. Hero m taken on values only in steps of 2, being oven or odd with N. When .V is large, it is morn convenient also to introduce, in place of m as a variable, t ho total net diaplacemont from the starting point. | ml, where / is the length of a step. In practical eases l is usually small relative to distances in which wc are interested; then it is con- venient to treat | as if it were a continuous variable and to define, the probability for it, Pi, by the statement that P { tff is the chance that £ 270 KINETIC THEORY OF GASES [Chap. VII lies in a given lange df The number of possible values of m included in d£ is d£/2l, since m vanes in steps of 2, henoe P( d( - P m dt-/2l and P( = P m /2Z, so that, wuting we find x - iVff, Pi = xv^ e (216b) The enoi m (215b) as compared to (213) can be i datively largo in the "tails” where |f| is large, but this does not mallei since P { itself is there negligible whon JV is large Foi the samo reason wo Fia, 01 — Tho random-walk probability curve, oq (2166), can also for mathematical convenience suppose £ to lango fiom — «> to + co. If we then evaluate the aveiugcs * 2 = f.y p **e> ffl = - ajtv.dfc recalling that /:. — 1, wo find: ' (S 2 ) >s = \ Iff = ^ X - 0 798 X. (216a, b) Thus X is of the natuie of a loot-mcan-squaro displacement, Tho Probability cuivc for the final displacement thus has tlio foim of an ciror cuivo, as illustrated in Fig 61 The most piobabio single net displacement, lather suipiismgly peihaps, is zoio; but tho absolute expectation of displacement, [£], and tho loot-mean-squarc expectation or standard deviation, X, both incioaso as 's/N* These expectations mcieaso loss rapidly than the numerical sum of the stops. FLUCTUATIONS Sec. 155] 271 however; the expectations for the fractional displacement, \%\/Nl or \/INj decrease as 1/V^, \/lN being in fact equal to l/y/N . The quantity P m or P % admits, of course, of the usual double interpretation, either as a probability referring to a single instance or as a distribution function for a large number of instances. If the random walk were repeated a huge number of times the various values of £ would be distributed (almost certainly very nearly) as indicated by Pf in eq. (2156), and the square root of the average of their squares would (almost certainly) lie very close to X, and the average of their absolute values to ]£]. 155. The Varied Random Walk. The results just obtained con- tain the gist of the solution of all simple dispersion problems; certain generalizations of them are required, however. First of all, suppose that the steps in a random walk vary in size but are numerous enough so that we can, without incurring appreci- able error, assume that there are many of each individual size. Then formulas (2156) and (216a, 6) will still hold but with a value of X given by X » (pNy\ (217) where T 2 denotes the mean of the squares of the variable step lengths. To see this, consider first the simple instance of N i steps of length h and Nz of length h. Let £i, £2 denote the separate net displacements resulting from each kind of step. Then the probability functions for £1 and £2 are, by (2156), Pt> = *jL 2x1* XaV^r "2X a » where Xi =* X 2 = Uy/Wz) and what we desire is the probability for tho resultant displacement, £ = £1 + £2. Now, when £1 has a certain value, £ will lie in a given range <$£ only if £2 lies in a certain range of equal size, <$£ 2 = 5£; and the chance that £2 should do this is P €l $£ 2 . On the other hand, the chance that £* itself lies in a range d£ x is d£ x , independently of the probability for £2 (the location of the specified range <5£ 2 shifting a little, of course, as £1 shifts position in d£x). Accordingly, the probability that simultaneously £1 lies in d£i and also £ in 6£ is P&P& S£ d£ 1, and the total chance that £ lies in 5£ is P e n ( p { i p £i s& di 1 = 27rXi\ 2( J- ill 2\i* JiL 2 Xa “d£x. 272 KINETIC THEORY OF GAtiEE [Chap VII Hcie fa = £ — £1 and is a vauablc, whoioas f is to be kept constant in integiating Evaluating the mtogial, wo amve again at (21 56) foi Pj, but with X 2 =* NJ-l + Nil* " Nl 2 } as stated m (217), What we have piovcd heie is a soit of addition thcoiem foi cuor cuives The addition of othei groups of stops can be effected in the same way, and cleaily in the long mn the same foim foi P* must lesult if the steps aic distubuted continuously in length accoidmg to some definite distubution law In the second place, howcvci, the steps may be taken 111 random dnections in two 01 even thice dimensions In such a case the dis- placements m any two dnections at light angles to each othei aie statistically independent Foi, if we consider fust a lot of steps whose components m two chosen dnections have always one of the foui sets of values (a, b) (-a, 6), ( a, - b) } (—a, —6), the plus and minus signs will occiu at landom independently foi the two components, and the lesultants of the two components of these steps will thcicfoio be quite independent of each othei The same thing holds ior any pan of values Accoulmgly, each component will have a piobabihty function of the same foim as (2156) but with X 2 = Z 2 iV, h standing for the conespondmg component of a step If l denotes the total length of a step, we can also wntc, because of the obvious symmotiy of tho situation, eitliei If — 01 l\ ~ -g-T 2 and X 2 - i 01 X 2 = §7W, (217a, b) according as the motion occuis in two or tluce dimensions, N being the total number of steps The mean values for any component of the displacement are then given 111 teims of X by (216a, b) Since tho vanous components aic statistically independent, wo can obtain the piobabihty that a displacement ends in a givon element of space meiely by multiplying togcthoi tho piobabilitios for tho sepai ate components, it is unnocossaiy to wnte down explicit formulas [but cf, (218a) bolow] 156. Dispersion of a Group of Molecules. The random-walk foimulas coulcl bo applied at once to the motion of molecules in a gas if we could assume that each molecule after colliding with another is equally likely to move off in any dnection We could then say that a gioup of molecules initially in the neighboihood of a point will be distubuted aftoi a time i a\ such a way that an element of volume d%dr)d{ t whose cooidmates lelative to the point of dcpaituie aro £, V) fj contains a fiaction P drj df of them, whole P is the pioducfc Sec. 167 ] FLUCTUATIONS of throe expressions such as (215?;) or 273 P = X 3 (2jr)«' 2X a . (218a) For X we should have, from (2176), in which N = vl/L or the number of free paths of mean length L that are executed by each molecule during the time t, v being the moan molecular speed and V the moan of the squares of the free-path lengths. These formulas are not accurate, however, because of the per- sistence of velocities that was discussed in Sees. 86 and 109. In consequence of this phenomenon, successive free paths are not statis- tically independent, but there is a moderate tendency for a succeeding path to favor the direction of the preceding one, and the dispersion is thereby increased. Accurate formulas for such cases are given in Furth’s book. In practical cases, however, it is better to apply the ordinary theory of diffusion; the persistence of velocities is then fully allowed for in the value assigned to the diffusion coefficient. The formulas obtained hero furnish, nevertheless, a convenient basis for a useful qualitative view of the process of dispersion, and for a rough numerical estimate in an actual case. The formulas cor- rectly indicate that the molecules remain permanently densest near their point of departure but show a dispersion from it increasing as the square root of the time. In using the formulas for a rough estimate, it is sufficient to sot I 5 = 2L S ; this would bo accurate if the distribution of the free paths were exponential. Equation (2186) in conjunction with (216a, 6) above then gives for the mean numerical value, [£f, and the root-mean- square value of any component of the molecular displacement According to these formulas, in air, where L — 6.40 X 10 -n cm and 8 = 46.9 X 10 3 cm/sec at 16°C and 1 atmosphere, Ti| = 0.36VT cm, t being in seconds. 167. Molecular Scattering of Light. Another interesting caso is that of the scattering of light, as in the case of scattered sunlight received from the clear sky, When a beam of light passes through a m KIN bint' TtIhUltY tip a\ShS l< MU' VII f yiH t out'lt molecule in ncI into fun oil vibration ami \ » Horn laiiud to not hm u houicc of ludmhon, mu l if I In 1 gu » w rmeln d, n» that if* mote cuIck are disl iibulcd hm if at iamtum< tin* pin »* of l I im \wu» * t mitn<I by each molecule, hi mg determined b\ flu* pin ♦ «*! f hr im nl< nt light ul Uh pnmtmn, will uirv from one molecule to the m \t m a imtdnm manner. In certain due* turn > the Im*mih » «*miftt pI h> the ddhonl UlttltM'llll'S IlCVClthelnn HUMP tuglthel IM » Mnleimifir lO'lfllll r Mini Nil give i I mp t ci ( he* ordinary trlwnfed hi am Mini In the ah, mphnn of t In* incident I team; Iml in nil other dorritnui n fandom didnhutiou of l dull vn plrnncH orcum, und tin* m uitf e* what wo > nil ^altered imliulioiL In midi dimlmiiN the \uuoin wu\r u*tnr» lomhme like tin* steps m u rune tom walk, mol, i im e the » m ig\ r- pio|*orhonul In Ihc HtpmicM of (Im mo veelure, the mean o uilfani miu^ < mm a mil Jtial equal tu the sunm of tin* rnetgifs *milhd h> I Im* ddlt mil mole cuIcm; jiihI uh aecmding In tiitOui l he nn un ol it* x\ nod llti a in (urn hy (*^17) equal* XV m Him total imiulfci n( drp< multiplied by the mcun Kqmue of thru* length Jt follows tlml (he molcculn of a imr m n can hr hinted m Hide pendent, aeul l rrer* of light Hinoim* of Him hot a utnplr nluhoii h easily found between the oidmmv iefrmii\r unit \ ami th* mhnuy of molecular weal leiing, mh follows,* Aermding hi eleetlomugucUc f heavy the refractive ueh \ \n M - V*- where r, in the effective didecltie fondant for light of frequency *; und ui tin li t, 1 1 | *l*wr ri whirr nv \* ilm amplitude of i In trie moment induced in the nmlei ulr l»v mill mnphtmle of lie electric vector in (lie heuiu, u being the mnuheif of nmlei u|c* prr iithie roue meter. Hence it, - (g* IMpth Un the othu hand, deifm- magnetic Uieniy give* for the number of erg* |ier mmol mUleml by a molecule vihrutmg with moment mnphtmle in - M i** ‘ r being the npeed of light Multiplying thin tpomhty by rr and tlwhU mg by r/Hr, which reineaenlH the mean energy imnhnt on unit area per ueeund in u monochromatic U*nm of unit mnphtmlo, ami iuni rtiug then the value j iih| ghen for 0*, m terum of a* we foal lor the fraction of the incident eneigy wallercd from each cubic millimeter of the gart M'f It A bmiKNir, *' Problem* of Muilnra Phy»trB. 44 ISiJt u Wi Site. 168] FLUCTUATIONS 275 8 ttV (ju* - l) a 32 7r a (M"l) 2 3 c> n- 8 nX' 1 ’ since g -|- 1 = 2 nearly enough for a gas and c/r = X, the wave length. Thin formula could lie used for the determination of n if better methods were not available. In optical experiments on scattering accuracy is hard to attain, but, measurements of the absorption of sunlight in the earth's atmosphere, and also of scattering by pure gases in the laboratory, seem to be in satisfactory agreement with the formula. The phenomenon is of special interest because it depends essentially upon the granularity of the medium and would disappear entirely if the latter were made indefinitely fine-grained (n = to). FLUCTUATIONS ABOUT AN AVERAGE So far we have dealt with the dispersion of particles that start out in a group from a given initial position, or with problems that arc mathematically of the same typo. A rather different class of cases of great importance is characterized by the fact that a statistical equilib- rium of somo sort exists, and wbab wo arc interested in is tho irregular local or momentary departures from it. The distribution in question may bo a spatial one, such as tho distribution in space of tho molecules of a gas, or it may bo temporal like tho distribution in time of cosmic- ray impulses. More homely examples from kinetic theory aro the spatial distribution of the separate molecular impulses that give rise to the pressure of a gas on the walls of its containing vessel, or tho distribution of these impulses in time on a small hit of tho wall. Most of tho quantities encountered in physics are subject to such fluctuations, and what wo ordinarily deal with is somo sort of average value; even tho length of the standard meter bar must bo continually fluctuating to a minute extent. Since in all such cases the basic mathematical theory is the same, wo shall first develop it in the abstract and then take up special cases, 158. Theory of Fluctuations about an Average. Suppose that wo have before us JV elements of some sort and that they aro distributed at random among a possible regions or ranges of value, which wo shall for convenience call cells. Tho average number of elements in each coll is then N/a, Suppose, now, wo ask for the probability that some particular number v of them occur in a certain coll. Since each of the jV elements has a chance 1/s of falling into tho given coll and a chance of falling elsewhere, tho olianco that a particular group of v of the elements falls into the given cell, and that 270 hiSKrtr tummy or ttisr: \ it , t ,.< vu III Uh* hhiih* I urn* h)| tlif u|ln*r A r **|| *II« nl » f’tll tl < will re, 14 (0 0 0 M»ilti|'!)iiiK Mil ' i \|i|j ■ inti fin IlM I r Ilf iliflWohl kioh|*m rimlniiunK *’ hmIi flr*f » m U mil uJ Hn* <Y i*loim»nH, mu'll nf \\ list'll kimiijm )m mum *|inl ► Inn* * 1 ** I* flu mu* Omt f nils in llu* toll, up obtain a i iIm* ImI il fliif ilnrt* mo jiiNt i* rlpuirnh in tlitn u l! -• .,/■ ..(')'(■ '.r 1210) TIip pinhulahl v l\ Irwin i it * umxnuuni *»lup ulnu rMjimK ^ nr 1 I V wtmlpvrr inti^rr In * brlwirn r I | ^ run! * i * uln ip * or tin* iivonil#* munhrr of rlniirhln \n i n II, if * | * iMu!« yral, /*, lalo Hint lb< mtixihimai vitliir I mill fnr v - r \ * nml for r - * i iS Uu* not of irn'intniiiK Mio- I I multiple » U\ (' - i 1 Fur N t< \/k .V »< I 1 I tt* I l)(s 11 rt I 1 sr | Jitfl I i miii'n A 1 ' -« «»»; lii-itri* /*,,i » /', nt'MinlitiR it* »• ur iut I II I * ? o f -- f i l Now for n iimximtiiii nl i> * v n wr mu >i lmv<< I’m I t K «K I l « I Jt* I t* I Itrnrp, nnntnltiiK i«» tlir* rttiulilitm jiml i|t«lur«<l, >'n f' l “I * f« I f 1 I "f P* 8 $ to I III pbywnil K|t|ill<*riltiMiN 4 hnurtpr, X nml aarr u ^mllv mnimmwLH Unit, iih in tin* vm* of llm mmlum vmlk, n morn nmo-moil u|»|irn\oimU* formula ijk rnimwiitly nmirulr U'l \\n unto rJItla in ilm f**rin Sec. 158) FLUCTUATIONS 277 Then, provided v/N is negligibly small in comparison with unity, we can replace the second fraction by N“/s r = P”, and if 1/s is likewise very small we can set - W - ’>(-7 + approximately, Then logP„ = — log (VI) + ^log v — P, and wo have as an approximation Poisson’s formula: P t = ^ e~K (220) Here v is the average number or expectation of elements per cell; and the maximum value of P v still occurs at v — v or else at a value v = v m such that 9 - 1 + i S F- ^ 9 + i* For practical use we require also so mo sort of measure of the spread of the values of v about its average value. A convenient quantity for this purpose is the root-mean-square or standard deviation, whoso value is 8 m [Av{v - P) 2 ]W = V~P, (221) Av denoting the average value of whatever follows it. To obtain the value s/p for 8, note first that <w hence in Av(v - vY - ^(r - vYP v = ^P v - 2-v^vP, + P by (220) 00 {< i 4 vil 278 A / \ / nr nnnhl tn r. tvW,\ In tin* Hiutn \u\\ i mi U* nlifmiM >1 iltr nllif nw mI*\ rm » fmnuih * Ifrurr .ltd H' T I i ar ! r * Tim* ns i h im ir ii ml A kii» t # ■ nl *‘ Iml Hm r« I’tfur t1« i mltmi V r|nr| » I \ ' i*, t Ii*t |n v ik i»f lln f /' tiimjii ui in -j) > > in I mu||\ hiniuk’UN, huf if hnnini * nlnhvilv imrimwi n> \ulli flu* umKUiUnlo ol r Funnily, in r mu j it *uH r* lmu« im*l i utn* ImI mtlv m wtlm ■» nf \* irlutnrly ilum I»> v, I In* |Hnlm1i)h! V fm ullnt Vnlii* > I?* mu |»fm Iinilly /nti, wv uni Miiiplih ill IiiiIImi \*\ m flu Milling r*X|tit*Hr*iciit fur lutf »•! horn FJI lAt Mmnu tin t wt nMmu (mm < U2Ui Itttf /V i Ing v J |mu **Vi i It pjr? i« j r \u wluHi vu» 1*1111 unli' hu (1m! if uu krr|* milv tin 1 «nmlli |«»%m«i uf rr ***„ uu i » | I * H H J I , #l K* >* «* mnl * i u* /V - r '* fjfttt- \ "2* r If i» in \ ny rlnm* In h, if rim l«« r# plm ml |<\ *♦ multi I hr* rmlii n\ hum m Hum ux|im HHinii! mnt fin n wv 1m\r n m**r t uru 4 1m 1\ Thu htmnlnhl itr*\ mhufi nf i mu >i , t»i ill \ *« im gii* it l#v *2211 Frmn it ih Hint iikmii tlml it Ihii r i>* Inrun I\ lm i njij»rn irtMr \ltlutmmil\ lrln(|\r|\ Uvn\ v I v it |h imp H r lu munII In Flu IU urn hliuuii pluin n| /V m »h»<Iuiu f<» ifnr t - *, 1 , 1,^, n*nl fi Km i f All»rrnr\i »IrnMl> ^ fmr np|#r<»MnmI mn lu nn mm rtirvr, llm nmxiinum fnlhuu ni 1 *i 280 KIN NTW 'I'll M HIV OF HAS UK l« nw, VII imml l>y the intermnlrcnlnr formi uml minimi nilwr uu-llmdii fur il.-t (juantitnlivo Lmnlnumt. l'limtiinlloiiH in tinm rnii I m * i>iniiliirly luindli-d. Tlmy ni'i' lunoi likely to lend Uidi-lrel-nlili' rflfrrlH in * , i»iiih*«'I im> wiili mnleridni 1 Imuiim; Miell offontH are ennily nlwiTVnlilo ill Mir rime of rlrrlmn emiir-inu, In wliiuli Mm Nniim hIiiMnIIciiI Mmnry npplirH. Huppimii, for example, Mini, |mrliclr;i uie cinillrd iiidrprndeiilly nf (•null oilier lit illi nvrrnur min nf r |K*r »nd nml Mini I In 1 iiiiinlirr nmiltnil during Himerwiivn < 1 <|iliil ililrmiln of I line in rniililril, Tlirn /’, nH given l>y (2211) repnwnlH llm frnrlinntil |mrl nf Minr imiiiiIhth M mb wilt in Mm long run lie fnmnl In Imvn n pnrlieiilnr viilun »•, In ninny OX]H!rim<mtn nr MiiH Hurt p 1h nnirli lr«i Mum I; lliie finpiruMy liupiKuiH, fur example in iiiwrYHlinim willi n liriger munlrr iipmi nomine myH. In mmh piihi'm, during llm imijniily nf llm inln vnln nf Minn lin pnrMnln III nil Ih ulmerved (/*„ r ' I, m-mlyl; dining n nmnil fmoMuu 7\ nr iWr' nf Mm iiilurvnlii JiimI. mm in nliiiervml; during l\ nr PtT> nf llmiii, two (i.e., n ,, ruim*l«l«iiii , i ,, ‘ mmiiruj; Hr. Tim fnotnr i u In l\ nhniilil lm noted. (1 might Imvr lieni llmuglil Mint Mm (iliuimu nf tv colnulilmmn wniilil Ihi iiimply /’?, niurr 1', in llm likflllinnd of niutli nf Mm wi| Hindu rvriiln; but in llm (ruin of llmuglil leading up In thin nxpnoluMnil I'liuli coincidence In milly rnunlrd Ivvirr, Hovurnl tinncH nf LIiIn general enrt Iiiiyi< lirrii nlmliril in ■irlull ux|Mirinmnliilly nml Mm Mrnnrullrul furiniiliiu Imvr Ih-imi elmwn In hold. DIFKUHION AND T1IJC IIKOWNIAN MOTION 100. The Brownian MoMon. In 1828 n linfuninl, Itnlmii llnmn, olmerved Mini |Hillrn grulim Himpendril in wider uml viewed iimli-r Mm nii(imHnn|Ni allowed nn irregular dancing lunlinn llmt imvrr crimed. KximrlmeiiMiig fiirMmr, lm fniiml Mini nny kind nf millirimiMy ••until imrtlnln exlilliltcd Mmmtnm iiinMun nmi rnnrli|i|i-il Mini Mm plmiiiiiuminn niiml lm dim In Home iinklimvn imiiiiiiinlr ranee. Tldu innlinii In mm nf Mm iiinnl liilrrcHling plmiiniimnu in pliyelru uml In mill'll, lifter it* dimwvomr, Mm llmwiilnn inoMoii. After vnrlnilH hypotliertea im In tin riiiinn liml Imni prn|Hwd, llm miMlnrn view narrililng it In liiulmilur ugllnlinu wim pul forward by JIoIhuux In 1K77 mid Inin 1 by (Inuy In 1888. tinny nlemcd Mud. llm mnlinn wan niiuffeHod l»y viiilnliniiu nf llm lulrueily nf llm light ffdlinn on Mm particle*, lail. Mint It iiirirmrrd in vigor wlmu Mm vmnwlty nf Mm KiippnrtiiiK Ilnid wiih derrrmn-d. Jlr win ponded, however, by Mm fimt Mint Mm np|iun’iil, velocity of llm parlirlrn in very nnirli iuMH tli nn would Ihi Mint nf n mm molecule of tlm Hunm iimu* according In klmiMn Mmnry; lm did not rrnllxc Mini Mm him ]iaili nf n llruwiilnn Sec 159] FLUCTUATIONS 279 As a final comment, it may be lomarkod that the random-walk formula, (213) above, is really a special case of eq (219), obtained by setting 5 = 2 In both cases the theoiy zests upon the fact that a selection occms, guided by piobability, between two alternatives In phenomena of dispel sion, howovoi, we aie inteiesled m the differential icsult, the negative steps being subti acted fiom the positive ones, wheieas in the type now undci discussion we aie mtciestod only in one alternative, counting up the ele- ments that lall into our selected cell and ignoring those that fall olsewhcie 169* Examples of Molecular Fluc- tuations. The fonnulas just obtained can bo applied at once to the distribu- tion m space of the molecules of a gas in cqiuhbiium, Any volume r that is small compaied with the whole volume contains on the average v = nr molecules, n being the mean density in molecules per unit volume; but momentanly it may contain any other numbci, The chance that it contains v molecules is P v as given in (220) oi (222) The loot-mcan-squaio do- Fig 62 ■ 2 A -Probability o fluctimtion viation fiom the average valuo nr, according to (221), is <5 = \/ivr. As the volumo is mex cased, the actual magnitude of the fluctuation inci cases as tho squaic root of the volumo, but its lelativo magnitude decieascs m the samo ratio Equal fluctuations occui at equal values of the pioduct of tho volume undoi consideration into the pressure. As a numerical illustration, a cube 1 n on edgo drawn in air undci standaid conditions contains on tlic avoiage v = 2 7 X 10 7 molecules, and this numbor fluctuates only by about [2,7 X 10 7 ]^ oi 6,200 mole- cules, io*, by per cent, If, however, tho pleasure is loweicd to 1 mm Ilg, ic , to a Goissler vacuum, v = 3,6 X 10 1 and 5 = 190 molecules or 0 6 per cent, while at 0,001 mm, oi in a cathode-iay vacuum, v = 30 and 8 = 6, oi 17 pei cent Such fluctuations in density cause vanations in the lefractive index, and the lattor may be regauled, if desned, as the cause of the molecular scattoiing of light by raiefied gases, a phenomenon which was treated abovo fiom an entirely different point of view, When a gas is bi ought close to its ciitical point, these fluctuations become onoimously large and give use to a ohaiacteiislic opalescence, at such densities, howevei, the character of tho phenomenon is gioatly influ- Sec. 161 ] FLUCTUATIONS 281 particle is irregular on a minute scale far beyond the reach of the microscope and is, therefore, immensely longer than the apparent path that we see, which is only a sort of blur or average of the actual one. 161. Theory of the Brownian Motion. An adequate theory of the phenomenon just described was first developed by Einstein, who pub- lished in 1905 the equivalent of our eqs. (224a) and (2245) below. I-Iis method of procedure was to establish' a connection between the properties of the Brownian motion and the viscosity of the medium, with a minimum use of molecular assumptions. Let x, y, z denote the cartesian coordinates of a particle and x, y, z, as usual, the corresponding components of its velocity. Then the ordinary viscous force on a suspended particle has the magnitude —ci)V, where v = (,x 2 + y i + i 2 )^ while y denotes the viscosity of the medium and c is a constant depending upon the size and shape of the particle; the components of this force can be written — ctjx, — c-rifi, —oyz. Hence wo can write as the equation of motion of the particle in the direction of x tniS — X — cyx] (223) here m is its mass and X is the instantaneous a-componont of force, exerted upon it by tho molecules of the medium over and above the ordinary viscous force —cysk, which represents tho average of tho actual instantaneous forces. X will thus be highly irregular in value and as often negative as positive. Now lot us multiply this equation through by x dt and then inte- grate it, thus: mj'g £x dt — ^Xx dt — dt. Hero in tho last term tho differential is dt — d(£ a; 2 ), while in the first term it equals (dx/dt)x dt = x dx. Ilcnco, if wo integrate tho first term once by parts, we obtain: A (mxx) — mJ'gV 2 dt = J 0 ‘^ x dt — h ct|A(a; 2 ), the symbol A denoting tho change in a quantity from i = fltol = i. In this equation the second term can bo written rn¥t and so increases ultimately in proportion to t\ for x 2 will have in tho long run some definite average value, I 2 . The first term, on the other hand, fluctuates rapidly because of tho factor x in it; furthermore, since the motion must be in reality of the nature of a random walk, our general results in Sec. 155 indicate that the factor x in that term should 282 kinetic rummy of a i ses vu momtso only ns \/t . (!uiwo<inonlly, in tin* Itmn rim tin* brut t i*rm Ikwoiuoh nculigiblo in companmiii with tin* hi fund, and fan bo dropp'd In j^'Xx ill Lhn inli'niand Xx likewise lliiiluaifs ia|tnlly, and we lmuhL porhapH oxpool, IhaL, by llio mndoiii walk pun. iplt*, (Ins lutoKini, also, hIiouIiI inotonso only as \/f In width, howeter, Him pi* notion of Un« fm'loi' x oiiiihoh llio inloKiund itself, who li lime tale . (in' pluen of Urn Hli'li lanulii lit lll(' liindnni walk, In toad to itinmoe a. \// t and Urn wlioln inli'Rial, (lioiefnio, inoienses al a fasti i ia(i*, it must, in fact, inerfitso an lln> IiihL ])ow'or of t, olsr I lie oipiuhoii would fut nidi uh willi a uniform valno of Ai 1 for all pinticlis, wlioious inod.no walk llicoiy IciuIh uh Ln oxpool wide xaiiahuiis m Him lullin' ipumliix fad iih now Mlippleinonl. I lie iiili’Kialiou jiml made b> imiauifi^ Llio lank (‘(jiialion for a Iiuko nmnlifi A r of differmil pitiful. » TIih avaraKinn pinooss (finis fmlliei (o supplies I lie Ills! Ifiio in tin* i ipia lion, wliicli will bo puniltvo for mime pal I if lim and neuulne for other* It uIho finally nuppiesses tho (liiid Ifim \ r ill, omen (Ida tmin, |m», niliHl la* positive for Homo pinholes and noKuliin for nlhms and mi ran only yield a mmlom-wnlk ifHiitual propoilionid to \ \ when wo mini Llio oiputtions, and lifin-f a i onlulailion In tin* uv'iutto pin portional to \/\/N, Wo LIiiih obi iiin finally, as llu* lomili of uu'Iukiiik imr a law* mmibor of jiartirli's, mPt JniAlx'l, A^J repimuilhiK lb'' avoingo of A(r 9 ) for all partinlm* Now »i7 4 > J mCP ! Y >|- P) I wif 1 , and by llio pnneiple of llio fijmpaililion of enmity lilt* moan k hint it* onorny of tin* Hiowmnnjiniliole Hltould bo llio mime as that «<f a naa molmilti, ho LI nil. i hw* | k'l\ llonoo, if wo assume (but llio pnrlii'loH Htart fiom llio oi iftm ho dial A(.r a ) » ,r®, wo have na tin* final roHiill: ( 22 'bi I Horo k in Llio rum fonalauL for one molooulo, I 1 r l/ni and represents Llio (dyntunio) mobility* of tin* paitiolo in llio fluid or its wtoady volocity of drift undor mill timing fmre, T ih llu* almniuU* lorn- * Nol l<» lot oonftiHi.il willi llm inoliilitv of na ion rnrrylaa a flmrge r ia no l)lcc (11 it; fluid, iih nidhmi Jl> dofliiud, whirl. In |r|f? Sec, 101] FLUCTUATIONS 283 perature, and q is the viscosity of the medium; x 2 represents the square of the ^-component of the displacements during the time i, averaged either for a large number of particles moving simultaneously or for many successive displacements of a single particle. The valuo of the coefficient c depends upon the shape of the particle. For a sphere whose radius a is much greater than the mean free path of the particles of the medium, c = Ottu, as first shown by Stokes; for such a particle the last equation becomes ^ vr_ 3 t T(tl) * (2246) The method of analysis that we have followed here gives us only x 2 , but not the distribution function for the separate values of x 2 . Anal- yses more complete in detail have also been given, which start by writing down an actual integral for the equation of motion; the final result is then obtained by an application of the random-walk theory. The whole process is, however, so obviously of random-walk character that without giving such an analysis we may assert that, in accordance with the conclusions readied ire Soc. 155, the prob- ability function for the position of a Brownian particle after tho lapso of a given interval is of the Gaussian error-curve form, This being granted, the actual function can then easily be written down if desired, care being taken to make it yield the correct value of ~x 2 as given in eq. (224a). For example, wo can write, as a special case of (218a) in Sec, 166, for tho fraction of a group of Brownian particles starting out from the origin which after a time t lie in an element of volume dx dy dz } P dx dy dz , where P « 1 (AirkTUt)* 5 4 kTUt * 9 V) this makes Iff” dx dy ds 1 ancl also (225) co ic 5 — f f JvW dx dy da ~ 2 kTUt, — to which agrees with (224a), Here, as wo shall see in Sec. 163, kTU can also be replaced by D } the co efficient of diffusion of the particles in the supporting medium; and formulas such as (225) are, in fact, more commonly obtained by an application of the ordinary theory of diffusion. 284 KINETIC TUMMY OF CASKS It me MI 162. Observations of Brownian Motion. Olixemitiriii* upend- rally (lmKn<>tl to nmlco a qunnl dative nf I he kinetic fin uiv inler- pintation of the Biowninn motnm weie iuMilulcd by IVmn m HHlfl, Tin worked with omulmoim nf nuniboKe m mtislie made muchly nmfmin ns to pnrtirla nisjo liy rrnliifumdmnK 1’eirln fust showed that the |uii I u'l»*s of mii li an enmlann tli«| nut r Hell lo elilliely to llie Imllum nf I ho \nm-nI Iml netimlly lenmmeil < li^l I i‘i I in i Milling In the buy * ,• of llie dial ill ml inn nf ninh’i nit ^ III U Foiei held, mi , e\|nehsed hy eq ( 7 llr) \ llienreiieid dialtibuhnn m lullin' to IIinho nlismnd bv him e. shown ut Kix, • (W By eoiiuliiiK pin I idea id dilTeieut In i^IiI * h , t • under the niioionope, ho icnlitd lliid tin* mi an t diMlidmlitm wiih exponent ml as them v reqnm ami * IK*./*! itHMimiiin if In lie |irnpnrlinn»l In i he wiw * aide to eidetiliile from Ilia ohm ivatinim the number *' * , of pint idea in n kiimii innlreiile nr .Yu Hud, /*■„ * lieiliK Hie Kiw CoiinIiiiiI fnr ll Rrnm innleeiile III , • \ IliiH way he found iV 0 > Oft lo 72 X ID”. Tim * , • neceMMiuy value of llie iuii«h «i wie< nlilaiued in • , , Hoveml iiiKeniniiH wiiyn, na do lerilied In Iiin interest } ,‘ t * hiK booklet.* * , I’erriu then obwiied (lie uiolioii nf iudU idual . "* * ■ parlieleH, projecting I hem for ibis purpose no a , , ' t nereen, and allowed Ihnt llie menu square of their / 4 ** , ,,V Imriwnilnl displacements dining » lime / was *« *, iippioxinuilely piopnrtiniml lo /, ni letptired by <• . tlm theory llml wn* develnped hi llie bud mi (ion; , ' *•,*• « ’ and lie nine showed Hint these displacements were f-*— i. o distributed ns nearly urcurdiiig In an error i urve n« imNlHlri'lmUon £ mM } il »"* ™Pwl«l nu’nmireil the din- gravity (ft ihnnratirnl (if I WH imitlf'h'rtfinri with thr* k " ,,w " viM(,,wil y "f Wider, Miiri. lent data for a second erdeulnliou of No or ltu/k by means of the Kinston equation, ( 22 <lb), The bent value found in this manner wn« ft H x HI 11 . ICvon the rotational diaplaceinenlH id tlm particle*, f„r which a similar theory holds, eoultl be followed in emtio nf llie drops of gamboge whose diameter waa around Id /i and which had \ noble sputa on them; fuun those ohservatioiiH ho deilueed A'» 1 E« 0 ft X III” *.1. ft Pbuhin, "Alenin," trnnnl l»y IlniumhU, mill. Bec. 182] 'fluctuations SB'S A theoretical track exhibiting the same general features as do some tracks actually observed by Perrin is shown in Fig., 64,* The successive positions of a particle at equal intervals of time -are shown by dots; these are connected by lines in the figure, but the actual intervening path would be irregular and very complicated on a fine scale that is even beyond the reach of the microscope. The fair agreement of Perrin's val- ues of iVo with those obtained by other methods, which were not very precise in his day, combined with the qualita- tive results that had been obtained by him and by others, completed such a striking confirmation of the kinetic theory that the last doubting Thomas seems to have been converted forth- with. Later work on tho Brownian motion has yielded values of Na in still better agreement with what wo now know to be tho true value of this num- ber; thus PospiSil (1927) t obtained from suspensions similar to those used by Perrin IVo 5=3 6.22 X 10 23 , as against the present value of 6.02 X 10 23 , In practice, ins toad of observing displacements in a given 'time, it is more convenient to note tho time re- quired for a particle as seen under tho microscope to cross for the first time either of two parallel cross hairs, after crossing a third hair placed midway Fio, 04. — Track of a Brownian pnrtiolo {n theoretic til trnok ro&om- blirjg obsorvod ones; of. footnote below) , between tho outer two. The rather complicated analysis necessary for the interpretation of such observations was given by FUrth,} *Tho theoretical Brownian track was drawn as follows. First, a random sequence of 800 digits was prepared by copying tho hundreds digits frpm tho numbers in a telephone directory, taking names a& they stood in alphabetical order, Tho row of digits was then divided into groups of throe, and successive pairs of these groups wore taken to ropresonfc x- and ^components of the successive displacements of tho partiolo. Tho distribution shown in Fig, 08 was constructed by moans of tho same sequence of digits, one group being taken na tho aj-component of a particle, and tho logarithm of tho next as its fl-componont moaaurcd downward, f Poswsil, Ann . Physik t 83, 736 (1927). t Fthmi, Ann, Physikt 68, 177 (1017), or his book, lac. cit. 286 KINETIC THEORY OF OASES [Chap VII The Bio win an motion is also visible in the case of oil drops 01 othei small pai tides suspended in a gas, and on a much largei scalo than in liquids because of the lowei viscosity Seveial cases of this soi t have been studied in detail, but the expenmcntal difficulties aie consideiablc, and the lesults have not always been enthely satis- factory fiom the theoietical standpoint. 163. Diffusion as a Random Walk. It is evident after a little thought that the oidmary pioccss of diffusion must anse as a con- sequence of Biownian motions executed by the individual pai tides A connection must exist, theiefoie, between quantities chaiactonstic of this motion and the oulmaiy coefficient of diffusion, this connection was ascertained by Einstein in the following way Let <p(£, t) d£ denote the piobabihty that a given particle during a time t undeigoes a displacement whose component in the ^-direction lies between £ and £ + d£ We need assume nothing about the foim of the function ip, except that <?( — £, t) = <?(£, t ) Then if we draw a small plane m the gas perpendicular to x, any pai tide staitmg at a positive distance % fiom the plane will have crossed it in tho Fio 06 — Diffusion ns n random backwaid dnection after a time t piovided ,vftlk foi this particle £ < — x‘, and out of n dr particles initially in an clement of volume dr located at a positive distance ^ from the plane, tho number n dr J_ 0 d£ will cross the plane towaid —a m time f (cf. Fig. 66) The total number thus ciossing unit aioa of the plane will be, theiefoie, N _ = f 0 *Mv) chfjya, t ) d£ = £ "n(.r) dx£”<p(g, t) d£ since ■ £, t) — (f >{ £, i) Heie n has been indicated as a function n(x) since we assume it to be nonunit onn, Snnilaily the numbei IV+ = dx ££<?(£, l) <*£ = £ a n(-x) dx t) d£ ci oss toward Thus, if wo leplace n{%) by n + % dn/dx } where n now lefeis to tho density of the pai tides at the plane, we find for tho net tiansfei of pai tides toward N^N + ~ N- = -2 ^J\dxj\(^ t) d£, the teims containing n itself canceling, 01 , after mtegiating by paits Shc. 104 ] FLUCTUATIONS with respect to a: and noting that 287 N = ^ *(*,*)«- -¥><*,*), 'sW ,,(s '‘ ) ‘ is ]L, *vfeo* -sr^e, o ^ o <« - 2 * tlx representing the mean squared displacement per particle in the direction of x. The integrated term vanished here because we may assume tlmtJ^*V(£, t) cl% vanishes as x — > co faster than x 2 increases. This number must now also equal — ID dn/dx by the definition of the coefficient of diffusion, D, Hence ~Dt ~ or e - 2 Dt, (226) (Wo need not worry about the backflow of the surrounding medium so long as the diffusing particles form only a small part of the total mass (cf, Secs. 106 to 108).) By means of this equation cither of the two quantities or D can be calculated from observations upon the other. For example, we can calculate £* for a gas molecule, whose displacements cannot be directly observed, although its value of D can be; or from the observed or theoretical value of F 2 or ^ wo can calculate the coefficient of diffu- sion for a Brownian particle. The equation also shows again that the mean square of the dis- placement must be proportional to the time, this conclusion following here from the assumption of a constant coefficient of diffusion, If we compare the equation with (224a), identifying x l with £ 2 , wo obtain again the result that D « /cl VU, as was found from a different point of view in arriving at eq, (170d) in See. 115, 164. Brownian Motion under External Force. Certain experi- ments involve a measurement of the rate of drift of a small particle through a fluid under tho action of an external force; for example, in Millikan's determination of tho electronic charge, measurements of this sort were made upon charged oil droplets suspended in air and either falling under their own weight or moving unclor the influence of an electric field. It is important to know what effect tho Brownian motion may have upon such observations. 288 hINKTK' 'I'fUHHO or (IASI'S (Ciur VII Fur Hucli it ]nu \\*i' can wnli' ih I In* equation tif motion of u>< i-eomdinale, in place of (228) above, mV X t 4 >)( I F, where /•’» in the component of a fom* F of c\l **i uul origin, winch we Hindi assume to In* steady Let us change hi new axes mm mg in such it way that, if t\ //', s' art* (In 1 new cnuidiual' i, we euu wule m terms of I lid tlllll' l r ' i x' | uj i ii, 1 ' i ri} with a Hiniilai* equation fm // ami z Then x F | u„ x ,r', nml Min equation of molnm for j' in mV' •' X a jr', tlio term /*', having illsappi'nictl TIiIk equation ha* mm exactly the Hiimo form in lei ms of Mu< now vui table x‘ an (228) liml in terms of tin* old onn. A. cm responding i i*suH in oliluiiii-il fur ;/ ami a Hence wc limy conclude Unit i claim* to llio now axes the paittcles will execute their uaual Utowuhtn motion umnoiltlii'il l»y tlm presence of Mm field The motion of Ilia pniliclcs is thus hi genei at llio vector mim of their ordinaiy Hiownmn motion and a uniform niolinii of drift uiidor tlio fount F at a veined y equal to F/t or In t ! F, where V is tins (dynamic) moliilify; one mol ion in simply sii|H’rpu*cd upon the otlior. Tim am age vector displacement of a group of particles in time I in, llii'icfori', Mimply l /!?{, mmlTi'i’h'd by Mm Brownian motion, which l>y itself given an average displacement of Hero in pi nr lice, however, it in much nunc convenient (o measure, not displacements in u given time, Iml nil her the lime required In go a given (linkmen, for example, the tune inquired l»y a parliele as aeen under the mieioHCope to liaverse the distance between two parallel eroHH hairs. If theie vveie no Itiowiiinn motion, thin time, fora din- lance a, would bn a/l’F, and mo would give immediately the value of (A It in important to invoHligate whether the brownian motion, which causes thcne tiinea of linnalt to flueluate, iiImo altera their mean value. ho distribution of such limes of transit ran be inferred from results already obtained above by means of tlio following argument, os was shown by Nrhradiugor.* Hupposo a group of N particles atari out at time l >*> 0 from a point, which wo shall lake as the origin of coordi- nates, and drift thereafter under a constant external force F at menu speed u « UF townrd the geometrienl plane x - a (ef. Mg. flfl) I lion at limn l the group will have become both dispersed, In eoiiw- * SmiuttntNOMit, Fhgmk. Xei U., 10, 280 (1015). Sec. 104] FLUCTUATIONS 289 qucnco of their Brownian motion, and also displaced a mean distance ut toward +x. Their density in terms of x will accordingly bo ff p dy dz, whore P in given by (226) in Sec. 161 with x replaced by x' or X — id; it can bo written Pt N (a— itQ* , e " (4.rDt)y> with D — hTU (of. end of tlio lust section) , Here p x dx represents the fraction of. tho particles tliat lie in the range dx, and tho value of the constant factor can bo verified by showing that J ^ p» dx — N. Now wo can find tho rate nt which tho particles mako their first transits across tho piano x — a if wo oan fnul a general expression for the total number of those tliat at any moment have crossed it at least once; tho latter wo shall call for brevity '‘crossed'’ particles. Obviously all particles tliat lie beyond' tho piano at a timo t havo crossed at ono time or another; hence at points beyond tho piano tho density of crossed particles is simply p«, and, in parti- cular, at tho piano itself, it is X"-" 6 a I'lo, 00. — Drift plus Brownllin motion. (p»)n N (4ir (« — Hi)* C . Thoro will bo others, howovor, which after Grossing toward tho right, i.o., toward +*, subsequently crossed bade to the loft again; lot their density bo denoted by p'. Some of them may have crossed back and forth several times. Wo know tho value of p' a only at x a, since there it must equal (p*)„; for tho crossed molecules actually originate at tlio plane and then quickly scatter away from it in both directions, so that thoir density, viewed as a function of x, must bo continuous at tho piano. If there wore no external force those particles would scatter equally in all directions, and thoir density at any moment would, therefore, bo symmetrical on tho two sides of tho piano. Tho drift duo to tlio force, howovor, disturbs this symmetry, and wo are, therefore, compelled to resort to some speoial device in order to find a gonoral expression for p' x , Wo may regard tho crossed molecules to tho left of the piano as having ontorod that region through tho layer of fluid next to tho plane and having undergone thereafter a combined Brownian dis- persion and drift. Their density will, therefore, bo uniquely deter- 200 ia Nitric riiiumr or <; i a/ a It 'mi* MI mined by the valuea at all tunes of llieir detent^ m I In* lion at (lit* plane, logothci with I lie fuel lhal llieii density is M *•( t *• and also vanishes eveiy where at l 0 If we eaa Imd a fmietnm Hint HatisfieH IheHa leipmcmenls and also iepie i eiits a ib -iiibiilimi *>el up By Brownian motion under the Hhmdliiinoiis mlhiomo « •! the nvlniiial foiee, this fnnelion will be tin* dialled den at v Now mieh a fnnelion h easily guessed A gioiip *»f A" iiimIh tiles n(mling at tune / 0 at a point > b would hn\e a dnisit\ o'. It (’Itr nt )'« (if h Hit at any point at time l (ef the similar o\pies.|on for /»* above j, and it in enmly vended that I, Ida funetion of r and t leditem to i >>,*„ at a n till provided wo Hot b - « 2n, It Nt 1 lenee />£ as *o delnn d n pn *t nts Ilia density of eionsed inoleetdeK also at points v\ht ie x - n We can then wnle foi the total iiiiiiiIh’i* of eiosaed niolei idea at lime t after mibatilulmK h •> 1 2ri and (r 2n tit) t/ m the (ii*Ht liiti'Kial and U nl)/(‘Wt) li < it in the aeenml and wnt mu ti ‘I ui a ut n ^ (<U)t) h ' f ll/tf) 1 *’ '1'he derivative of Huh oYpieHsion with lewpeet to I is then, hually, the number of parlielea that, enwa the plane for the find time in a second, it ia (IN' (It AT vV ♦Ml fit <»' tltx tit fit tlH til mV 1 t*t «iMl uh Tho mean time of traiiKit from the origin to thr> plane Is * ? tsa « r * tit V-lr/^/w Vt U 4 (H fl * u f J Iiifi ih oxuclly wlmt llu* iHr 4 !! 1 1 time* would In* jf all iwrlirlifa miuum! ut Iho uniform drift mO u. Tho pmumrn of iho Biowniun motion iIuin only flu 4 timkiitjar of many olmoivulionM of tho liitu'H of limtNii in nnlor lo til ii run u uuml avoruRc*, no cnrierlinn in the final ramilf irnnir horn# mpuml, * Tho hitc'grnl ran bo rmlunod tci f */ ’*iU CHAPTER VIII PROPERTIES OF GASES AT LOW DENSITIES In our treatment of transport phenomena we assumed the mean free path to be small compared with distances in which we are inter- ested from the physical standpoint. Under such ' conditions the properties of the gas depend essentially upon the frequency and character of the intcrmolccular collisions, As the pressure is lowered, however, with any given experimental arrangement, there must come a time when this condition is no longer satisfied, and a departure from the laws of high-pressxivo behavior would then be expected. As the density sinks, intermolecular colli- sions must lose their importance, and finally there must come a stage when such collisions are actually rare in comparison with collisions of gas molecules with other bodies such as the walls of the containing vessel Each molecule will then act independently of all the others in giving rise to the properties of the gas, A gas in this latter condition will be said to exhibit free~molccale behavior, in order to have a convenient term of reference. The term “molecular” has often been used in the same'senso, contrasting with the “mass” or “molar” character of ordinary gaseous behavior. Experimentally it has been found that at low densities gases actually do exhibit novel properties/ As the density is lowered, the gas seems to lose its grip, so to speak, upon solid surfaces; in viscous flow it begins to slip over the surface, and in the conduction of heat a discontinuity of temperature develops at the boundary of the gas. Curious force-actions may also manifest themselves, such as are responsible for the niotion of those radiometer vanes that are fre- quently seen spinning in the sunshine in jewelers' windows. It is convenient to discuss the theory of all these phenomena as a group. If we knew more about the interaction of gas molecules with solid and liquid surfaces, it might be logical to begin with a study of the laws of this interaction. We know little as yofc, however, concerning these laws, and hence it seems preferable to follow the historical order and to take up in succession the subject of viscosity at low density, then the conduction of heat, and finally that of thermal creep and its application to the elucidation of the radiometer. In dealing with the first two subjects, it will be convenient to take up 291 SJ02 KIN I 1 TU' THI'OltY OF <> I SI S‘ flint the initial tli'jim Line from oulmury law* ie< Ha* deieutv i** lowered, hi id | Ik>ii lhi> behavior of I la* Rus when in I la* fin* nioloi till* toudilmu Cii'i'p and tin' radiometer cffeeH, cm (la* other Imml, mo * fleet i\e|y limited lo ili'iiHilicH llml hic* only mmli intelv low, MOTION IN RAUKKIin OASIH 168. Viscous Slip. In 1H7H Kumll nml YVuibuiR* pnfoimcd a mhu*s of expriituenls ui*on tin* dampiiiR of a mIiihIiiir do*k by 11 Mir JtnuidinR rus and found llml ill low precunes tin* dumpuiR «l> ereaml; UiIm cffi'i'l, ||u>y iwri lietl to u nlippinR of (In* Rim mi’i (In' wall* of Hit* lulm, hui'Ii u 4 hint Hoiiii'limi'H been supposed In ni't'iii with lupinls Their iiilm in Kill ion of (In- itlu'iioiiu'ium 1ms been loolirnnd l»y Inter work, both expeiimenlul nml llieoirlieul Pioni nimbly anynueh niippuiR wmihl In* proporfioiml lo l In* voloi uy Rmil ion t next to llu- wall of tlio tube, at leant ho Ioiir itr* ll»H gradient ih ho ml 1 . Aecoiilinnly, if we wule en foi llml eoiiipotii'iit of the mim* velocity wlueli vanes mill di ' u/ds for i(h giudicnl, taken positive in the iliicKioii away fiom (lie wall, (In* velocity of slip h relative to (In* wall will have the clnection of v n anil can be w alien u * ■ t CM 7 ) ds where f Ih a eimnlanl and in eominoiily ealled (In* ctrjl'tnnil »>/ oltp, OlivimiHly l ie|iieHenln a length; il ean lie picltiied by noting llml tin* motion Jh the name an if t I k* wall were displaced backward a dinfa f with tlio velocity gradient extending uniformly right up to jtero velocity at the wall. Iviunll and Wailnng found llu* magnitude of f to be of the order of tlu< molecular mean free path in tin* Ran, nod, like the latter, invtunely piopoilionai lo the pleasure, Inntead of the appioaeh Dial we have chosen here, (lie process oeomrmg in the ga« next lo (he wall in fictpicnlly nnalytted in (erma of the fmcen, following the proceilure employed by IlelnihnlU in the cane of lii|iiidn 'I'lie vihcouh (lad ion in llu> Ran, which is » j(U< ei ds in teiinHtif theordiimiy coenicienl of vineomly v, iinint, in uteady mol ion. bo t't|iial lo thi* tract inn or force per unit men exerted by tin* Ran on the wall; und for the latter we can wule e», whom t i« n constant ealled the coefficient of external fuel ion of the Ran on the wall, Thun tu ydv/dz, and by comparinoit with (!W7) we nee that K V ' M * ICiiNitr ami VVAitiumn, Ann I'hymk, Ififl, 337 (187.1) Site. 106] PROPERTIES OF GASES AT LOW DENSITIES 293 V — ► 1 dx 166. Steady Flow with Slip. Formulas pertaining to viscous phenomena are easily corrected for the existence of slip. We shall illustrate this by obtaining the correction to Poiseuille’s formula for the steady flow of a gas through a long straight tube of circular cross section.* In such a tube the velocity v of the gas across any cross section is a function of the radial distance r from tho axis (cf. Fig. 67). Consider the gas inside an inner cylinder drawn coaxial with tho tube and with length dx and radius r. To make steady flow — possible, the net force due to the pres- F, °' 07 - — F,ow nlo " E 51 tubo> sure p on tho ends of this mass of gas must be equilibrated by the viscous drag over its sides; hence, equating forces in the ^-direction along the tube, dv _ r dp dr ~ 2 dx’ and by integration r 2 dp , n v = ‘ 4 . c. ax Now, when r - a, the radius of tho tube, we have v ~ u, the velocity of slip, or, by (227), . dv .dp n dv -irr“ -j- = — 27rrij t> dx dr dz f.dv dr Hence, according to tho expression just found for dv/dr, when r ~ a, v — — $ (fa/v) dp/dx. Tho formula found for v reduces to this if we give to C such a value that ■ i We then obtain, after multiplying through by tho density, which can be written p/RT in terms of tho gas constant R for a gram and tho absolute temperature T, for the total mass of gas flowing past any oross section of the tube per second, J*£l( 1 4- 4 8 i)RT \ ^ dj dx Now in the steady state Q m must bo tho same at all points; and, according to Kundt and Warburg, we can write £ — £1 /p, where £1, * Tho flow la fissumofl to bo slow enough to avoid turbulence, Qr, - JL f“ “ RTjo ■ Znrv dr 204 kin uric r it unity nr uasfs MU in independent of p and ho of .1 {tin* lenipetahne being Mummed uniform). Then, if we nmlliplv (In’ lu-l I'lpmlmii fh tough by dr ami integmlo along the lube, willing f ilr /, dm length of dm (uhe, wo obtain IQn v a 4 K nltT /*S> in) * Jh and pa being dll' piommuioh III (ho end* of (ho lulio. III , if we irplio e Qm by Q»\ - liTQ m oi (ho amouiil of gmi (1ml pu^ien per • ihiimI n- meieuued by its pi' valuo, and il wo aim nidudime Iho nte-ui of (h<< ond piOHMinoH, /) (p, | p a )/2, wo have Q>" ' JJj (p I 'I /•»! (22S» Abnolulo unilH havo I mm nnmimoiI (hioughuttf m ihnlmmg Ihi* otpm lion, but it in obvinunly ponmnniblo in o\pio**ung Q v y <h uutplnv uny othor unit for Iho piohhimo, piovhloil Iho hiiiiio uml m iiUm imoit fur (Pi - p»). In thin foi inula iho lorm *1 (i/n umy bo logunlotl iim mu inoiomoui that inunt bo athlotl lu Iho uohnil moan pio»noo io allow fur Iho advmilagouuM offooH nf nhp* Al If i\\* pioKHiio** iIuh |tim may bo lolativoly largo, but tho foinmlu ilsclf U UM\ lu fad whon iho ralio £i/V a or fA* w nut Mimill, fur thon ntuutnl uomlihonM nmuol do\Hop in tho oontuU pint of tho lubo, uml, futlhoimmo, iho nuwiluro uf Ilia wall ih thou hkoly lo bo of inipoitunoo Pvoblnnn * J. Ju iho ulomly How of a gim hot worn two parallel piano hu rf nr ok, Iho volooily lining owuyuhoio m ilm muuo direihun, hIiow that tho aiuouuir of gun (ntm*fVi k i*i I por aernml (ur nirli tuiti of width porpomlioular lo iln volooily, moanuml in lorm* of ifw pV value, in (oxorpl, of coin ho, unur (.hr imIrok ol Iho phtuoaj <i,,v ' 12 »}/ \f 1 t, 1 )*/' 1 dd, vm>n w I wing tho diHlniu-e between I hr* mnfnreH and l (heir length in flie direction of Now, t; (la* UHnmiiy uf die gen, p iIh mean pienanm and Pi -pa Iho total dlnp in prewure, f ,/p f, dm alip dm! in me ut each Hiirfaeo; « g.H. unilH am mwiinmd, 2. If one plain ih al rnnl and Ilm nllmr moving tangent nilly al uniform upend U, hIiow dial in (lie notation jiint dnliimtl dm vita mi* drag upon each plale ih (except near dm edged, and provided (he plain* aro relatively elnw> (ogetimr and the gna prmnne iinifmin) Sec. 107] PROPERTIES OF GASES AT LOW DENSITIES 295 tv + 2f (229) dynes per unit area. (Hint: The viscous stress must be the same across any plane parallel to the plates anti hence equal to ij dv 0 /dx.) 167. Maxwell’s Theory of Slip. It should be possible to calculate the magnitude of the slip distance £ in a gas from kinetic theory. On this point the theory developed by Maxwell in 1879, although not perfect, is still the best that we have. Ho utilized for the purpose the results of an elaborate analysis that he had previously made of the stresses in a moving gas, but his reasoning can also be thrown into a very simple form. * Consider the usual case of a gas having a mass volocity Vo, whoso direction is everywhere the same but whose magnitude varies in some perpendicular direction; let us take the direction of the velocity as that of the y-axis and the direction of its variation as that of x. Lot the gas bo bounded at the left by a fixed plane surface perpendicular to x, and beyond a certain distance from this surface suppose that tho velocity gradient dvo/dx is sensibly uniform. Then just next to the surface we. can group tho molecules into two streams, of which one consists of molecules that are approaching the surface, and tho other of those that have just struck it and aro now receding from it; and we can view the viscous drag on tlio surface as arising from the difference between tho tangential momentum brought up by tho approaching stream and that carried away by the receding ono. Maxwell now makes the rather bold assumption that the approaching stream is of tho same character as it is in tho midst of the gas. If the same thing wore truo of tho recoding stream as well, and if the gas at the surface wore on tho whole at rest, it would then nocossavily be true that tho impinging molecules wore reflected on the average with their tangential components of velocity just reversed; for it is these components that are responsible for tho viscous stress, and in tho midst of tho gas each of tho two streams gives rise to just half of this stress. Such a law of reflection is very unlikely, however. Let us accordingly mnko with Maxwell the more general assumption that on striking the surface tho molecules give to it, on the average, tho fraction / of their tangential momentum (or of their tangential momentum relative to tho surface in case tho lattor is in motion). To restore tho viscous force to its proper value, wc must then allow tho gas to slip over the surface. Lot us, therefore, assume with Maxwell that tho approaching stream exhibits a velocity gradient * Qf, Millikan, Rhys. Ret)., 21, 217 (1923), 296 KINETIC THEORY OF OASES [ClIAl\ VIII extending unifoimly up to some value v 0 — u at tho suiface (of. Fig 68). The tangential momentum biought up to unit area of tho suifnco in a second by the approaching molecules can then be analyzed into two paits Relative to a fiame of lefeience moving with velocity u, momentum will be biought up equal m magnitude to that which is tiansmitted elsewheie in the gas by the eoncsponding molecular stream, 01 to \ t; dvo/dx pel unit area pci second, n being the coefficient of viscosity, to this must then be added the momentum due to tho slip velocity w, of amount } mmm, -} nv lepiesentmg by (72a) in Sec 37 tho number of molecules incident poi second on unit area of a piano in a gas containing n molecules per unit volume Pro os —Velocity gradient nfmr whose mean speed IS V Accoiding to our assumptions, thoie- fore, wilting nm = p, the density, and equating momentum givon up to the surface to that tiansmitted across parallel planes in the gas, we have , (l dv 0 , 1 _ \ dvo '(s’* Hence u must have the value u ® 2 - f ij dvp J PV cH ’ and for the coefficient of slip, as defined m (227), after inserting p = p/RT and v = 2(2RT/ir)M horn (66a) in Sec 30, we find ? (230 a) or, if we insert q = cpvL fiom (126a, b) in See 86, f = 2c L. (2306) Here T = absolute temperature, R *= gas constant for a gram, p ~ pressure, L = mean free path, and c is a number lying between 0 491 and 0 499, so that veiy neaily 2c — 1 168. Discussion of the Slip Formula According to (2306), f is always of the oidei of one mean hoe path and must, like the latter, vary at a given tempcratuie m niveisc latio with the pressure, as was originally found experimentally by Kundt and Warburg 297 Sec. 108] PROPERTIES OP GASES AT LOW DENSITIES The value of /, the transfer ratio for momentum, will presumably depend upon the character of the interaction between the gas molecules and the surface; it may vary with the temperature. We can imagine a surface that is absolutely smooth ancl reflects the molecules "specu- larly” with no change in their tangential velocities; in such a case / — 0 and f « oo , viscosity being unable to get a grip upon the wall at all. On the other hand, we can imagine the molecules to be reflected without regard to their directions of incidence and therefore with complete loss of their initial average tangential velocity. They might, for instance, be reflected diffusely according to the same cosine law that holds for the diffuse reflection of light or for the effusion of molecules from a hole [of. (73a.) in Sec. 37], being distributed, therefore, as if they came from a maxwellian gas at rest relative to the wall. In this latter case we should have / *=* 1, all of the incident momentum being given up to the wall, and f = ^VRT = 2cL, (230c) so that s' is almost equal to L. Maxwell suggested that diffuse reflection might result from free penetration of the gas molecules into interstices in the surface, where they would strike a number of times before escaping. An approach to such reflection would result also from roughness of the surface, except that at largo angles of incidence chiefly the tips of elevations would bo visible to an oncoming molecule and something like specular reflection should occur, Another possibility is that tho molecules might condense on tho surface and then ro-ovaporato after coming into thermal equilibrium with it; in some cases there is, in fact, definite experimental ovklenco for such an occurrence. In the general case, Maxwell himself interpreted a fractional value of / as meaning that a fraction/ of the surface reflects diffusely and tho remainder specularly; but such a special interpretation is obviously unnecessary, Even a value of/ exceeding unity is conceivable, indicating that tho molecules are reflected on the average with a partial reversal of their tangential velocities; for example, uniform reversal would bo produced by reflec- tion from a rectangular-zigzag surface. In comparing (230b) and (230c) with expressions given elsewhere it must not bo overlooked that L is here calculated using tho modern formulas, (126a, b). So many different formulas have been used for L during the last thirty years that it is scarcely sufficient, in writing equations, to define a certain symbol as standing for the mean free 298 KINETIC THEORY OF OASES [Chap, YIH path without specifying its assumed relation to the viscosity 01 to some other measuiable quantity It should be lemcmbeied, fuitheimoio, that the analysis loading up to (230^, 6, c) is fai fiom ligoious, this was, in fact, emphasised by Maxwell, At least the letwmng sticam of molecules is almost coi- tainly modified fiom the maxwelhan foim in quite a diffoicnt manner than is the corresponding stream in the midst of the gas, It may be of intoiest to note that the slip speed u will not usually bo the same as the actual mean velocity of the gas at the wall 169, Observations of Slip, Intel esting dneet measui ements of slip weie made by Tnniriazeff,* using the method of two coaxial cylin- ders with the gas between them In this method the inner cylinder is suspended elastically, and fiom its steady deflection when the outer one is l evolved about it at constant speed, the viscous toiquo oxertccl upon eithei cylmclei by the gas is calculated, and from this in turn tho viscosity In Timii layoffs apparatus both cylmdois wcie nickeled, and air, cm bon dioxide, and hychogen weie employed in turn Ho obseived that as the piessiue was leduccd fiom atmospheric, tho torque at fiist remained constant, illiistiatmg tho constancy of tho ordinary viscosity; then it diopped lapidly as the slip distance f became compaiable with the inteicylmder distance to* A founula equivalent to (229) in Sec 166 was deduced on tho assumption that the nanow space between the cyluideis could be tioated as if bounded by planes, and this formula was found to hold closely even down to pressures at which it might be expected to fail, tho obseived value of f was compaiable in magnitude to the mean fiee path in the gas (about 6 8, 5, and 10 X 10~° cm at atmospheiic prcssuie foi tho three gases in the Older named) The most accurate measiu ements of slip, howevei, are undoubtedly those made by Stacyt and by Van Dykef under Millikan's supei vision They used the coaxial-cylinder method but employed tho accurate formula for it, tho theoiy usually given, which itself requites a little thought,! is readily modified to allow for slip and then gives for Iho torque on unit length of the mnei cylmdoi, which is suspended at lest on an elastic suspension, while tho outei cylinder involves around it at constant speed 8?r Vfofr * Timiiuazbff, Ann Phyail 3 40, 971 (1913) t Stacy, Phys Rev , 21, 239 (1923), Van Dyke, Phys Rev , 21, 250 (1923), t Cf Newman and Sbarlbj, il General Properties of Mattel, M Sec. 160] PROPERTIES OF OASES AT LOW DENSITIES 299 in terms of the viscosity y of the gas, the radii 7*1 and r 2 of the cylinders, and the speed of the outer one, v, in turns per second. The cylinders were made of brass but could be coated with oil or shellac to obtain the slip on such surfaces as well as upon the bare metal. One observation was made at atmospheric pressure, at which the slip distance is negligible, and then another at a pressure slightly above 1 mm, which was low enough to produce a considerable drop in the torque and yet high enough to make tho ordinary slip theory applicable, [f/fa — n) being fairly small]; from these two observations both the value of £ corresponding to the low pressure and the value of 7 i could be calculated, and f for a pressure of 76 cm was then calculated on the assumption that it is inversely proportional to the pressure. In a paper in the same volume of the Physical Review* Millikan gives a table of values of Maxwell's reflection coefficient / which were cal- culated by substituting in the equivalent of our eq> (230a) in Sec. 167 the values of £ obtained by the observers just mentioned and a few others. His table is repeated below, with the addition of the cor- responding values of £/L, tho ratio of the observed slip distance to the mean free path as calculated from (230!)) with 2 c set equal to 0.908. The oil referred to in the table is the watch oil that was used in Milli- kan's well-known oil-drop work on the electron. 1 / (%) i L Air or COj 011 rnficMnotl brass or old shellac 100 1.00 Air on mormity 1 100 1.00 Air on oil , , , , T , * 80,6 1.23 COa on oil . 02 1,17 Hydrogen on oil 02.5 1.16 Air on glass 80 1.24 Helium on oil 1 87. 4 1.29 Air on fresh shellac 70 1.D3 Tho value, / =* 0,89 for glass, was calculated chiefly from Knuclsen's data for H 2 , 0 2 , and CO 2 on the assumption that £ is proportional to tho mean free path* Knudsen himself, however, found no evidence of slip of these gases in tho free-moleculo case; and, as regards Ha, this con- clusion was confirmed by Gaede. A direct study of tho slip of air on glass would seem to be of interest* Such values of /must be received with a certain caution, however. In the first place, the difference, 1 — /, although commonly said to * Millikan, Phys. Rev., 21, 217 (1023). 300 KINETIC THEORY OF GASES [Chap VIII lepiosent speculai leflection of the fi action 1 — / of the molecules* presumably lepiesents in leality only a ceitain piepondeianco of fonvaid directions m the scattoimg piocess, in the second place, Max- well's formula foi f cannot be said to be ngorously established until the state of the gas next to a solid boundary has been more accuiatoly woilced out (The lattei uncertainty, of couise, does not affect the values of {/L ) As to the lattei point, however, it n>ay be noted that Blankenstem* obtained in a smulai way values of / ranging from 0 98 to 1,00‘foi H 2 , Ho, an, and C0 2 reflected from polished oxidized silvei, and obtained values only 1 to 3 pei cent lower when he repeated his observations at picssuies of 0 0005 to 0 002 mm, at which the free- moleculc foimula (2326) should hold Since the lattei foimula is not subject to the same unceitamty as is Maxwell's, this agreement of values of / obtained at high and low pressures tends to confhm Maxwell's foimula foi f at the lnghei piessure 170, Free -molecule Viscosity, The concept of slip as usually understood is applicable only when the layer of gas is many moan flee paths thick so that ordinary viscous motion can come into exist- ence m the moie distant pait of it When this condition is not satisfied, the phenome- non becomes moie complicated; its theoiy has not been worked out for the general case, The situation becomes simple again, how- ovei, in the extreme iicc-moleculo case in which the density is low enough, or the gas layei thin enough, so that the collisions of molecules with each other may be entiiely neglected m compaiison with theii impacts upon the walls, This case is easy to treat As a first example, consider two parallel plates sepaiatcd by a distance w that is very small as compared with the mean fiee path in the gas between them, and lot the upper plate be moving tangen- tially with the velocity U (cf Fig 69), Under these circumstances each molecule, aftei staking one plate, moves at constant velocity until it strikes the other. Hence, if ui is the mean tangential com- ponent of velocity as the molecules leave the lower plate, this will also be theii mean component as they arrive at the uppei, and simi- larly we oan write th for their mean tangential component as they leave the upper or amve at the lower, Then, if / 1 , / 2 denote the coefficients of momentum tiansfei at tho lower and uppei plates, respectively, defined as in Sec, 167, the -u* Ml, (At rest) Fig 09 — Viscoua drag on a ldato * Blanicbnstidw, Phya Rev , 22, 582 (1923) 301 (tec. 170] PROPERTIES OF OASES AT LOW DENSITIES momentum given to unit area of the upper plate in a second is - XJ), (231) where Ti is the mass of gas that strikes unit area of the plate per seoond. An amount of momentum equal to this is lost by the mole- cules thomselves, and this loss can obviously be written — tt ? ). Hence it must bo that Mi - Mj — ' /a(«i “ XT). Similarly at tho lower plate, which is at rest, we find Ut — Mi — filli. From theso two equations we find that Mi «=» /»(! -/.) h + A — A A u, lh = A A + ft — AA U. Now lot us mnko the usual assumption that the molecular velocities in the gas are distributed very nearly in the maxwellinn manner, corresponding to some absolute temperature T, This will certainly be true so long as U is small ns compared with the molecular speeds. Then, by (726) in Sec. 37, r a = p/(2x.ftT)» in terms of the pressure p and tho gas constant It for a gram; hence expression (231) for tho momentum given to tire upper plate por unit area per second can be written in tho form — ZU, where AA V ~ jTi +/,”!7. (2 «RT)» (232a) The expression — ZU also represents, of course, tho momentum in the opposite direction that is given per second to tho lower plate. If tho plates are alike (A = ft — /), ' ■ 7 / V * ~ 2 - / (2ri25T)15 If / = 1, as for perfectly diffusing plates, this becomes simply (2326) z ~ (2 JiTp' (232c) Hero p, R and Z are all in e.g.s, units. Tho coefficient Z thus defined might be called the free-molccule viscosity of tho gas between tho plates. We note that it is independent of their distance apart. A little re (lection shows, in fact, that, so long as intormolecular collisions may be neglected, Z must always be inde- 302 KINETIC THEORY OF OASES [Chap, VIII pendent of the magnitude of the solid bodies m contact with the gas and determined only by their lelalivo shape On the other hand, at constant tempeiature it is piopoitional to the picssuie or tho density, in contrast with the ordinal y viscosity As a fuither companson we may note that the ordinary viscous drag on unit aiea would be, by (229), r\U/{w + 2f) oi, by (126a, b) m Sec 86, cpvLU/(w + 2f), oi, by (66a) in See 30 and p = pRT, 4cpLU/(w + 2f)\/2 RT, wlieio c is close to l A For / ^ 1 this latter expression exceeds ZU calculated fiom. (232 b), at least so long as L > w, so that, unless f exceeds unity, fioe-molecule viscous foices are well under those calculated by the oidinaiy foimula Free-moleeule viscosity was put to use by Langmuir in an mstiu- ment that he devised foi the measuiemont of veiy minute picssmcs He suspended a disk on a torsion fibei above a second paiallol disk that was kept m constant rotation The steady deflection of tho upper disk, due to viscosity of the intervening gas, was found to bo exactly piopoitional to the gaseous piessure, piovided tho mean fiee path was many times the distance between the disks, and this deflection served, therefore, as a measuie of the piessuie after the instalment had been calibrated at one known piessuie 171, Free-molecule Flow through Long Tubes, The most impor- tant type of gaseous flow for practical purposes is that through long tubes Let us suppose that the tube is cylindrical but has a cross section of any nomeentiant shape, and that its walls have a pcifectly diffusing surface Let the piessure bo maintained at diffoient steady values at the two ends, the tempeiature being unifoim Consider the flow of molecules acioss a cioss section BC of the tube; and consider fiist the flow acioss ail element of area dS of this cioss section (Fig 70), These molecules that cross dS come fiom various points on the wall of the tube, where they underwent reflection; lot us select those that come from an element dS' on tho wall distant -c fiom the plane of BC , and also distant r' from dS in a direction making angles 6 with the noimal to dS and 6' with the normal to dS', icspoc- Sbo, 171] PROPERTIES OF OASES AT LOW DENSITIES 303 tively. As molecules strike dS' } they will leave it after diffuse reflec- tion in the same manner as if they came from a gas in equilibrium with the density n' and mean speed d of the gas in the neighborhood of dS r , and by (73a) at the end of Sec, 37 4tt n'V dS ' do) cos O' of them will pass downward through dS per second; here do) is the solid angle subtended by dS at dS f and has the value do — dS cos 0 r' 2 Now x s r f cos 0 , Furthermore, if we draw in BO a line’ of length s from dS to that point D on the tube which lies on the same generator as dS f , and then draw the normal to the tube at D , this normal making an angle e with the former line, and if wo then project these lines upon the cross section through dS* as in the figuro, wo see from tho geometry that r* cos O' = s cos e, since r f cos O' is the projection of r f and hence also the projection of the broken line (dS-N-dS') upon the normal to dS' , Also, r' 2 — s 2 + a 2 . Hence, if we write dw dx for dS\ dw being an clement of the periphery of the cross section through dS' t wo have for tho total net number of molecules passing downward through dS per second cos 0 cos O' dS f *== dS 4.T C w n*x dx C 7 J-. ^ + ar J SCOHtdw > tho integral in dw extending around the periphery of a cross section distant x from BG but being obviously independent of x. Now, if we limit ourselves to small density gradients, wo can write n , , dn n representing the density at BG. Then in tho last integral tho term in n vanishes, whereas that in dn/dx contains tho integral f" x* dx _ 1 ('“ J_ „(** + *•)» 2j_ ( dx x l + s 2 -i tou- i r 2s Furthermore, projecting dv) onto dS } wo see that dw cos € = s dtp, where <p is tho angle between 8 and any fixed line of reference drawn in BC through dS, Heitee we have finally, for the total net number of mole* cules that pass through dS upward or toward x ~ 4- co per second, 3u4 KINETIC THEORY OF OASES [Ciiai> vm dNm ~l(f 0 2nsdip )^ dS ’ and, lutegiating again, foi the whole number passing upwaicl acioss BC pei second, Let us now multiply this equation thiough by m } the mass of a molecule, and then intioduce m it v =» 2 s/^RT/tt fiom (60a) m Soo 30, and the piessuie, p ~ nmRT, R being the gas constant foi a giam The result is the mass of gas passing any point of the tube poi second : Qm = {2rRT)» [/‘“’JT* d<e ] £ In the steady state Q m must be constant along the tubo, hence m the free-molecule case the pressure giaclient must bo uniform, in contrast to its linear vanation in the high-density case of Poisouille We can, therefore, replace dp/dx by (pi ~ p*)/l 9 where l is the longth of the tube and p t — p 2} the cliff ciencc of pressure botween its ends* For piactical use, howevei, it is more convenient also to multiply thiough by RT and so obtain the amount of gas passing per second measiued in terms of its pV value, for tins wo thus find, finally, (233a) In this equation any units of pressure can be employed for p i} p 2f and Q, R must, howevei, retain its c g s value 172* The Long-tube Formula. The integrals left standing in tho last two equations lepresent quantities characteristic of the cross sec- tion of the tube, which can be calculated m any given caso For a circular cross section of radius a, the calculation happens to be easy, although even here I s d<p is not independent of tho position of dS and the flow is, therefore, not quite uniform over tho cioss soction Writing M 1 ' sd(p = d<pf sdS , and then fixing <p momentarily while we cany out the integral in dS } let us diaw oaitesian axes as in Fig 71 with the ongm at the center of the cncle and the 2/-axis parallel to the line of length s Then we can wnte dS = dx dy , and, x and y denoting cooulinates of a point m dS> Cue. 17'2] PROPERTIES OP OASES AT LOW DENSITIES 305 J f>a r'(a*~x l )M sdS — I dx I [(a 2 — s 2 )W — y] dy = J-a J — (o s — ■r.< Hence s dS ^ 167ra 3 /3, and (233a) becomes, for a circular tube, (233ft) 2 j (a 2 — x 2 ) dx ~ - a 3 . This formula, like (233a), is limited in its application by the double condition that the diameter of the tube must be small in comparison both with the mean free path and with distances in which a consider- able change occurs in the density of the gas. There arc also end corrections which can be neglected only if the tube is very long. Formula (233ft) was first obtained by Knud- sen,* but ho used an unreliable method and some of his results were wrong. Formula (233a) for the general case was obtained soon afterward by Smoluchowsld,t following the reasoning that wo have given. Knudscn's method was to equate the momentum. imparted to the tube by molecular impacts in a second to the difference in the pressure forces at the ends. This is correct, of course, blit then in calculating the momontum, ho assumed Maxwell’s law to hold approximately, and as we have scon in our treatment of ordinary viscosity and heat conduction this assumption may introduce a considerable error in calculations of differential effects. We have ourselves employed an expression for the effusion of molecules that is appropriate only to the equilibrium state, but we were not com- pelled to subtract from it another quantity almost as big as itself, and the resulting error is, therefore, only of the seooncl order in the density gradient. In experimental work, however, the most significant thing is likely to be the rate of flow as measured by volume , since this controls the relative rate at which pressures undergo alteration. The rate of flow in these terms is, roughly, 2<J/(pi + p %) ; hence, from (233ft), wo reach the important conclusion that for a given value of the ratio jn/p 2 the flow in terms of volume is independent of the density of the gas. * Knudsbn, Ann , Physik, 28, 7J5 (1900). f Smoltjohowbkt, Ann. Phynk t 83, 1559 (1010), 300 KINETIC THEORY Ob' OARER ICjui* VIII To illustrate tho order of magnitude of fiee-nmleride flow, Jl may he lemavked llml according lo the foimola a Inillt containing a litci of mr at a pressum nnywhoie below 0.01 mm ami neeted to a high vacuum through a lube 30 cm long and 2 mm in diameter will half empty itself in ft liltlo over thieo minutes, This is a fmily long time, and it imlt- ealCH Unit at low density equidiiiulinti of the piessuie thiough small openings is n compai atively slow pineess The situation in much better, however, than if tho mdimtiy I’nisctnllc equation ((22B), with f] ca ()] hold under these conditions; in that oust* the time requued would bo noarly two bourn It was pointed out by HmoluchowHld (lor n't.) that the foimula is eaaily generalized to cover the case in which only a eeilam fiat linn/ of tho moimiloH am Healteied diffiiHcly, while a fiaetion I / aie specu- larly reflected. In that ease the formula for Q becomes, in place of (233b), Q** J 2 ; * (MtT)» (/», /ij). (233r) Tho now fftolor (2 —/)// ichuIIh from the fact that the net number of molecules crossing any cross section lt(\ being determined by tho density gradient along tho tube, depends upon the mean distance from 11C at which they cxpei ienco then last diffuse lellectum fiom the walls, specular reflection meicly handing them on with their component of velocity along the tube uualteied, and this mean distance enn be shown lo bo ineieased in the ratio (2 - /)//. 173. Flow through Short Tubes. Tho results just obtained are, of course, accurate only for IiiIicm of indefinite length. !u pi set ice the end oormetioiis required fur Lillies of finite length may likewise lie of interest, and in sumo eases, for example, in working with molecular bourns, the value of tho freo-molootiie How through a short tula* may la* needed. For such quantities only approximate values Inn e as yet been obtained, but some discussion of these may lie of mlotoal. (Suppose two vessels containing gas in eqmlihimin at temperature T and at very low pressures pi and pt, respectively, are connected togel her through a round lube of length l, which lias peifcclly diffusing walls, and lot tho radius a of this tube bo veiy small as nun pared with the moan free path in tho gas. Then, when l/n is very large, the rale of How through tho lube measured in Ioiiiin of p V is given by (233b) above and ho is proportional lo l/l, At the opposite extreme, on the other hand, when l **> 0, the lube reduces to a circular opening in a thin plate and, according to oq. (72d) in Bee. 37, the differential rate of free- 307 Sec. 173] PROPERTIES OF OASES AT LOW DENSITIES molecule flow through it, measured in terms of pV, is <3 = a\\irTlT)Hvi ~ p»). Now it is easy to invent a formula that passes into these two forms as limiting cases. The simplest one is Cushman’s formula: 0 = a*(%wRT)U 1 + * {l/a) (pi - ?Ja). (233d) For a very short tube, however, this formula is easily seen to be not quite right. A short ringlike tube, as in Fig. 72, acts to decrease the flow as compared with a plane opening of the same cross section by intercepting molecules that would otherwise pass -through. The effect of this interception can bo found, to the first order in l/a, by the following argument. If the gas had everywhere the same density ni that it has in the left-hand vessel, then by (72a) in Sec. 37 a total of $ nifh X 2«tJ or * miihal molecules would strike the wall of the tube per second, and almost half of these would have come directly from the left-hand vessel; after striking the tube, only half of these in turn would eventually pass on into the right-hand vessel, whereas, if l were 0, all of them would do so. The presence of the tube thus decreases the number passing through by a quarter of the number that strike its wall or by $ TWifhaf. On the other hand, the total number passing through in the absence of the tube is \ wiSiTra 2 . Hence the tube reduces the flow in the ratio Fra, 72, — Effusion through a ring. 1 _ | Tffliih al _ i „ I L. J nif)i7ra 2 2 a This result suggests as an approximate formula valid for small l/a, in place of (233d), «-TT5W (l, ‘ (m,) For largo l/a , however, this formula is certainly wrong in turn, by a factor of The problem of the short tube was subjected to thorough study by Clausing in his thesis,* No accurate formula could be obtained in * P. Clausing, Dissertation, Amsterdam, 101S. 308 KINETIC THEORY OF GASES [Chap. VIII terms of known functions, but he works out a close approximation and also gives a table of values representing the ratio Q -i- a 2 (|- irRT)'>~('pi — p a ) as a function of l/a [page 130, values from his oq. (203)]. His results agree, naturally, with (233e) for small l/a ; a compact empirical expression that reproduces them within 1.5 per cent for all l/a is 20 + SI Q = 20 + — + s (- ~ ' a a*QirRT)*(pi - P 0. As compared with Clausing^ values of Q, use of the shorter formula of Dushman, (233d) , may incur an error of nearly 12 per cent. 174. Observations of Free-molecule Flow. The only investiga- tions that cover the free-molecule case are the elaborate studies of the flow of gases under a pressure gradient made by Knudson* and by Gaede. f Both of these investigators studied the molecular flow of gases through circular glass tubes, for which Knudson developed a formula equivalent to our (233 b) above. Gaede took the further precaution of freeing the tube thoroughly from adsorbed gas by pre- heating, and kept all water vapor frozen out in a side tube immersed in liquid air. A critical quantity in such work is the ratio of L, the mean free path, to the radius a of the tube. Using a capillary 0.206 mm in diameter, Gaede found in the case of hydrogen agreement within 1 per cent with the theoretical formula as represented by our eq. (233 &), the pressure ranging from 0.0001 mm ( L/a = 8,700) up to 0.001 mm (L/a = 870), and in the case of nitrogen agreement within 2 per cent up to 0.002 mm (L/a = 230); but at 0.008 mm (L/a = 108 for H 2 , 58 for No) the observed flow was in both cases smaller by several per cent. Experi- menting with less refinement, Knudsen had got a similar agreement using H 2 , 0 2 , and C0 2 ; in the case of H 2 he found the formula to hold even up to pressures at which L/a = 0.6. Since the only special assumption made in deducing the formula, in addition to the general results of kinetic theory, is that of diffuse reflection by the walls of the tube, the latter assumption seems to be definitely confirmed by these experiments for H 2 or N 2 reflected from glass. Both Knudsen and Gaede investigated, also, the manner in which at higher pressures the transition occurs from the free-molecule formula ♦Knxjdsen, Ann. Physik, 28 , 75 (1909); 35 , 389 (1911). t Gaede, Ann. Physik , 41, 289 (1913). Sec. 175 ] PROPERTIES OF GASES AT LOW DENSITIES 309 to that of Poiseuille. In Fig. 73 is illustrated the general course of Q, the rate of flow measured in terms of pV } as a function of the mean pressure p for a given ratio of the pressures at the two ends of the tube. Starting out at low pressures along the free-molecule straight line FM, the curve for Q approaches asymptotically the quadratic Poiseuille curve P as the mean free path becomes less than the tube diameter. Both Knudsen and Gaede found that the curve had a form like J rather than like K , the ratio Q/(pi — p<P) exhibiting a minimum when the mean free path became several times the tube diameter. Finally, Gaede investigated the flow of hydrogen between two parallel plates placed only 0.004 mm apart and found QJ (pi — p<P) to be as much as 50 per cent below the theoretical free-molecule value at a pressure of 23 mm, the mean free path L being then just about the same as the distance h between the plates. The drop from the theoretical value began, how- ever, at very low pressures; from 0.019 mm to 0.265 mm it amounted to 18 per cent, whereas at the latter pressure L is still about 83 times the width of the slit, and under such conditions, even if we assume every collision to remove both molecules entirely from consideration and suppose, also, that the molecular paths might somehow have an effective average length of as much as three times h, we can reach only a possible theoretical drop of or 4 per cent. The entire observed drop in Q/(pi — p<i) below the theoretical value was ascribed by Gaede to the formation at higher pressures of an adsorbed layer of gas on the walls of the tube, which he supposed might increase the resistance to the flow. It is hard, however, to see how an adsorbed layer could do anything except increase the amount of diffuse reflection, and the latter is already assumed to amotmt to 100 per cent in deducing the theoretical formula. Further experiments on this point would seem to be worth while. 176. Stokes’ Law for Spheres. An interesting special case that deserves brief mention before we leave the subject of slip is the steady motion of a sphere through a viscous medium under the influence of a steady force, such as its weight. Stokes showed long ago that if the ordinary laws of hydrodynamics hold, and if the velocity U of the sphere is not too large and there is no slipping of the medium over it, the force required for steady motion is F = — 6?r TjaU 310 KINETIC THEORY OF GASES [Chap. VIII in terms of the radius a of the sphere and the viscosity 77 of the medium. His deduction is easily modified to allow for the occurrence of slip;* in terms of the slip distance £ (Sec. 165) the modified formula is -a to the same degree of accuracy to which Stokes' law itself holds. This latter formula should be valid for a gas so long as the mean free path L is much smaller than the radius a . The opposite extreme case of large L/a was studied in detail by Epstein, f He showed that in such cases the force on the sphere is F = — cx.vpa 2 U , where p is the density of the gas and v the mean speed of its molecules, while the constant <x depends on the law of reflection from the surface of the sphere; if the reflection is specular (or if the molecules condense on the sphere, spread uniformly over it, and then evaporate again) a = ax = 47r/3, whereas if the reflection is diffuse a ranges from 1.442 a x when the sphere does not conduct heat to 1.393 ol x in case it conducts perfectly. Experimentally, Millikan J found values for charged oil drops moving through air of very low density equivalent to a = 1.365 ax, while Knudsen and Weber's results for glass spheres in air correspond to a = 1.353 an. Smaller values like these can be accounted for by assuming the existence of a small tendency toward specular reflection. The intermediate condition in which the mean free path is of the same order of magnitude as the radius of the sphere is difficult to handle theoretically. In this region, Millikan, and also Knudsen and Weber, find that the empirical formula j? _ QTTjaU 1 + (L/a) (A + fits the data well. In terms of mean free paths calculated from eq. (1266) Millikan's data for oil drops in air require A = 1.23, B = 0.41 c = 0.88. At such low pressures that L/ a is large this formula becomes approxi- mately j/ 80 that ^ and B must related to the coefficient a in the preceding formula thus : a = 7-5 — r » (A + JBjpLv *Cf. A. B. Bassett, “Hydrodynamics,” vol. II, p. 271, 1888; H. Lamb, “Hydrodynamics,” Sec. 337. t Epstein, Phys . Rev., 23, 710 (1924). J Millikan, Phys . Rev ., 22, 1 (1923). Sec. 176] PROPERTIES OF GASES AT LOW DENSITIES 311 At high pressures, ( L/a « 1), on the other hand, the formula becomes F = — — — and comparison of this equation with one 1 + — a just above shows that f = AL. The value A = 1.23 agrees, as a matter of fact, with the ratio f/L = 1.23 obtained by the revolving- cylinder method and cited in the table in Sec. 169. THERMAL CONDUCTION IN RAREFIED GASES 176. Temperature Jump and the Accommodation Coefficient. In analogy with the phenomenon of viscous slip it was suggested long ago by Poisson that at a wall bounding an unequally heated gas there might be a discontinuity of temperature. He wrote for this assumed discontinuity an equation equivalent to T K -T w = g^, (234) where T w is the wall temperature and T K , as now understood, is what the temperature of the gas would be if the temperature gradient along the outward-drawn normal to the wall, dT/dn, continued without change right up to the wall itself. The constant g represents a length and may be called the temperature jump distance. Upon Warburg’s suggestion Smoluchowski* performed experiments in search of this effect and found it; he showed also that, as kinetic theory would lead one to expect, the jump distance g, for which he wrote 7 , is inversely proportional to the pressure and so directly proportional to the mean free path L. He found g = 2.7 L for air but g — 11L for hydrogen, in terms of modern values of L; the latter high value he ascribed to a difficulty experienced by the very light molecules of hydrogen in exchanging energy with the molecules of the wall. In developing a theoryf of the phenomenon Smoluchowski intro- duced a constant to represent the extent to which interchange of energy takes place when a molecule of the gas strikes a solid (or liquid) sur- face. There has come into common use, however, a slightly different constant introduced later by Knudsen. J This constant, which he called the accommodation coefficient and denoted by a, can be defined as standing for the fractional extent to which those molecules that fall on the surface and are reflected or re-emitted from it, have their mean energy adjusted or “accommodated” toward what it would be if the * Smoluchowski, Ann. Physik, 64, 101 (1898). f Smoluchowski, Akad. Wiss. Wien, 107, 304 (1898) ; 108, 6 (1899). t Knudsen, Ann. Physik, 34, 593 (1911). 312 KINETIC THEORY OF GASES [Ciiap, VIII returning molecules woie issuing as a stream out of a mass of gas at the tempeiatuie of the wall. If E % denotes the oneigy bi ought up to unit aiea pci second by the incident stieam, and E r that can led away by these molecules as they leave the wall aftei inflection from it, and if M m is the eneigy that this lattei stream would cany away if it earned the same mean eneigy pei molecule as does a stieam issuing fiom a gas in equilibrium at the wall tempeiatuie T„, then a is given by the equation E, - E r - a(E, - E„) (235) Pro 74 — Tomporuturo gra- dient near a wall Knud&en himself prefoned to attach a tompoiature to each of these streams of molecules, just as in the leveise way wc have just associated an eneigy E w with a temper atm e T w ; ho wrote the equivalent of tho equation, T % - T r - a(7\ - ay Heie T % is not necessarily the same ns T K above, which lopiosonts the lesult of exit ap- pointing the tomperatmo gradient in the gas up to the wall; nor aie T% and T r connected m any simple way with tho mean energy or temperature T v of the molecules that me actually piescnt at any momont in tho layer next to the wall Tor these various tempera- tures the theory developed bolow suggests some such relationship as that shown m Fig 74 177 Theory of the Temperature Jump. An approximate theo- retical expression foi the l elation between tho accommodation coeffi- cient a ancl the jump distance g is easily obtained from kinetic theory by completing an argument of Maxwell's in a way that is analogous to his own method of connecting / with the slip coefficient Wo shall alter Maxwell's reasoning somewhat, however, by introducing Knud- sen's ideas in legard to the behavior of the heat onei gy. Maxwell assumes that the stream of oncoming molecules is tho same right up to the wall as it is in the midst of the gas and conesponds to a tomperatiue grading uniformly clown to tho value at tho wall which we have called TV These oncoming molecules will then bung up to unit area of the wall in each second both the heat content of a niaxwel- lian stream issuing from a gas at tempeiatuie TV and the excess oneigy which they carry as their contribution to the conduction of heat, for which we can write ^ K dT/dn in terms of the thermal conductivity K of the gas Now the translational energy cairied by a stream issuing fiom a gas at temperatuie TV is 2T2TV eigs pei gram, as is stated in Pioblom 4 at Sec. 177 ! PROPERTIES OF OASES AT LOW DENSITIES 313 the end of Sec. 37, R being the gas constant for a gram; this is % times as great as the mean translatory energy of a gram of gas in equilibrium at the same temperature, the difference being due to the fact that the faster molecules both issue in larger numbers and carry more energy. The total energy brought up by /S' g of such a stream is accordingly S(2RT k + U IK ), where U,k is the internal energy of the molecules in a gram at tempera- ture T k. A similar stream at temperature T m would transport 5 ( 2282 ’. + Ur, a) units of energy, The difference of these two expressions can be written with sufficient accuracy S(c v + \ R)(T k - TJ), . (236) d U where Cr = f B + and represents the specific heat at constant volume (cf. Sec. 143). If we then add to this latter expression the excess energy carried by the incident stream as its contribution to the conduction, we lmvo the difference between the incident energy and the energy carried away by the molecules on the assumption that they leave as a max- wellian stream at the wall temperature T Wt or Ei - E„ - \ + s(ov - T„), where S now stands for the grams of gas brought up to unit area in a second by the incident stream. By (72&) in Sec. 37 S — p(RT k/27t) w = 'p/{2ttRT)'A nearly enough, in terms of the pres- sure p and the temperature T of the gas at the wall, for which we need not distinguish here between T„, T K , and T„. Furthermore, by (197) in See. 141, R = c p — Cr — (y — l)c t . in torms of the ratio y of tho specific heats, so that cr + $ R — $ (y + l)<v. (237) Hence wo have finally E ( - E, »U> -K~ 2 dn + 5 (y + 1) Cv(T k - T w )p (2 wRT)K ' On the other hand, tho not energy actually delivered to the surface can be equated to the total heat conducted across a parallel plane out in the gas, Hence 314 KINETIC THEORY OF GASES (Chap VIII *• - * - K % E r representing the energy carried away by the reflected stieam From these equations together with (235) we find K ar dn CvfiT K ” Tw) (2t t RT)» y whence, by (234), g = 2—2 (SfcrB20» CL K (7 + l)cvV (238a) Or, if we wish to exhibit the relation of g to the mean free path L, we can do this by introducing the viscosity ij = cpdL fiom (126a, b ) m Sec 86 , let us, however, replace p by p/RT and v by 2(2RT/w)M fiom ( 66 a) in Sec 30, which gives Then 4 Wk (2rRT)» 2 — a 4c K £ CL 7 + 1 V C Y (2386) Here 0 491 gc^O 499, so that 4c — 2 very neaily. As a special case it might happen that a « 1 ; then g should be only a little larger than L } for 1 <7 <| % or 2 < 7 + 1 <2 7, and usually 1 5 < K/j}Cv <25 (of Sec, 103) We should have a » 1 if the gas molecules were adsorbed on the surface as they struck and woie then subsequently re-evaporated at the tcmpeiature of the suifacc, or if the surface were so irregular and cavernous that most of the molecules struck it a number of times before escaping, oven though the accommo- dation coefficient for a single impact might be considerably loss than unity If a fraction y of the molecules wore specularly reflected, the average coefficient of accommodation for tho lemaindei boing a\ } it is easily seen that a =z (1 — <p)au It must be recognized that in our deduction we have tacitly assumed the accommodation coefficient to have the same value for the internal molecular energy that it has for their tianslatory energy Now accoid- mg to the principle of the equipartition of energy it is a fact that these two kinds of energy are distributed independently in any state of equilibrium, but there exists no gcneial leason to suppose that m cob Sun. 178 ] PROPERTIES OF OASES AT LOW DENSITIES 310 lisions they possess tho same mean rates of transfer. The formulas could easily bo generalized by introducing different coefficients for tbo two typos of energy, but the accuracy ho far obtained in experimental work scarcely justifies tho introduction of this complication; moreover, an experiment of Kmulsen’s described in Sec. 181 below supports tho simple assumption made above. Wo have likewise assumed in all of our discussion so far that ordinary conditions occur in the gas at points a few moan free paths away from the surface. Whenever this condition is not mot, tho results of tho present section are not applicable. Problems, 1. Show that the heat conducted per unit area per second through a gas of conductivity K lie tween two parallel plates separated by a relatively small distance <1, per degree difference in temperature between them, is K d + </i T (7a (230o) (except near the edges), Q\ and (/» being the temperature jump distances at the two plates and gi /<l, Qt/tl being assumed rather small. 2, Show that if the plates take the form of two coaxial cylinders of radii n < r 3) for tho inner cylinder A K n (2306) 178. Free-molecule Heat Conduction between Plates, The oppo- site extreme of conditions, in which conduction occurs between two surfaces so closo together, or at such low pressures, that collisions between molecules are rare, was likewise first treated by Hmoluehowski,* but Knudsen's treutmontf scorns to bo a little more satisfactory and will bo followed hero. Consider first a layor of gas between two parallel piano surfaces or plates at temperatures T i and Ta, In such a layer Knudson divides the molecules into two sets, a stream moving with a Component of velocity away from the Hist plate and a stream moving with a component directed toward it; and lie assumes the distribution of velocities to be approximately nmxwellian in each stream but to cor- respond to temperatures T{ and respectively (Fig. 75)4 f-et P[, * 8moi,uohowhki, Akad. 11 't‘a*. Winn, 107, 304 (INDR); Phil. Mag., 21, 11 ( 1011 ). t Knudamn, lac. cil. j 8omo support Ih lout to tills lumiunplkm by Hie obHorvatlmui of Oriwloln mid van Wyk described In Hoc. 182. KINETIC THEORY OF OASES [Chap VIII 810 E$ denote the lespectivo actual amounts of energy earned acioss unit aiea per second by these two streams, on the othei hand, let E h Ez denote what these energies would be if the molecules earned the samo mean energy as they do in maxwellmn sti earns at temperatures T\ and 7*2, respectively Then the first at) earn is continually falling upon the second plate and theieby feeding its molecules mto the second stieam, hence, if ai, a* aio the accommodation coefficients for the two plates, by (235) oi E % — E t ~ a(E x — Eu) } we have in the picsent instance Je',T,' |e£t£ E{ -E'z = (h(E{ - E*) (240a) ‘ 2 Similarly, consideration of the levoiso process v/ 7 7/ 77 J7 r ;/7 7 V7 77 //7 ? vs at the fiist plate gives Fro 76 — Free-tnolooulo conduction of heat E' t - E{ = at (Ei - Ei). (2406) Fiom these equations we find foi the net amount of heat given to the second plate, 01 abstracted fiom tho fiist one, per unit area pei second H - E{ - E't = a i&z ~h ^2 — a-iaz {Ex - E % ) Now Ei — 2?s, being the diffeience in the cmronts of energy in two equal maxwelhan streams at tempeiaturos Ti and Tt, must be given by (236) above with Jbcand T v _. replaced by Ti and 7b, lcspectively, so that Ei - Ei - S(c r + %R){Ti - T s ) Here S must lepresent the grams of gas earned acioss unit aiea per second by each of the actual sti earns in the gas under considciation, being obviously the same foi both m order to prevent accumulation of gas at one plate From (72 6) m Sec 37 we find for its magnitude S — I p1 5 i — \ where p'i, pj are densities in teims of mass and v[, v' 2 mean molecular speeds in the two stieams, the factor is K instead of M as in eq. (726), because here p lepresents the density of molecules moving toward one side only, or just half the density m the equivalent maxwelhan gas This can also be written 8 = i (p'A + P %) = i p'v', wheie p' is the total density of the gas and S' the mean speed of all of its molecules Using p' = p[ + p'z, we find from these equations that Brio. 1781 PROPtiimtiS OF GASM AT LOW PMSlTlES SI? It is more convenient, however, to introduce into the expression for 8, in place of the mean speeds, the corresponding temperatures, Tit T* We can convert (240a, b) into equations in terms of these temperatures merely by replacing E{ } JS f 2) Ex, E 2 by ? T i, T^ } T u Tv, respectively, to which they are proportional; solution of the resulting equations then yields the values ff = "" Q'dT^ ,jy _ aiTy + ai(l — a*)Ti _ ( 24 : 1 a b ) 1 ai + ds *“ ^2 ’ 2 ai + ft2 — aia 2 ^ * Let us also write T* for the temperature of a nmxwellian gas in which the mean speed is v\ Then, since d * VT 7 , the last equation in 9 f gives VF zivr 1 ^ Vf)' (241c) By means of these equations F can be found in terms of the plate temperatures 2\ and 1\ anc! the accommodation coefficients, If T[ and F 2 do not differ much, T 1 will lie close to their moan, and if ai ~ «a this is tlio saino as the mean of r l\ and '1\. Of course, if a, = a a = l, we have T{ => Ti, T 2 = T 2 . In the equation, S — £ p'S 1 , let us now replace v' by 2{2RT f / t) v> according to (06a) in See. 30. Then substitution of the resulting value of S into the expression found above for E\ — ISt, and then of this value of Ei — Ei into the expression found previously for the rate of heat transfer II gives us finally II = am /mA"/ Oi + a 2 — aia 2 ^ \ 2ir / \ cv 4 - - Ti). For convenience lot us write II = A,,,,,,^': — , l\) j A„ iai denoting, therefore, the conductivity per - unit area of the space between the plates. Then, inserting also p' — p'/UT' in terms of the pressure p' of the gas and using (237), wo can write as our final result for free- molecule conductivity between two plates A<, ‘ a * “ a, 4 Z - am Au ’ Au ~ 2 ( y + ^ TpSvF)*' (242a ’ Hero p' is the pressure of a nmxwellian gas having the same density as the gas between the plates but a temperature T‘. If the sur- rounding gas is at a different temperature T, its pressure p should be related to p' by the equation 318 KINETIC THEORY OF GASES [Chap VIII for the diffusive balance between the two masses of gas is controlled by the quantity pv and should, accordingly, be the same as it is in theimal transpnation between gaseous masses [cf eq, (76) m See 39 above] In the two cases a\ — a 2 — a, and a x = a, = 1, respectively, the formula reduces to A oq - — ~ A u or A rtL = aA u (242 c, d) & — d From these formulas we note at once that at a given temperature A p' ancl is independent of the distance between the plates Thus under these conditions the conduction of heat, like the viscosity, follows veiy different laws fiom those obeyed under ordinary circum- stances It is, in fact, easily seen that in any free-moleoulc case the conduction must be pi oportional to the pressure of the gas, and depend- ent only upon the shape, but not upon the size, of the bounding suifaces 179. Free-molecule Conduction between Coaxial Cylinders. One other case, that of coaxial cyhndeis, was likewise taken up by Knudsen This case is impoitant because cyhndncal suifaces aie commonly employed in experiments Here a new fcatiue enters in that some molecules will strike the outer cylinder seveial times before sti iking the inner, which results m raising the effective accommodation coefficient of the outer one; in fact, if its radius is made indefinitely Iaige, the molecules will come completely into equilibrium with it before striking the inner cylinder again, and it will therefoie behave as if it had an accommodation coefficient a *= 1, There is, however, one curious imaginable case, pointed out by Smoluchowski, in which increasing the radius does not increase a; If the outei cylinder reflects a certain fraction of the molecules specu- larly and leflects the remainder as it would for a = 1, and if the inner cyjinder is exaotly centered, then it is easily shown that A is independ- ent of the lelative sizes of the cylinders, and so must have the same value in terms of unit area on the inner cylinder as it would have for parallel planes. In general, the conduction of heat will depend a good deal upon the distribution in direction of the molecules as they return from the outer cylinder For the case of diffuse xofloction as to directions, with Sec. 179 ] PROPERTIES OP GASES AT LOW DENSITIES 319 an accommodation coefficient a as regards energy at both surfaces, Smoluchowski deduced a formula which seems to be right and can bo obtained in the following way. In obtaining a result accurate to the first power of the temperature difference, we may assume the total density of molecular impacts to be the same on both surfaces even when those are curved; this is exactly true when the gas is in equilibrium, and since conduction depends on the impact rate itself and not on its differences, any slight departure from equality at the two surfaces can produce only a second-order elfect on the conduction. Now all molecules that leave the inner cylinder strike the outer; but, if ri, r% are the radii of the inner and outer cylinders, respectively, these molecules, constituting the "first” stream in the terminology employed above, form only a fraction ri /r% of all of those that strike the outer one, since r\/ri is the ratio of the areas of the two; the remainder of the molecules that strike the outer cylinder come from the outer one itself and so belong to tho “second” stream. Accordingly, we can apply (240a) to the process going on at the outer cylinder, provided wo replace El by E{ — + E'J 1 — — V The r 2 \ rzj resulting equation, ? (K - F' t ) - a 2 \^E[+(l - - E % ], t 2 L 7 & \ 7 2/ can be written in the form ^ ~ Ft) - “AF'i ~ F*)l V 1 ? 2 / but this is equivalent to tho original equation (240a) with a a replaced by «a(h - • Equation (2406) holds unchanged. Hence in tho general result obtained from those equations, which is oq. (242a), wo need only replace a s by tho same expression, and then sot «i — a 2 = a, since the surfaces are hero assumed to bo alike. Wo thus find for tho beat conducted per unit aroa per second from tho inner cylinder, II - A rtr ,(Ti - Ta), 1 + (1 - o)(n/r.) A “’ (242e) where 320 KINETIC THEORY OF GASES [Chap VIII T* m the expression given for An in (242b) being obtained from (241a, 6, c) by making theie also the changes just desoubed m a g and then m <t,\ and a 2 For Vi =» ? 2 this formula passes into (242c), as it must For r 2 — » co it becomes Area = 0 A 11 ; (242/) and in this latter case T f 2 — T % while T[ = aTi + (1 — a) Ti foi tho tempciatuie of the stieam leaving the innei cylinder 180 Observed Variation of the Accommodation Coefficient. Tho most illuminating observations of the accommodation coefficient for a gas in contact with a heated suiface are undoubtedly those lcpoiled lecently* by Blodgett and Langmuir, by Robeits, and by Michels The method employed in all these cases consisted in measuung the heat loss fiom a wne stretched along the axis of a cylindiical tube kept at a fixed temperature, in this method the observed resistance of the wne selves to measuie its tcmpeiature, while tho powei spent in it measuies the boat loss. The pressure was made low enough so that the mean flee path was at least six times the diameter of the wiie, in oider that the equivalent of our eq (242/) taken together with (2426) might be employed in calculating the accommodation coefficient a . The last two of the investigates named simply assumed that the impinging molecules had the temperature of the tube, but Blodgett and Langmuii assumed them to have the mean tempeiature of the gas at a distance of one mean flee path from the wire, and then calculated the temperature drop from this pomt to the tube by means of the oidmaiy theory of mass conduction, which resulted in a conection in their case of 5 to 10 per cent. All of these investigators found that tho value of the accommoda- tion coefficient depended gieatly upon the past history of the filament This effect was studied in gieat detail by Blodgett and Langmuir in the case of a tungsten filament of diameter 0 00779 cm surrounded by hydiogen at a piessuie of 0,2 mm; the filament, 40 cm m length, was stretched along the axis of a tube 6 4 om in diameter immersed in liquid ail. They reached the conclusion that a = 0 64 when the tungsten was leally clean, but that this value holds only at tempera- tuies above 1000°C because at lower tempciatuies a film of adsorbed hydrogen forms on the tungsten and lowers a, even to 0.14 under ♦Blodgett and Langmuir, Phys Rev , 40 , 78 (1932), Roberts, Proo Roy Soc , 129 , 146 (1930), 136 , 192 (1930), 142 , 618 (1933) Michels, Phys Rev , 40 , 472 (1932) Sec. 180] PROPERTIES OF GASES AT LOW DENSITIES 321 certain conditions. Furthermore, if there was any oxygen in the tube, a film of that gas or of tungsten oxide seemed to form, and this lowered a to 0.2 or even to 0.1. Such a lowering of a by a gas film is not easy to understand; they suggest nothing that might explain it. An effect presumably due to a gas film was likewise found by Roberts and by Michols in the case of the rare gases, but it was in the ' opposite direction. Using mostly a tungsten filament, they found that when it had just been “flashed” at an elevated temperature, a was much reduced, but as time passed it increased, at first from minute to minute, and then more slowly for many hours. The value found immediately after flashing was considered to represent the accommoda- tion coefficient for clean tungsten. Several ways can bo imagined in which a layer of adsorbed gas might assist the transfer of heat from the gas to the tungsten and so raise a\ for example, heat energy might first be imparted to ail adsorbed molecule and then transferred to the tungsten as this molecule vibrates under the forces which hold it on the surface, or the presence of the adsorbed molecules might cause the impinging one to strike either the adsorbed molecules themselves or the tungsten several times before escaping again. Another possibility suggested by Roberts is that the clean metal may reflect specularly or may diffract a considerable part of the incident molecules, just as Stem and others have found a great deal of reflection and diffraction to occur when molecular beams are incident on certain crystals, and it might well be that this effect is greatly diminished by the presence of adsorbed gas. Roberts found also that for helium on tungsten a increased mark- edly after prolonged heating of the tungsten ; the initial difference between the clean and the gassy states of the surfaco still persisted, however. This effect of prolonged heating he ascribed to a fine-grained roughen- ing of the surfaco caused by the attendant evaporation. The prin- cipal values of a obtained by him are collected along with others in a table on page 323. Those described as referring to clean' tungsten wero obtained by making several observations in quick succession just after the filament had been flashed and then extrapolating to aero time, while those referring to gas-filmed tungsten wore obtained after the filament had stood overnight, The decrease in a observed at low temperatures suggests that perhaps at absolute zero it may prac- tically vanish; tins would be reasonable, since at that temperature the molecules of a solid must be frozen into a very rigid lattice. The much higher values found for argon would naturally bo ascribed to the heavier mass of its molecule, wero it not that Roberts obtained about the same value for neon (M « 20) as for helium (M = 4); the truo 322 KINETIC THEORY OF GASES [Chap VIII cause is, peihaps, moie likely to be found somehow in the stronger attractive force-fields of the argon molecule 181 Magnitude of the Accommodation Coefficient. Also included in the table on page 323 aie some values of the accommodation coeffi- cient a obtained in the pioneer investigations of Soddy and Beny* and of Knudsenf, as lecalculated with bettei founulas by Smoluchowslci;$ because of impeifoctioiis in the experimental method they aro of inter - est chiefly because of the variety of substances investigated. It is quite olhciwise, however, with Knudsen’b much latei investi- gation made m 1930, § in which he measiued the heat loss thiough hydrogen and thiough helium fioni a platinum stnp, first with both sides blight and then with one side blackened with platinum black. From these obseivations and the equivalent of cq (242/) the accom- modation coefficient foi both soits of platinum suiface could bo found Fuitheimoie, lie measiued also the foice on the stnp when its sides weie dissimilai, and then compaied this foice with a calculated valuo obtained on the assumption that the lecoil fiom molecules abounding liom a suiface is piopoitional to the kinetic eneigy that thoy carry away This companson was assumed to give the value of tho accom- modation coefficient foi tianslciloiy energy alone The latter eaino out within 2 pei cent the same as the coefficient foi the entno oncigy as determined fiom the theimal measurements; this agreement ho inteipieted as lending support to the assumption made in tlio tlieo- letical work as to the equality of the coefficient for all kinds of molecu- lai eneigy The lesults quoted m the table fi om Dickins 1 paper || wcie obtainod at much highei pies&ures, at which the othei t} r pe of theoiy should bo applicable (Sec 177) He measiued the heat loss fiom a platinum wne of ladius 0 00376 cm sti etched down the centei of a tube cooled by water Vanous gases weie employed in the tube, and tho pressure, ranging mostly fiom 1 to 10 cm (13 to 62 in the case of II 2 ), was high enough to keep the mean fico path under one tenth of tho diamotei of the wne Oui eq (2396) should apply to this case. In it the term g 2 /i\ can be neglected in companson with g i/?i, since r 2 /r i = 89, and g\> the shp distance for the gas in contact with the wne, being inversely pioportional to the piessiue, can be written g n /v wheie gu is at most * Soddy and Berry, Roy t Soc Proc , 83, 254 (1910), 84, 676 (1911), IKnudsen, Ann Physil , 84, 693 (1911); 36, 871 (1911) t Smoduchowski, Pint Mag , 21, 11 (1911); Ann Phy&ik , 35, 983 (1911). Kntjdsen, Ann Physik, 8, 129 (1930 v II Dickins, Roy Soc Proc , 143, 617 (1933) Sac. 181] PROPERTIES OF GASES AT LOW DENSITIES 323 a function of the temperature. With these changes (2396) can be written 1 _ 1*1 i 1*2 , ffll X = K los n + W<-’ K being the thermal conductivity of the gas* Prom this equation it is evident that 1/A plotted against 1/p should be a straight line, since K is independent of the pressure; from its intercept K can bo calculated, and from its slope, gn i and from this the accommodation coefficient* Dickins' data gave good ^straight lines when plotted in this way except at the highest pressures, where a drop in the curves indicated a more rapid increase in A, which he ascribes to incipient convection* Correction was made for the small radiation losses. The wire temperature ranged from 10 to 40°C with the tube kept at zero, so he also extrapolated the results to 0°C. Accommodation Coefficients On platinum: IIo No Ar H, CO 0, N, NaO CO, NHa so 2 CII< CjIIi (1) 0.38 0.24, 0.18 0.159 0.02 Iff? (2) 0.88 0,34 0.80 0.82 1 u 0,78 0.06 0,73 0.49 (3) Hi on glass (rocalo, by (7) Ho on clean fresh tungsten 0.07 Smoluchowski) 0.88 Ho on clean long-heated ( 4 ) Ho on glass, 130°C (0.32) tungsten 0.18 (6) On bright platinum: IIo on gas-filmed fresh t 0.19 h, 0.32 Ho on gas- filmed long-heated Ho 0.44 tungsten 0.65 On Pt-blftckonoci Pt; Ho on clean fresh tungsten, h, 0,74 22°C 0.067 Ho 0.91 — 78°0, 0,040; — 104°C 0.026 (0) II, on oloan tungsten Ho on nickel 0.086 (1000°C) 0.64 (8) IIo on clean tungsten (old?) 0.17 Ha on tungsten: lie on gas-filmed tungsten Ha film 0 fe 0,14 (old?) 0.82 Oa or oxido film O.l to 0.2 Ar on oloan tungston (old?) 0.63 Ar on gas-filmed t (old?),, , 1,00 (1) Soddy und Botry, mostly aa roc til ou In tod by Smoluoliowakl; (2) DlcUine, ronnloulntodi (3) Knudsan, 1011 ; (4) Ornaloiti jvnd vnn Wyk. Enpcoitilly reliable; (6) Kiiudnon, 1030; (0) Blodfsotl and Langmuir; (7) Koborla; (8) Mlohols. Unfortunately, however, the formulas used by Dickjns in calculat- ing a from g aro unsatisfactory; he employed an old formula suggested tentatively at one time by Smoluchowski, and combined with it $24 KlN’P/ria TifAo'RY OF GA888 ICftAl* Vlll Meyer’s old formula foi the viscosity The values of a that would be obtained by substituting liis values of g, obtained in the manner just descubed, into oui formula (238a) can be found by multiplying the value that he gives in his papei foi the quantity a - by '(7 + 1) tins gives the value of ~ that corresponds to out (238a) a a Values of a obtained in this way fiom Dinkins’ data ale given in the table above They aie smaller than the values calculated by Diclans himself, but aie at the same time considerably larger than the oldei values calculated hom the data that weie obtained at much lower picssuies by Soddy and Beny This latter disci opancy is not surpris- ing, howevei, in view of the impel fection of the older work; fuither- moie, it must not be foigottcn that oui fonnula (238o), while piobably the best available, is itself subject to some uncertainty It appeals on the whole that oui knowledge of the actual value of the accommodation coefficient is not yet veiy extensive The fact that a as calculated fiom the theoietical foimulas never exceeds unity and is nevei absuidly small indicates, however, that we must be on the right tiaclc m oui analysis of those phenomena, 182 Spectral Emission by an Unequally Heated Gas, The illumi- nating expenment reported by Oinstem and van Wylc* deserves mention because it furnishes a direct test of the assumption that wc have been malting m regard to the distribution of velocities among the molecules These investigators passed an electric discharge thiough a thm layer of helium at extiemely low prcssuie between two glass tubes, of which one was hcatecl electrically to 650°K while the othei was kept at 370°, ancl obseived the shape of a spoctial line emitted by the helium in a direction peipendicular to the tubes With this anangement, one half of the obseived line comes fiom molecules that last stiuclc the hot tube and the other half fiom those that last struck the cold one, which had been roughened with CuO It was found that the half line from the colclei molecules, when interpreted by the usual Dopplei theoiy (cf Sec 35), corresponded exactly to a max- welhan distribution of velocities at a temperatuie of 400°K, whereas the other half, corresponding roughly to a temperatuie of 480°K, was not quite maxwellian but exhibited a slight lelative deficit of low-speed molecules A depaituie from the maxwellian foim to the extent observed in this experiment would, however, mtioduce no serious eiroi into tire theoiy * Ornspuin and van Wyk, Zexts Phystk, 78, 734 (1932) Sec. 183] PROPERTIES OF GASES AT LOW DENSITIES 325 From their data they calculate an accommodation coefficient of (480 — 400)/(660 —400) = 0,32 for helium on glass. They recognize, however, that their observations really furnish information in regard to the energy of the molecules which are present at a given moment in a given volume, rather than of those which strike a surface in a given time. 183. Theoretical Calculations of the Accommodation Coefficient. Up to this point wc have treated the accommodation coefficient as a constant to bo determined experimentally and have dealt with its relation to the process of heat conduction. The value of the coefficient itself, however, must depend upon the properties of the molecules, and a number of attempts have been made to calculate its value on the basis of somo hypothesis concerning the mode of interaction of gas molecules with solid or liquid surfaces. In terms’ of classical theory Baulo showed* that, if all molecules concerned behave like clastic spheres and have random directions of motion, and if «ii denotes the mass of a gas molecule and tq, v[ its speeds, respectively, bofore and after striking a molecule of the surface, while wi 2 , i'a denote mass and speed of the latter molecule before col- lision, then for the average squares ml + w| , 2 ct 8 3 (mi H- M 2) 2 1 (»ti -1- m a ) 2 2 ' The mean loss of energy by the gas molecules is thus - mi in which ^ m 3 w| may also bo regarded as representing tho mean kinetic energy of a gas molecule at the temperature of the surface. Accord- ingly, comparison of this equation with the defining equation for the accommodation coefficient, eq. (235), shows us that, if only kinetic energy of translation had to bo considered, and if each gas molecule struck the surface only once before returning into the gas, the coeffi- cient would have tho value 2 m on 2 a “ (wj H- m a ) 2 ’ This expression has tho maximum value a = A when mi — »h and becomes small whenever tho molecular masses aro very unequal. Larger values of a than A could occur for hard spheres moving as * Baulk, Ann, Phyrik, 44, 146 (1014), 326 KINETIC THEORY OR OASES (Chap VIII classical mass points only if the gas molecule made seveial impacts with the suiface befoie escaping, as it might well do if the surface were veiy rough on the molecular scale, or if the gas molecules were heavier than those composing the suiface The collect theory must, however, be a wave-mechanical one; and an attempt to develop such lias been made foi helium in contact with tungsten by Jackson ancl Howaith * Aecoidmg to modem ideas, the atoms in metallic tungsten aie auanged m a tightly bound ciystal lattice having, foi a total numbei N of atoms, 3N degiees of ficedom To simplify then calculations, howevei, Jackson and Howaith zoplace this lattice by a continuous block of mateiial; and they assume then that an appioackmg helium atom has with the suiface mutual poten- tial eneigy V — C<r hv > The lattei law of foioe agrees well in foun with what we know of atomic fields at short ranges On the othci hand, it lepiesonts lepulsion at all values of the distance y of the center of the appioachmg atom fiom the suiface of the tungsten, wheieas theio is undoubtedly m icality at the laiger distances an attraction of van clei \Yaals natiue; in the particular case of helium, howevei, the effect of the lattei fences can bo shown to be pietty small The 3iV possible modes of vibration of the tungsten mass weic then tieated m the mannei introduced by Debyo for handling the specific heat of solids The impact of the helium atom excites these modes in vaiymg degrees, and the wave-mcchanical tieatment leads eventu- ally to a piobability foimula for the eneigy with which a helium atom is leflocted, as a function of its incident velocity; fiom this formula the accommodation coefficient for a maxwelhan stieam of atoms is then calculated In the Debye thcoiy the stiffness of the solid is repiesenfced by a eoitain eharactenstic tompciaturc; foi this the authors inseit the value 205°K as given by Lindemann's empincal 1 elation between this quantity and the melting point The constant C in the potential energy was eliminated by adjusting the theoretical cuivc to fit the data at one temper atiue. In this way tho authois secuicd a good fit with Roberts' data for three teniperatuies as given in the preceding table The constant in the potential-energy function, V — Ce~ bt/ } was assumed to have the value b = 4 X 10 s ; this is not very different from the correspond- ing constant m Slatei's formula for the mutual field of two helium atoms, which is, as stated in eq, (177a) in Sec 121, 2 43 5 29 X 10~° 4 6 X 10 8 * Jackson and IIowakth, Roy Soq, Pioc , 142, 447 (1933) Sec. 184] PROPERTIES OF OASES AT LOW DENSITIES 327 Such an agreement is encouraging, ginee we should expect these two constants to be at least of the same order of magnitude. An extension of the theory to include the effect of the attraction at larger distances, so that it should apply to neon as well, lias been given recently by Devonshire on the basis of work by Lennard- Jones.* THERMAL CREEP AND THE RADIOMETER 184. Thermal Creep. One of the most striking and peculiar phenomena at low pressure is the radiometric force that acts in an unequally heated gas upon any foreign body suspended in it. A fairly satisfactory explanation of this force has been achieved in recent years, but it is rather involved, and accordingly it is conducive to clarity to consider first the fundamental process that was pointed out by Maxwell as the probable cause of the phenomenon. In seeking an understanding of the radiometric action, Maxwell first investigated the stresses in an unequally heated stationary mass of gas and showed, for his special type of molecule repelling ns the inverse fifth power of the distance, that the stresses are unaffected in the first order of a temperature gradient. The same conclusion can bo drawn for any type of molecule from our own first approximation to the modified velocity distribution; we wrote for this/o where /o = f, = Cv x (% - (W)*~** as in eqs. (1416) and (1486) in Secs. 95 and 97 above, and if those functions aro substituted in the expressions for the transfer of normal and tangential momentum, +/ s ) and ^t> x v v ~ j'v x v u (fo +/«) (Ik [cf. (124) and (126) in See. 85], the contribution of f, to the integrals is found to be nil. At an unequally heated boundary of tho gas, however, Maxwell showed that a special offeefc was to be expected in the form of a steady creep of the gas over the surface from colder to hotter regions. Tho cause lies in tho fact that, when the gas is hottor over one part of tho wall than over an adjacent part, molecules impinging obliquely upon it strike with higher average velocity when they come from the hottor region than when they oomo from tho colder, and so are kicked back more strongly by the wall (except in the special case of specular reflection), with the result that tho gas acquires tangential momentum directed toward tho hotter side. Maxwell’s result for tho rate of tho resulting creep can easily bo obtained from our approximate dis- tribution function for a conducting gas. * DiwoNBiinus, Roy. Soc. Proo., 168, 200 (1937). 328 KINETIC THEORY OF GASES [Chap VIII 186, The Creep Velocity. Near a bounding wall let the tempera- tuie of the gas vary so that it mci eases unifonnly at the lato T* ~ dT/ds in a dncction inclined at an angle Q to the noimal to the wall (cf Fig 76) Let us take the a-axis outward along the noimal and place the ay-plane so that the giadient T f is paiallcl to it For an approxi- mate theoiy let us now make an assump- tion coiiespondmg to that made in ti eating tempcrnliuc jump, le, that all of the molecules which have a component of mo- tion toward the wall aie distubuted in velocity like the same gioup m the midst of the gas Then distribution function will then bo / = / 0 + f 8) with the values of /d and ft stated in the last paragraph, except that in the expiession foi /, we must now leplace v# by the component of v in the dneotion of T* oi v x cos 6 + v u sm 0 ♦ The total component ot momentum in the y-dnection of those molecules that stnke unit aiea of the suiface in a second will then be [cf (125) m Sec 85] ~nm f v x v y [A + C(v * cos 6 + v y sin &){% — dK } dK standing as usual for dv x dv y dv ef but the integral over v* extending only ovei negative values Heie the A and cos 0 tcims go out by symmetiy as v v ranges from — co to + ; but the sin 0 teim gives* Now under our assumptions 0 must have the value proper to a steady state in the conducting gas several fiee paths out fiom the wall, and accoidingly oui theory of conductivity gives for it the value obtained fiom cq (154a) m Sec 100 with the value of DXv x v 2 inserted fiom (154c) m Sec 101 In the lattei equation dT/dv repiesents the tcmperatuio giachent, which we are now denoting by T We obtain m this way G - '3 1 \5ir *nA 4 7t\/2 4 0 7 T For S vo it is best to employ the value given by (156b) at the end o( Sec 101, * Cf table of integrals at end of the book Sec 185] PROPERTIES OP GASES AT LOW DENSITIES 329 _1_ = 32V2 V S vo 5rr mv foi the viscosity y is in most gases moie simply lolalcd to the tians- latoiy molecular motion than is the thermal conductivity If wo then also nisei l A ~ /3 8 /V 4 , 0111 oxpiession for the stieam density of y-momentum loduccs to 3 y r . „ Air /3 2 i> T Snl °‘ Now suppose that m lobounding fiom tlio wall these molecules lose to it at least pait of then avciage tangential momontum Then tho layei of gas noxl to the wall will expouence a steady diam of momentum, and this is not countci balanced by a not inflow from the remainder of tho gas ; ioi we have soon above that tho flow of tangential momentum acioss any plane diawn in tho midst of tho gas vanishes This layei will accordingly bo sot into tangential motion, and it will then set adjacent laycis into motion by viscous drag upon them Suppose tho Anal lcsult is that the whole mass of gas moves at uniform velocity u towaid y = + Then tins motion will add to tho flow of tangential momentum up to the wall a siipoi posed stioam of density i p$u units of momontum per unit aioa per second [of (726) in See 37] To keep tho gas moving steadily the two stioams of y-momentum must just cancel each othor. This will bo tho caao piovidod u 3 n T' . A - -ssr-o ~m Sill 0 IT ppW 2 3 »| RdT ip dy’ (243) the socond form following by means of (65a) or (IS = 2/ a / tt , p ~ pliT and tho obscivation that T' sin 0 = dT/dy. Hero p = presume in tonns of dynes, It — gas constant for a gtam of the gas, y — viscosity, and dT/dy ropiesonts tlio giadiont of tho tempoiatuio along tho bounding wall. In most cases, howovor, tho sun oundings will prevent tho whole mass of gas from moving uniformly. Under such circumstances velocity giadicnls will bo sot up m tho gas, and those may then he accompanied by viscous slip over tlio surfaco, such as was discussed in Secs 165 to 169 above To tho first order, howcvoi, tho velocity of slip will simply bo superposed at the boundaiy upon tho velocity of thermal creep u as given by (243) It is to viscous stiessos indirectly called into play in this manner that Maxwell asenbed the indiomotiie forco. 1 330 KINETIC THEORY OF OASES (Chap VIII It should not be forgotten that formula (243) is subject to con- siderable uncertainty because of the aibitrary assumption upon which it rests in regard to the foim of the departure fiom Maxwell’s law near the boundary To give some idea of the rapidity of the creep, we may insort m (243), for air at 16°0, ij - 179 5 X 1CT S and B - 2 871 X 10 s , then, at atmospheric pressure or p = 1 013 X 10 s , we find fi r P u - 3 8 X 10 -4 -r- cm/sec d V This is small, but if we lower the piessuie to 1 mm Hg and make the temperature gradient over the surface dT/dy — 10° per centimeter, we find w = 2 9 cm/sec. 186. Thermal Pressure Gradients and Transpiration. Thermal creep will obviously have an effect upon the motion of gases in tubes, and in some cases this may be of practical importance. If the tube is unequally heated, theie may be a lesulting flow of gas along it, even in the absence of a pressuie giadient, oi, if the circumstances are such as to prevent such a flow, then a steady pressure giadient may be set up with the gas at rest. Measuiements of gaseous pressure by means of exteinal gauges at a diffeient temperature may requiie correction for pressuie differences aiising in this way in tho connecting tubes. Such effects constitute in part what has been called thermal transpiration, which has already been discussed to some extent. As usual, the two limiting cases of high and low density can bo handled easily but require diffeient methods, whereas the intei mediate case presents great difficulty. Let us consider first the free-molecule case, assuming the pressuie to be so extremely low that the mean free path is many times the diameter of the tube The appropriate foimula can then be found immediately by modifying the calculation given in Sec 171 above; we have only to allow V to vary with the temperature along the tube, replacing the relation vn' = -f- x as there wiitten by „ d(nfl) tl V 5=5 UV *T X ■ ^ We then obtain for the mass passing per second, in place of the expres- sion previously found, Qm “ “ivS [/ dS £* 8 H as (v?) Sue. 187) PROPERTIES OF OASES AT LOW DENSITIES 331 or, in a round tube of radius a, where f ds r s d(p = 167i-a 3 /3, Qm ■— 4 /2?r a d p 3 V R a dx y/T As a check, we note that, when T is constant, these equations yield (233a) and (2336) again upon multiplying through by RT. The condition for zero flow is then that p/s/T be constant along the tube. We have thus a typical case of the phenomenon called thermal transpiration, which has already been sufficiently discussed (Sec. 39). 187. Thermal Gradients at Moderate Pressures. Let us next allow the density of the gas to be high enough so that the mean free path is small as compared with the diameter of the tube, and that this diameter in turn is small as compared with its length. Then it is easy to extend Poiseuille's formula so as to allow for the presence of a temperature gradient along the tube. In our deduction of that formula as given in Sec. 166 wo have only to add a thermal term in the boundary condition for the velocity at the wall of tho tube, writing in place of v tho more general expression .dr , ^ dr 4 p dx in which tho second term represents the velocity of thermal creep as given by (243). Tho velocity is then found to be v = - <■* + I ,3 tjRdT, ^ 4 p dx' and integrating as before, wo find for tho mass of gas transferred per second past any point of a long round tube of radius a, whore tho pressure p and tomperaturo T have gradients dp/dx and dl'/dx, respectively, ■k a 4 p 8 t)R T dp . 3ff pa? dT dx ' 4 T dx * (244) R being the gas constant for a gram of the gas, n its viscosity, and f its slip distance at tho wall of the tube. 332 KINETIC THEORY OF OASES [Chap VIII Foi the case of steady flow the distnbution of pie&suie and of torn- peiatuie along the tube can then be found by setting Q m equal to a constant, and the late of flow itself can then be calculated Only the static case of no flow will be considered luilhoi heic, however In this case the tempeiatiue giadient must be accompanied by a piessuie gradient whose value, found by setting =* 0, is dp dx dT dx (245 a) We have here a situation m which the gas is oieeping steadily along the walls of the tube and is at the same time flowing back Unough the v - cential pait, as is suggested in Fig 77, the ' y pie&suie giadient is necessary m order Lo , maintain the return flow .. > To integrate the last equation wo need to dT dp know the mode of vanation of ?? and f with d7 ' Tx ^ x Now f is nearly equal to the metal flee Fl ° 7 U:S r ° gra “ P ath ancl 80 vaues louftWy inversely as the density, or inversely as the prcflsuio and dnectly as the temperatuie; y also vanes with lempeiatuio, blit m a less simple mannei An accuiato lesult can bo obtained m any actual case by means of numerical integration A lough estimate can, Jiowevei, be obtained simply by lcplacing rj 2 by its aveiage value y 2 ovei the actual range of tcmpeiaturo, and also sotting Vi — fi> the value of f fox p = 1 dyne/cm 2 , we may then givo to f L its value foi the aveiage tempeiatiue, which will bo denoted by If we then also leplace p by £ (p x + P2), eq (245a) can bo integrated at onco, the lesult can be wiitten thus ”■ - *• " W, (Tl ro, Pa = o (Pi + V*) + 4 fir (2465) Vh Ti and p 2y T 2 standing foi values at the two ends of the tube. This foimula can be used 111 estimating the order of magnitude of possible piessure diffeionces due to tempoiature giadients in vacuum appaiatus, such as might, foi instance, falsify measurements of the piessuie As a numerical example, forah atl5°C,7j » 179,6 X lO 4 ^, I? = 2 871 X 10 G , both in c g s units, hence if a = 0 5 mm and if wo use the value of y 2 at 15° foi wo have = 222, or 1,25 X 10 4 if the piessure is expressed in millimeters Hg If then, in particular, we make the tempeiatuie difference Ti — Tt — 100°C, Sac. 188] PROPERTIES OF OASES AT LOW DENSITIES 333 omit 4 and choose for the mean pressure p — ^ (pi + p 2 ) — Pt — Q* 0.5 mm, which brings the mean free path to about a/6, (2456) gives for the pressure difference due to creep pi — pz — 0.026 mm, or 5 per- cent of the moan pressure. The relative pressure drop due to creep varies rapidly with the tube diameter and the pressure p q , being inversely proportional to the square of both these quantities; it may thus be quite negligible under some circumstances, and very appreciable under others. A few data for a critical test of these results are available in a paper of Knudscn’s.* Without going into detail, it may be said that his observations seem to agree roughly with theoretical predictions in so far as those should apply, 188. The Radiometer and Photophoresis, It was discovered by Fresnel in 1825 that a small body suspended in a gas is. sometimes sot into motion when light falls upon it. The effect was exhaustively studied by Crookes (1874^-1878) and hosforined the subjeetof numerous recent investigations. Often the body in question takes the form of a wheel carrying vanes blackened on one side, which revolves continu- ously when illuminated; or, to measure the force, a light vane may ho mounted on one end of a crossbar, with a counterpoise or another vane at the other end, and the bar may then lie suspended from a torsion fiber so that its dofloction can bo read with mirror and telescope. An analogous motion of microscopic particles suspended in a gas was observed in 1914 by Ehrenhaft and was called by him "photophoresis.” Some particles move toward the light, others against it. The laws governing all of these phenomena appear to be sub- stantially the same. The force is found to be staidly proportional to the intensity of illumination, so that mechanical devices of the sort described can be used to measure a beam of radiation; for this reason they have come to be called radiometers. With increasing pressure, the force rises to a maximum at a pressure of about 1 mm Hg in the case of a disk of ordinary sine, or at several hundred millimeters of Hg in the caso of photophoresis, and then decreases rapidly. , It has been pretty well established that in all cases the light acts by heating the suspended body; the samo effects can, in fact, be produced by establishing in the gas such a temperature gradient as will give rise to the same temperature differences at the surface of the suspended body. The general rule is that hot surfaces behave as if repelled by the gas. The movements toward tho light which are often observed in photophoresis are ascribed to greater heating of tho far side of a * Knvdbbn, Ann. Phytik, 81, 205 (1010). 334 KINETIC THEORY OF GASES [Chap VIII transparent particle, this latter effect has been successfully imitated with a radiometei carrying disks of molybdenite, one of whose surfaces was fresher than the othei Vanous opinions have been expiessed from timo to time as to the origin of the radiometric foice A tempting hypothesis at first sight is that it is due to the reaction fiom gaseous molecules rebounding with higher velocities fiom a hot suiface than fiom a cold one; but this is quickly seen to be untenable when we leflect that such molecules, upon reentering the gas, must drive it back and theieby thin it out until umfoimity of pressuie is lccstabhshed, whcicupon the force on the hot suiface will become the same as on the cold one and no radio- metric action can occui The cause must, thmefoic, be sought in some secondary action The effect has very commonly been regarded as occurring at the edge of the radiometer disk, whcio conditions in the gas must be far fiom uniform, expenments designed to show that it is simply proportional to the length of the penmeter failed, however, to yield this result Recent theoretical and experimental studies liavo now made it pretty cleai that most, if not all, rachometiic phenomena are due, in one way or anothci, as Maxwell suggested in 1879, to tho thermal creep of the gas ovei an unequally heated solid (or liquid) surface, as descubcd m Sec 184 above It can be seen easily that this creep must give nso to forces on the surface whenever the resulting flow of gas is hindcied m any way In a simple two-vane ladiometer, for example, tho gas will creep around the edges toward the centers of the blackened and thexofore heated surfaces, and must then flow out and aiound somewhat as suggested in Fig 78(a), in which the vanes VV aie supposed to be transparent but blackened on one side, this eirculatoiy motion is then hindered by viscosity, and consequently the gas accumulates somewhat over the blackened surfaces and exerts a slightly increased pressure on 335 Sec, 189] PROPERTIES OF GASES AT LOW DENSITIES these and so pushes them back, thus tending to produce revolution about the suspension S. If, on the other hand, the vanes are alike on both sides but are given a cup-shaped form, as in Fig. 78 b, or are fitted with points, the edges of the cups or the points are observed to move toward the light. Presumably these parts are more effectively cooled by the gas than are other parts, so that a circulation is set up as suggested in the figure, and those eddies which reach out to the sur- roundings are effective by reaction in moving the vanes. Most prac- tical cases can readily be understood in this way, The existence of such streams in the gas as we have here postulated was shown directly by Gcrlach and Schiitz. * They suspended a tiny vanelct near the radi- ometer and observed that it became deflected, presumably by the action of the streaming gas, in the right direction. Another interest- ing experiment pointing to the same conclu- sion is that of Czerny and IIettnor,t who mounted a movable disk parallel to another disk along which a temperature gradient was maintained (Fig. 79). They observed that a tangential force acted on the movable disk in J ia ' 7 tfiomaUroop. U ° such a direction that it could be explained as~arising from viscous drag by the gas as it creeps along the unequally heated disk. 189. The Quantitative Theory of Radiometer Action. Qualita- tively the creep theory of radiometric action is completely successful. A quantitative calculation of the force, however, presents, unfortu- nately, a difficult problem. One has first to solve the thermal problem in order to find the distribution of temperature, which is determined, under the given conditions of illumination or of boundary temperature, by the conduction of heat through the gas and through the disk itself ; allowance must also bo mado, if accuracy is desired, for convection of heat by tho creep motion itself. Then one lias to solve the hydro- dynamical problem of tho streaming as determined by the velocity of creep as a boundary condition; and, finally, from this tho total pressure on the disk is found by integration. Tho complete problem lias been solved only for the ideal case of an ellipsoidal disk, circular in principal outline but of elliptical cross section, which, if thin, should present some approximation to n flat * GnRiM.cn and Sente, A cits, Physik, 78, 43, 418 (1932); 79, 700 (1982); 8t, 418 (1933). f Czerny and Hhttnuu, Hails. Physik, 80, 268 (1024), 336 KINETIC THEORY OF GASES [Ciiap VIII disk The best tieatment is that of Epstein * His result foi the foice F on the disk in teiras of the temperatuie difteicnce A T between the centeis of its sides can be wntten, foi a thin disk, F - -3,r^A T, R being the gas constant foi a giam of the gas, q its viscosity, and p its piessuie AT is heie assumed to anse eithei fiom a temperature giadient, which at gieat distances is paiallel to the axis of the disk and of umfoim magnitude dT/d% } 01 fiom a unifoim beam of liglit of intensity J ergs/cm 2 /scc, the values of AT foi these two cases aro ( 1 ) a dT 1 ■ 2fca d% } 7T ha 5 (2) AT = 2 aJ v K + -h,~ 7T 0 Heie a is the ladius of the disk, S its thickness, and h,i the conductivity of its matenal, while A>„ is the conductivity of the sun ou uding gas If the disk is nonconducting (k,i = 0), the foice under llliumnation is and is thus piopoitional to tlie ladius, or also to the peiimoter, of llio disk This latter fact is suggestive of an edge effect, but tho analysis shows that the foice is in leahty distnbuted ovci tho sin face At the opposite extieme of a veiy highly conducting oi oxliemoly thin disk (k,ia/k 0 5 laige),the foice is F = ~3ir RrfSJ pki and so independent of the radius For a quantitative test of Epstein’s formula, Gorlach and SchtUz (be ctl) constiueted an almost nonconductive disk by mounting platinum toil on both sides of a mica disk 9 mm acioss, the foil on tho side towaul the light being smoked Their obsoived foiccs exceeded those which they calculated fiom Epstein’s foimula. Their cal- culations weio made, howevoi, fiom a formula into which Epstein had conveited his lesult by mseitmg n = 0 35 P vL,f but in this they insoi ted * Epstein, Zeits Physik, 64, 637 ( 1029 ) n ll r b<i K er aS * r , ul ° nofc t0 ln trocliioo L into tho statement of any beei e r p loyed Ca " b ° aV0lclecl ’ bocau8 ° 80 mfwl y diffemnt formulas foi it havo Sbxl [Zeits Physik, 62, 249 (1928)1 obtains a final oxpiessionfor the foice tliat 8m. 160 ] PROPERTIES OP OASES AT LOW DENSITIES 337 values of L, presumably taken from a table, which were obviously calculated from the modem formula, i) = OAQdpvL. If the latter is employed in converting Epstein's expression, the theoretical forces in Gerlach and Sehtita’s experiment come out nearly twice as large; the experimental force is then only about half as large as the theoretical in air, whereas in hydrogen it exceeds the theoretical value only by 20 per cent at the higher pressures and drops below it at the lower. The pressure ranged from 0.1 to 1 mm in both cases. In view of the many uncertainties this degree of agreement between theory and experiment must bo regarded as vory encouraging, particu- larly because, as remarked by Gcrlaeh and Schiitz, an overestimate by theory is to be expected, since in the theoretical treatment cooling by creep convection is neglected, and also L/S was assumed to be small; the latter was not true in the actual case, differs from ISpstoin's, whon n ~ 0, only bccauso ho writes his result is really limited to the ease n ** 0 because of an assumption that is made in regard to the value of K or h g . CHAPTER IX STATISTICAL MECHANICS In this chapter we shall finally take up foi discussion the basis of those statistical punciples which we have alieady employed seveial times without pi oof This foims the subject mattci of what is called statistical mechanics It is a branch of theory which is very abstiact and also, because of its difficulty, incompletely woilccd out Wc liavo space heie only to survey buefiy the mam line of thought, confining our discussion foi the most pait to those topics which boar on the theory of gases 190, Nature of Statistical Mechanics. In statistical woilc wo are concerned, not with complete knowledge of the state of a dynamical system, but only with ceitam bioad features of its behavior which happen to be physically significant Veiy often these features aio of the nature of an aveiage of some soit; for example, wo have alieady noted that quantities such as piessuie oi density repicsont avciagcs, taken over macroscopically small spaces and times, of molecular quantities that vaiy lapiclly Often it is convenient to deal with probabilities, but these can likewise be identified at will with certain averages, for example, the probability that a molecule is in a certain element of volume dr is the same as the fi action of the timo during which it is there when the total intei val undei consideialion is made indefinitely long We may even have to aveiage ovor vauous possiblo molecular motions which aie macioscopically indistinguishable For convenience we shall lefer to all such aveiages and probabilities as ultimate statistical features of the system Statistical mechanics can accordingly be said to deal with the ultimate statistical featuies of the behavior of dynamical systems when subjected to specified conditions Now, since the molcculai motions are regulated by mechanical laws, we should nakually oxpcct to be able to denve these featuies from those laws alone, without being compelled to introduce further postulates representing addi- tional independent laws of nature To be sure, piobabihty can novel be deduced from ceitamty; probability must somewhere be intiociuccd if we wish to get it out again, But it will be obseived that, in all of the conclusions concerning probability which aie diawn from 338 Sec. 191) STATISTICAL MECHANICS 339 statistical mechanics, the element of probability is introduced or postulated in describing the situation to which the conclusion refers, and what we deduce from mechanical laws is only the relation between two probabilities. A simple example would be the calculation from mechanical laws of the chance that a clock pendulum, viewed at a random moment, is close to the end of its swing; here we introduce the idea of probability in posing the question, when we specify that the pendulum is to be viewed at a random moment. We shall begin with the classical treatment as worked out by Maxwell, Boltzmann, Gibbs, and others, and shall then take up the modifications that are required by wave mechanics. The classical theory, besides its theoretical interest as a limiting case, serves as a guide for the wave-mechanical treatment; and its results seem also to be correct within the range of experimental error for the translatory motion of the molecules in ordinary gases. A. CLASSICAL STATISTICAL MECHANICS 191. System Phase Space. In the theory of mechanics it is shown that the motion of a dynamical system having s degrees of freedom can conveniently be described in terms of s generalized coordinates qi, • • • q.„ together with s corresponding generalized momenta, pi, p 2 > • • p»; the latter become tho ordinary components of momentum, mx, my, mz when cartesian coordinates x, y, z aro taken as the q’a. Each phase of any motion of tho system is then represented by a sot of values of the s q'n and tho s p's; and during a Eio. 80 — Systom-apnoo trn- particular motion these values vary as hofcory for <i lulling body, definite functions of the time. For example, in the caso of a falling body wo may take as q tho distance the body has fallen from rest; then the momontum is p — mq = gi, and q ~ k gt 2 , To many minds it is helpful to think of those variables as tho coordinates of a point in a space of 2s dimensions, in which the q’ s and p’s play tho role of cartesian coordinates. Such a space is com- monly called a phase space; wo shall call it, more specifically, for a reason that will appear lator, tho system phase space or simply system space. As tho system executes a particular motion its representative point traces out a certain trajectory in system space. Geometrical language such as this will bo used freely in what follows. Any reader who finds it distasteful can easily roplaco our geometrical statements by equivalent analytical ones; for instance, a "trajectory in system 340 KINETIC THEORY OF OASES [Chap IX space” can be mtei pretecl as meaning meiely a succession of sets of values of the and p's, In the example of the falling body the system space is obviously a plane (so long as we leave all lateial motion out of account), and the trajectory of the representative point on this piano is tho parabola q =* p 2 /2g } obtained by eliminating t fiom the equations p = gt (of Pig 80) 192 Representative Ensembles. Instead of contemplating directly the succession of phases thiough which a single system passes in the couise of time, Boltzmann showed that it is more convenient to contemplate an equivalent distribution of phases belonging to cliffoicnt systems, all considered at the same moment Foi this pm pose we contemplate a huge collection oi ensemble of leplicas of the oiiginal system, each of which is executing tho same motion but is at any given instant m a diffeient phase of it If these systems aie pi open ly spaced they will give us a complete picture of the bchavioi throughout time of the given system In the ensemble we can also include, if wo wish, other sets of systems lepicscnting in tho same way all diffeient motions that the system can execute unclei the given conditions, The statisti- cal featuies of the behavior of a single system can then all bo obtained by suiveying the ensemble at a given moment Prom this point of view the fundamental pioblem of statistical mechanics is the determination of the correct ensemble to lopreaonl a dynamical system under given conditions The most important case is that of statistical equilibrium This ease is characterized by the fact that the long-time behavior of the system is definite and independent of time The ensemble that lepiosonls it must accord- ing y be a steady one, it must be lepiesentecl in system spaco by a steady density of the moving points, then number m any clement of the space dO = dq i dq* dq s dpi dp 2 * • dp a being p dG f where p does not change with time, although the identity of tho points that are momentanly m the element may change The simplest type of system for which such an ensemble can i oachly be set up m full detail is, peihaps, the harmonic oscillator, tho study of which led Planck to the quantum theory. The system space for a simp e oscillator is, like that of the falling body, a plane, on which its coordinate q and momentum p figure as vanablos For tho kinetic energy of the oscillator we can write | ap\ and foi its potential energy, i bq , a and 6 being constants, hence during an undisturbed oscillation we have, by the conservation of energy, £ ap 2 + | = E = const, Sec* 193] STATISTICAL MECHANICS 341 This equation defines for an undisturbed oscillator the. trajectory of the representative point on the system plane; the trajectory is obviously an ellipse (of. Fig. 81). If we now sow moving points along this ellipse, they will represent oscillators all moving with the same energy E but in different phases of the motion* We can space these points in such a way that they give us at a glance a quantitative picture of the statistical features of the motion of any one oscillator; to do this, we need only allow fresh points to start out one after the other at equal short intervals of time from- some chosen location on the ellipse, continuing this process until the first point has gone round once. The points will then form a steady onsomble, for the number of points on any segment of the ellipse will be constant although their identity changes; and it can be shown (cf. Liouville’s theorem, See. 194) that the fraction of them on any segment is the same as the fraction of the time spent by any one point on that segment* It must be noted, however, that in ono important respect this example is misleading. The trajectory correspond- ing to a definite energy is here a closed curve, but that is not generally true. Fia. 8i,— Trnjootorios for u siinpla Unfortunately, open trajectories confined to a finite region can occur only in systems of two or more dimen- sions, and for these system space is at least four-dimensional and so is difficult to picture, 193, The Ergodic Surmise, The most important case of equilib- rium is that of a systom that is either isolated or restrained by a fixed force-field, such as the field of the vessel holding a gas; and only this case will be considered in detail* The point representing a system under these conditions cannot wander widely in system space but is confined to a surface defined by the condition that the energy has a certain constant value; this is called an ergodic surface. In the example of the oscillator described above the ergodic surfaces are obviously the elliptical trajectories there described* In setting up an enscmblo to represent the system, wo nfe then confronted at once by the fundamental question whether all systems whose representative points lie on the same ergodic surface h^vo the same ultimate history, and must, therefore, be included in the ensemble, 342 KINETIC THEORY OF OASES [Chai* IX Physical intuition would answoi this question in the affirmative, The ultimate behavior of an isolated mass of homogeneous gas, for instance, is not obseived to depend upon its initial condition. Much effoit has been expended m the eiioit to establish such a conclusion rigoiously, but so fai, unfox tunately, without success, at least in teims that are of any use in physics At tins point, thoioforo, it is necessary to intiocluce some soit of hypothesis, or lather a sui misc } since we are piesumably dealing with a consoquonco that should follow ligoiously fioin the laws of mechanics and theio is, tluuefore, no room for any new hypothesis expressing an independent law of natiuc The situation is complicated by the fact that special cases can bo cited in which the question must be answoicd in the negativo. As an example, a mass of gas inside a peifoctly smooth sphoiical vessel, if undistuibed, would letain its initial angular momentum foiovcr Replicas of the gas, lepiescnted by points on the same orgodic surface but having difteicnt angular momenta, will, thciofoio, cxocuto radically diffeient motions, the path of the rcpiescntativo point for each one will be confined to a small part of the crgochc surface Again, in a rectangulai box, haubsphciical molecules composing a larefiod gas can be so staited that each moves foievoi back and forth at constant speed along a fixed line, in such a caso theio would bo a permanent distribution of velocities which might bo anything but maxwollian, Such cases, however, will be dismissed at once by Lho physicist as very exceptional The slightest scratch on tho suiface of the sphoiieal vessel, 01 the slightest deflection of one of tho moloculcs in tho box will undoubtedly result in bunging about tho normal distribution Accoidingly, we shall assume that in general tho affix mativo nnswor is the collect one We have not space to discuss heio tho famous “ergochc" and “quasieigodic" “hypotheses" or surmises which luivo been suggested in suppoit of this conclusion, but shall simply adopt as true the following statement, which lies close to physical intuition: Ergodio Suimise , The ultimate statistical features of the behavior of a dynamical system m statistical eqmhbi mm m a fixed force-field are m geneial independent of its initial condition , cxcopt in so far as this condition consists of the geneial lostnctions that dotermino tho equilibrium Whether a given case is 01 is not a “geneial ono" romains at present, unfoi tunately, a question that has to bo clodded in tho light of physical intuition and ultimately by experiment As a special case, of course, the foice-field may be oveiy where aero, tho system being then isolated, As applied to a gas, foi example, this means that whafcovei initial state of motion or inequality of temperatuie, etc,, we may give to tho Sec. 194] STATISTICAL MECHANICS 343 gas in the beginning, lias no influence in the end on its statistical behavior* Of course, there will be certain features of the initial state that do have such an influence; for instance, if the system is isolated, its energy is fixed by the initial state, and this determines its subse- quent history* There is, however, an immense variety of different initial states corresponding to any given value of the energy, all followed, in the long run, by the same ultimate macroscopic behavior. This principle being adopted, we are guided to a determinate choice of the proper ensemble to represent a system by the following famous theorem : 194. Liouville’s Theorem. If any portion of system phase space is sown thickly and evenly with moving points representing a dynam- ical system in different possible states of motion , then the laws of motion are such that the density of these points in system space remains constant This theorem is most easily proved from the equations of motion in the Hamiltonian form, which are: Qi (i = 1, 2 • • • s); (ff) here — dqt/dt, etc,, and II is the Hamiltonian function of tho ff's and p } & } which may also contain the time explicitly* For tho sake of vividness we shall call II the energy, but it must be understood that the theory developed in this chapter is applicable to any caso in which there is a Hamiltonian function, whether the latter represents tho energy or not* The proof of Liouvillc's theorem follows so closely the deduction of tho Boltzmann distribution equation, (87) in Sec. 51, that we shall leave the reader to construct most of the details, p denoting the density of tho moving points, there will be p Aflfi A(?a • * • Aq a Api Ap% • ♦ * Ap a of thorn representing systems in which lies within a range Aq if etc* As time goes on, points will cross each of the two faces of the element Aqi ♦ • ♦ Apt which are perpendicular to the praxis at a rate pq i Aq 2 ♦ • • Ap i} in which ptfi has tho value proper to that face; the net outflow across these two faces will thus be Aqi Aqz • • * A p,* 344 KINETIC THEORY OF GASES [Chap, IX Adding up expiessions of this soit for all pans of faces, equating the sum to the net loss of points fiom the element, and passing to the limit as all A’s 0, we obtain as a gcnoial diftoiential equation for p dp , 'Sjf «(/>?.) | _ n j t + + % « 1 Now suppose that at a given moment p is unifoim throughout a ceitatn legion Then the last equation gives in this legion di p ^Ld?, + c>pJ u > each teim m the sum vanishing by the Hamiltonian equations of motion Thus p lemams unifoim and constant, as tho theorem asseits * In consequence of this theoiem, a laigo (stiictly speaking, an infinite) group of systems, stiewn unifoimly over any legion m system space, lemams unifoimly distiibutecl over a legion of tho same size, although the location and shape of this legion may change continually The legion may, of course, include all of system space As an example, we may tako tho falling paiticlo mentioned m Sec. 101 above, whose eigodic "siufaoos” aio parabolic cuives on tho system plane to which system space reduces in such a one-dimensional case An aioa such as that dotted at A in Fig 82, strewn Fra' 82— A uniform eprond umfoimly with movmg points, represents parfcicio^ m pointa for a fallmg many leplicas of tho falling body dropped J " from various initial heights and at various tunes The area occupied by these points changes its shape as time goes on, as at B } but the reader should bo able to vonfy without difficulty that it does not change m size, 195, The Ergodic Layer and the Microcanonical Ensemble, Lot us now leturn to the considei ation of a system which is cither isolated or subjected to a fixed foice-fielcl, and attack the problem of finding the propei ensemble to lepresent its statistical behavioi, * 11 p *- const and unifoim” is a solution of tho chfToiontial aquation, satisfying the stated initial condition, and them is only one such solution Sue, 105J STATISTICAL MECHANICS 345 Under the conditions specified, II does not contain the time explicitly, and it then follows from the Hamiltonian equations that II remains constant. The representative point in system space moves, therefore, on a fixed ergodic surface defined by the equation, II (.p, q) — const, Suppose, now, we draw two such surfaces corresponding to values II — E and II — E + 8E, enclosing between them a thin slice of system space which we shall call an ergodic layer; let us fill this layer uniformly with moving points representing the systems of an ensemble (Fig. 83). Then these points will remain permanently in the layer; and by Liouvillo’s theorem they will remain distributed uniformly over it. The statistical properties of such a microcanonical ensemble, as it was called by Gibbs, are obviously the same as those of uniformly distributed fixed points in the layer, and so can be studied by geometrical methods. Furthermore, for tho same reason the statisti- cal properties of such an ensemble do not vary with the time; and because of this steadiness we can show, on the basis of principles already laid down, that they are also sensibly the same as the statistical properties of any one system in tho I’m. ss, — Dlngmin layer, in so far as these are not appreciably an orgodio affected when the energy of the system is varied a little, We can, in fact, infer what we shall call the Principle of Statistical Equivalence, 'The ultimate statistical proper- ties of a system in statistical equilibrium , either in a fixed force-field or isolated, are the same as those of a microcanonical ensemble containing the system, and are also the same as those of the ergodic layer of fixed points containing the representative points of this ensemble, In par- ticular, the fraction/ of tho time during which n system possesses some property P is equal to the fraction <p of the systems in the onsemblo which possess that property at a given moment, and it is also equal to the fraction of the volume of the ergodic layer whose fixed points represent systems having tho property P\ and similarly tho average of any quantity Q for the system is equal to the instantaneous average of Q over the systems of the ensemble, or to its average as a function of position in tho ergodic layor. By volume is meant, of course, Jdqi • • • dq,dp\ - * • dp,, and the average of Q over the ergodic layer is Q = jQ(q i • > • p,) dq, , • • ■ dgu/j'dqi ■ • • dp,, all inte- grals extending over the layor, The principle is limited to fractions for 346 KINETIC THEORY OF GASES (Chap IX to aveiages Q, which do not vary sensibly when the onorgy of the system is changed by a veiy small amount SE To piove this pimciple, suppose, fiist, that each system possesses some propel ty P duimg a fiaction/ of the time Then / will be tho same for all systems, in consequence of the cigodic bin mine laid down m Sec 193 above, and the numbci of “system seconds” during which P is possessed by all v systems in the ensemble duiing a long time t will be vjt But this number can also be wntten (pvt, where <p is the steady fraction of the systems of the ensemble that possess P at any given moment It follows that f - <p, In the second place, tho aveiago value of any moleeulai vanable Q dunng time t for any one system is (Q) t = -J Q di, and this by the ergodic suimise is tho same foi all systems, hence, summing ovei the ensemble, K©* - dt But 2Q = KQ)* wheio ($)„ denotes the steady average of Q over the ensemble Hence, v{Q) t — v{Q)vj§dt - and (@)* — ($),» As a homely example of the same argument, if at a locopUon all guests spend the same length of time in the refreshment room, and if at every instant half of them axe thcic, then it must be that each individual guest spends just half his time in that loom This pimciple of statistical equivalence furnishes the basis foi the entire tieatment of systems m equilibrium Its great advantago lies m the fact that the determination of statistical quantities is thoroby reduced to mtegiations ovei system space We shall now consider the application of the pimciple to some cases that have an impoitant bearing on the theoiy of gases 196, The Point-mass Perfect Gas As a first example, consider a set of N similai monatomic molecules, without internal motion, constituting a peifect gas m a ngid vessel For their Hamiltonian or energy function we can wnte 1 * H - + p» + py + u »> of which the first teim lepresonts the kinetic cnoigy expi ossod in terms of the caitesian components of momentum, p x> p ut p„ and U<> is a potential-energy tcim expies&ing tho lestvaimng offool of tho vessel and so has the value zeio except when a molecule comes Sec* 196 ] STATISTICAL MECHANICS 347 exceedingly close to the wall. Of course, there must in reality be still a third term in II representing interaction between the molecules, in order to guarantee statistical equilibrium, but we shall suppose this term to be so small that its effect is otherwise negligible. For such a system the volumes in system space that we need to evaluate in order to apply the principle of equivalence can easily be found directly. To find the entire volume of an ergodic layer, it is con- venient first to evaluate the volume of all of system space up to a limiting energy E } which is represented by the integral v{E) « f dp* i dp v i dp g i dp x 2 ♦ ■ ■ dp tN dx x dy x dz i * • • dz N) ii<e integrated over all values of the momenta and of the coordinates of the JV particles such that the Hamiltonian function II < E , The indicated integration over the coordinates themselves can be carried out at once; for the integration pertaining to any one particle gives simply the volume V *=» m dx dy dz of the vessel. Iionce N o ■ — V N Jdp xl • • • dp,n, 2m 2) (P*; + P& + V%) < ; = i (the inequality specifying the range of integration). Let us introduce here 3iV new variables p\, p t , • • « pm, each equal to one component of momentum divided by \/2mE. Then the integral becomes on r w J dpxi ’ ’ • dp t N = (2 mE) 2 J dpi • • • dpm, < 1. T=1 The new integral occurring here is independent of E] let it bo denoted byCV.* Then m <t » CMV«(2mE) 2 . Since in any practical case iV is a huge number, we see from this result that <r varies with J? at a prodigious rate. On the other hand, writing </ for dcr/dE, we have V = 3JV V 2 E‘ * It is the volume of ft unit sphere in 81V-dimonsioiial spaoo or [of. (292)| 348 KINETIC THEORY OF GASES [OlIAl 1 IX which, besides being of oidinaiy magnitude, is compaiatively steady in value It follows that E can be vancd by an amount sufficient to change a by an enormous ratio without altering <// cr appreciably- For such a variation in the neighboihoocl of some particular value Ei we can write, theiefoie, as a close appioximation, a '/<r “ 3N/2JCu and, integrating, 3 NK <t(E) « De 2hl , D being a constant, We have then, finally, fox tho volume of an cigodic layei of indefinitely small thickness 6E } replacing E i by K after diffeientiatmg, Sa = &*($) 5E “ 3W 2 E a(E) 5E . 197 The Molecular Distribution, Molecular Chaos. Now, an a statistical pioblcm to be solved for the gas under consideration t suppose we ask for the probability that, when its total cneigy ih a certain molecule, which we may call 1, lies at a given moment in small element dK «= dp* dp v dp z dx dy d% of tho momentum-cooidimitc phase space foi one molecule, its eneigy being ^ mv 2 in teims of Hh mass m and velocity v By the pnnciple of statistical equivalence, this probability will be equal to the fiaclional part of the volume of the ergodio layer within which this paiticulai molecule lies within dK To find Uiih fraction, we note first that the volume up to E of that pait of syntom space in which molecule 1 lies in dK is got by changing the integral for cr in such a way as to limit the variables of this molecule to dK\ the eneigy of the remaining N — 1 molecules, which we shall donoU> as a vaiiable quantity by II n must then stay below E ~ \ mv 2 . Tim volume in question can, therefore, be written I « dKj dp x 2 dp v v ♦ * * dz N) II r < E — ^ mv 2 . Let us use the notation / a(jy-i)jg dp dpvt • • dz Ny « a x e 2i?l I1,<1S approximately, in analogy with the expression last written for c r(7V). Then I . ^(E - | mv 2 ) dK = dK, (Cf Fig, 84 ) Sec, 107] STATISTICAL MECHANICS ’ 349 Wo can now write for the volume of that part of the ergodic layer in which molecule 1 lies in dK (shaded in the figure) dl_ 4E SE = 3(iV - 1) 2E — 8QV— ai(E)e 4 « dK SE , Ei being replaced again by E after differentiating, This in turn forms a fraction N-l <n(E) N a{E) 6 dK of the whole layer, whoso volume 5<r or <r'(E) 8E was found above. Here the factors in front of the exponential are constants so long as E is fixed; and in the exponent, E/(N — 1) is physically indistinguishable from E/N, which equals ^ w# and so, by (256) in Sec. 15, also equals •§ kT in terms of the Boltzmann con- stant k and the absolute temperature T. The probability that the first molecule lies in dK can, therefore, be written _ WV J Ce W dK, Fig. 84.- molooulo -Looftlismtion of a in ayBtom apaoo. C being a constant dependent on E and V . Comparing this expression with that in eq. (57), See. 28, we see that wo have here deduced Maxwell's law from statistical mechanics, in so far as that law expresses probabilities for a single molecule; since the space coordinates do not appear in our result, the probability is independent of position in the vessel. By fixing several molecules in cells dK i} dK a • • • , respectively, one finds in the same way that the probability of such an arrangement of these molecules is proportional to i waPi 1 q 2 kT 2kT ’ * ’ If wo thou consider different possible positions of just one of these molecules in its phase space, whilo keeping tho others fixed, we obtain the same typo of probability function for this one as wo found above when all of tho other molecules were left free. The probability for one molecule is thus the same, wherever other molecules are assumed to be located, or however they are moving, In this conclusion wo have a proof from statistical mechanics of tho principle of molecular chaos, which was made the basis of the deduction of Maxwell's law in.Seo, 21. 350 KINETIC THEORY OF OASES [Chap IX Besides obtaining the pzobability for one molecule, or a fow of them, the analysis heie given can be extended so as to show that the molecules as a whole will neaily always be distubutecl in a manner not differing essentially from the maxwellian distribution This is done by showing that such a distnbution is, with a negligible cnor, charactonstic of almost all of the points in the eigodic layei. This lattci conclusion is reached a little moie easily, however, as a lesult of the combinatorial aigument in the next section 198, The Loose Many-molecule System. The most general typo of system for which a complete statistical theoiy has been obtained as yet is one that consists of very many subsystems intei acting with each othei only veiy slightly, or only occasionally Tho only piactical example of such a system is the gcneial poifoct gas whose molecules have internal as well as tianslational degiees of ficcdom; for this reason we shall fox convenience call the subsystems, in general, molecules , and shall lefer to any system of this type as a loosely coupled many- molecule system It is, howevei, of Iheoietical intciest, in connection with the canonical distribution to be discussed latei, that the loosely coupled subsystems, of which the whole system is composed, need not be capable of moving aiound like gaseous molecules, It is possible to treat this moie gcneial caso by tho plmao-spaco method that we have just employed for the point-mass gas * There exists, however, an older method duo to Boltzmann which serves better to exhibit the tiue situation, and which also enjoys tho advantage that it can be earned over into wave mechanics, whoro tho phase-space method becomes inapplicable, this othei method, which might bo called that of molecule space , is usually employed and will be followed hoio. To begin with, let us assume that the subsystems 01 molecules are all alike, and let then number be N Then each molecule by itself can be desenbed m teims of a ceitain numbex s of generalized coordinates together with an equal numbei of gcneialized momenta, thico of these coordinates refei, of couise, to its centci of mass, the others to its rotation and internal condition These 2s variables may be logarded as the cooidinates of a point in a space of 2s dimensions, which wo shall call the inoleculai phase space, oi molecule space The instantaneous condition of the entile system, which wo liavo been lepresentmg hxtheito by the position of a single point in system space, will then be lepiesented equally well by the position in molecule space of N points, one foi each molecule (cf Fig 85). A lopi oscillation of this soit was employed m Sec 17 foi the translational velocities If we weie to take as variables in the space there desenbed the thieo * Kiujtkow, Zeits Physik, 81, 377 (1933) Sec. 198| STATISTICAL MECHANICS 351 components of momentum, instead of the components of velocity, and then add three more dimensions for the coordinates themselves, wo should have the six-dimensional molecule space for a monatomic gas, as already employed in the last section* In the general case, wo have then to add still other dimensions corresponding to the internal motions. From the macroscopic point of view, oil the other hand, the state of such a system is determined by the character of the motions which its molecules are executing, but does not depend upon the identity of the molecule that is executing each particular mo- tion, since the molecules are all alike* Ac- cordingly, only the distribution of the M points in molecule spaco is macroscopically significant, without regard to their identity. In order to give a precise meaning to this rather vague term, distribution i, let us adopt the customary dovico of dividing the whole of molecule space into small cells of equal volume, which we suppose to bo numbered off in a definite order, and let us define a distribution by saying that it is specified by the numbers Mi, N* • * ■ of molecules whoso representative points lie in cell 1, cell 2, and so on. Many will, of course, be zero; and obviously = N, (246) i the summation extending over all the cells. Now, usually a given distribution as so defined can result from many different arrangements of the individual particles, an arrangement consisting in the assignment of each moleculo to some definite cell. The number of possible arrangements for; the distribution Ni } • * * will be, in fact, the number of combinations of M* things taken N i, M 2 1 • * at a timo or Sysiem space Molecule space Fia, 86. — Moleculo sp ft 00 and ey stem space. M m Ml JVitiV 2 ! • • •’ (247) As a conci'efco example of tho same thing, the number of ways of divid- ing 20 books among two shelves so ns to put 12 on one shelf and 8 on another, paying no attention to the arrangement of eacli set on its shelf, is 201/(12! 8!). 352 KINETIC THEORY Ov OASES [Chap IX The anangements , on the othei hand, have in statistical mechanics the impoitaiit piopcity that each one conosponds to the same volume m the big system space This becomes cleai when we lcfloct that, as the lepiesentative points foi all of the N molecules moving inde- pendently exploie then cells m molecule space, the single point icpre- sentmg the entue system in system space tiaces out a region in that space whose volume G is obviously the pioduct of the volumes of N cells in molecule space, 01 G — < 7 ^ m tcims of the volume g of a cell * The volume G is thus obviously the same foi all anangements This fact leads to veiy impoitaiit conclusions It enables us to infei fiom the pimciplo of statistical equivalence foimulated in Sec, 195 above that each airangemcnt occurs equally often, or with equal probability, among the systems of a miciocanomcal ensemble (piovided it can occrn at all) , no anangemont is favoied ovei any othei. From this wo can then diaw the further conclusion that the probability of a given distribution of the molecules is simply pioportional to the numbei of anangements that give use to it, or to the number M whose value is given 111 oq (247) 199, The Most Probable Distribution. In this conclusion we have acquired, foi a loosely coupled many-molccule system, a means of compaung the piobabilities of the vanous possible distributions of its molecules 111 the concsponding molecule space, Now among these chstubutions theie will be a most probable one, and it turns out that this one is of tianscendent importance Among the systems of the ensemble, appreciable depaituies fiom the most probable distribution aie, m fact, laie; and for this loason Jeans called it the normal distribution, By the principle of equivalence, this dis- tribution will then almost always be veiy close to the actual one in any single system; loi the distribution that is most piobable during the motion of a single system cannot change much in chaiactex with a small change 8E m the energy, and so must be sensibly the same for all systems in the ensemble. Accordingly, this normal distribution can be used in place of the actual one in making statistical calculations pertaining to a single system, Therein lies its gieat importance. To find the most piobable distnbution, we need to determine the AVs so as to make the numbei M , or log M } a maximum, subject to the condition that the total numbei of molecules is N and that the total enoigy is approximately equal to E, This is not easy to do with dpx, dqi dq N * (f dpi dp, dqi dpi,dg ,+ 1 • < • • (Jdpw-D. * <*?,) dqi, r,) Sec. 109] STATISTICAL MECHANICS 353 completeness, but if some of the Ni s arc large their relative values can easily be found by using Stirling's formula for the factorial [cf, eq. (214a) in Sec. 154]: log (iV'f!) = (N{ + i) log N f — Ni + log v^jr- In varying these large Ni we can also treat them as continuous variables without appreciable error; and we can then keep E actually fixed, since the variation of E over the small range SE cannot shift the location of the maximum appreciably, Let the energy of a particle when in coll i be then the total energy is* %N^ = E, (248) summed over all cells. For a maximum we must have, accordingly, varying only the large Nf a in (247), 0 = d log M « - 2d log (N (!) = - S^log N { + ^ dN (l in which the dN{ are arbitrary except for the two conditions, derived from (246) and (248), that %dNi = 0, 5}c«UV<- 0. i i Multiplying these last two equations by a and j3, respectively, and adding to the preceding, in which wo may drop the very small term l/22Vf, we have then 5)(log Ni 4* « + 0e<) ~ 0| i and the usual argument (cf. Sec. 27) then leads to tho conclusion that for any i log JV< + <x + = 0, Thus, wherever its value is largo, Ni = (249a) C being written for <r“. Hero C and 0 are tho same for all cells, their values being fixed in terms of tho total number of particles and the value of tho energy by eqs. (240) and (248). * Variation of tho onorgy as tho points movo about in their oolls is lioro neg- lected. Tito rigorous handling of this point is tedious but can bo effected, 354 KINETIC THEORY OP OASES [Cixap IX It will be shown in tho next section that 0 = 1 fkT } T being tho absolute tempeiatiue and k the Boltzmann constant or gas constant foi one molecule Poi convenience of lefercnce we may assume this lesult heie in advance and write N t « Ce « Ne~ KT j 2 « (2m) the last sum extending ovei all cells, and this second form of tho expiession lesultmg from substitution m the equation ^JV, = N. X Sometimes, howevei, it is moio convenient to employ in place of the distribution for the system as a whole a probability function for tho individual molecule Since all of the molecules, being similar, must spend equal times in any given cell, the piobability that a given one is in cell ^ is obtained simply by dividing eq (249a) oi (2496) by N, since, howevei, tho molecule is also equally likely to bo in any pait of the cell, it is moic convenient to go ovei now to elements of molecule space Wo can wntc then as the piobability that a given molecule is m any given element dg oi dq \ dq B dp i * dp Bi wheie its eneigy is e, oi the fi action of its time that it spends in such an element, P dg where p * Citr* - 0 X G w « — e — (249c) f <T^ dg Here the integral in tho last exponent extends over all of molecule space, and Ci = C/Ng oi may be icgai ded as a new constant whoso value is determined by the condition that It is this lesult that was cited above m Sec. 53 as the classical Boltzmann distribution fonnula, eq (89a), and was theie shown to include Maxwell's law as a special case 200 The Most Probable as a Normal Distribution. By extending the calculation just made, it can also leadily bo seen that laige values of N t diffcnng appreciably fiom those given by (249a) or (2496) must bo laic, provided only that the total number of pai tides is laige Foi, if the laige N t aie changed by small amounts 6N t from their most piobablo values, which aie such as to make the first-oi del teims m 5 log M vanish, we find fiom (247), upon pushing the calculation to tho second oidei in the jJWs, that tho change in log M is approximately Sec. 201] STATISTICAL MECHANICS 365 5 log M — M itself is then changed in the ratio e 5ln * M from its maximum value. Now if we imagine N ancl IS to be increased toward infinity at the same relative rate, a brief contemplation of eqs. (246) ancl (248) shows that all largo Nt must increase in the same ratio ( C changing but not ft ) ; hence, for given fixed values of the relative variations, dNi/N^ we have 5 log M ->—«>, Thus, as the molecules become numerous, any particular distribution differing from the most probable one by a given relative amount, as measured by the values of 6Ni/Ni } becomes relatively rare. On the other hand, the number of different distributions having 6Ni/Ni in the neighborhood of given values is readily seen to change with N and E in roughly the same ratio for all values of 6Ni/Ni, Hence the probability of the occurrence of any given range of values of 8N</Nt that does not include zero decreases indefinitely as N — > Thus we roach the advantageous conclusion that in a loose many- molecule system distributions departing appreciably from the most probable one occur only very rarely; and accordingly, as was stated above, in deducing statistical behavior of such a system, we can take the actual distribution to bo the same ns the most probable or normal one. 201, Some Generalizations of the Loose Many-molecule System, The restriction of the preceding equations to largo values of Ni may be awkward in actual cases because it may bo impractical to construct cells in molecule space that are largo enough to include, say, 1,000 or more molecules. This is so, for example, in considering the concentra- tion of water vapor near a charged smoko particle of ultmmicroscopio size, which wo might wish to study in seeking an explanation of cloud formation due to such nuclei. Most practical cases of this sort can bo covered by remarking that wlmt wo are chiefly interested in is only the average expectation of molecules in a cell, which is, of course, indistinguishable from the most probable number when this is large. The distribution law already found can readily be extended to the average number in any cell, provided there are many other cells with the same value of the asso- ciated energy 6;, and provided that the group of such cells as a whole contains a largo number of molecules. In the example cited, for instance, we can suppose that there are many smoke particles in the vessel, each having a cell in a certain relative location near it, and all of these cells can then be grouped together. 350 KINETIC THEORY Ol>' OASES (Chap IX Since any molecule moving aiound in a gioup of n cells sweeps out n times as much volume in system space as it docs when it is con- fined to one cell, the piobability of a distribution with N{ molecules m a group of in cells, JVj in a gioup of /is, and so on, will be pi oportional, not to M as given in (247), but to (/*!*' M or iVl - V' * Pioceeding as befoie, one finds then as the noimal distribution, in place of (249a), foi the ,?th group of cells N[ = frCe-P". The molecules are, however, equally likely to bo m any one of the various cells of a gioup Hence, for the avei age number -A , of molecules in cell i, which equals IVJ/fq foi the gioup containing this cell, we find _jl Ne~ kT N, = Cerf>« = Ce kT = — — -> w (249d) which is of the same form as (2496) above, and for tho probability function P foi a given molecule in its phase space, which is $x/N 0i wo obtain again eq (249c) above Furtheimoie, consideiable geneializatton of our results is also possible in legaid to the natuie of the molecules themselves Instead of being all alike, they may be of sevoal different kinds , piovidod there are many of each kind In that case M is simply tho pioduot of seveiai fractions like that m (247) above, e g , M - jVl N " I iVJUV.I ■A'i'I.V','1 ■ where primes distinguish quantities lofening to tho different kinds of molecules The equations following (248) then become ° = - 2 ( lo s N/ , + <w' - 2 ('°« N i + 2f?) dN " • ■ ' = o, 2}<wr-o, ••• % t X< dN * + X ( " dN < ■ = °; 4 * and when we multiply the last of these by /3 and the preceding onos by a\ a n • and add to the first one, and then equafco to zero the Sac. 202] STATISTICAL MECHANICS 367 coefficient of every dN'i, dN" • • - , we obtain log N\ + a> + P e 'i ~ 0) log -|- a" + j8e" = 0, * ' ‘ > whence N[ = CV*', N? - C"e-t«" Thus there exist equations of the same form as (249a, b, c, d ) for each separate kind of molecule, with different constants O', 0" • ■ • corresponding to the varying numbers N', N" * • • that are present of the different kinds. The constant 0, however, which again turns out to equal 1/kT, is the same for all, owing to the fact that interchange of energy is possible between all of the molecules and there is, therefore, only a single equation expressive of the energy condition. Except for this latter feature, each land of molecule is distributed as if the others were not present. Finally, it does no harm if a relatively negligible part of the mole- oules are of miscellaneous character, perhaps even only one of a kind; for then, assuming a certain energy for these few, we can apply the preceding theory to the remainder. The quantity E in (248) is then, strictly speaking, to be interpreted as representing the variable energy of the remaining molecules alone ; but no appreciable error is incurred if it is taken to represent, as before, the fixed energy of the whole system. In Sec. 206 below wo shall go further and, by establishing a connection with the method employed for the gas of point masses, show that actually the same distribution law holds for such odd particles as holds for the others. Before proceeding further, however, wo must first establish for loose many-molooule systems the connection of 0 with the thermody- namic temperature. 202. Introduction of the Temperature. The temperature was introduced into the theory of the perfect gas in Sec. 13 . by a method that is rigorous and can readily be generalized so as to apply to the general many-molooule system. In order to identify any quantity with the thermodynamic tempera^ turo T, wo must show that it possesses two essential properties: it must have the same valuo for any two bodies when they are in thermal equilibrium with each othor; and the heat dQ absorbed by a body during any small reversible change must take the form T dS, where S is a quantity, the entropy, having a single value to correspond with every equilibrium state of the body. In mathematical language, therefore, 1/7* must constitute an integrating factor for the reversible heat. 358 KINETIC THEORY OF OASES [Chap IX Such an application of thermodynamics implies a clear distinction between eneigy that enteis a body as heat and oneigy that onteis as work This distinction, which is made without any tiouble in ele- mental y physics, becomes fai from obvious when the system is regarded as exhibiting puiely mechanical behavior It can be leached, howcvoi , from the dynamical standpoint in the following way As viewed macioscopically, the effects of the sui roundings upon a system aie of two lands In pait they can be lepicsontod by introduc- ing into the Hamiltonian of the system certain paiameteis, ai, aa ' • , which repiesent physically interesting quantities averaged over molec- ular fluctuations, such as the volume of the vessel containing a gas; the Hamiltonian function can then be wnttcn II(p h pa • • , q u • , «i, a 2 • ) Any eneigy lost by the system to its sun ound- mgs in consequence of changes m these parameters can then be expiessed m teims of then changes and is called woik done by tho system Theie may then be m addition ceitain nicgular foicos exeited by thesunoundings on the system which cannot bo icpiosonled by such paiameteis but must nccessanly entei into tiie equations of motion as foices of external ongin All eneigy imparted to the system by such foices, which we shall call theimal foicos, constitutes heat absorbed by the system Usually a paiticulai moleculai foice has to be divided arbitrarily into two paits conespondmg to this distinction. As ail oxampio, tho force exeited by a molecule of the wall on tho enclosed gas can bo lesolved into a component lepiosenting its averago coiitnbution to the pressure and anothei megulaily fluctuating component, and the former, which depends only upon the mean position of tho wall and tho posi- tions of the gas molecules, can be taken account of by intioducing into the Hamiltonian for the gas a suitable potential-energy term con- taining as a paiameter a the mean position of the wall In oidei to identify the tempeiatuie, it will suffieo now to considot a loosely coupled system in which the molceulos aio all alike; the subsequent extension to more goneial cases is so easy that no details need be given Let us suppose that such a system is subjected to a small thermodynamic piocess dunng which the «'s undergo a small change, and duimg which the thounal foices may or may not impait a httle eneigy to the system Let this piocess be earned out so very slowly that the system lemains always vciy noaily in equilibrium, so that the process is leveisible in tho theimodynamic sense Then, resuming the notation of tho last few sections, we can wiito for the eneigy of the system at any instant E = summed over all the cells in molecule space, e , being the energy of a moleculo when m cell SBC. 202 ] STATISTICAL MECHANICS 359 i and A r , the mean number of molecules in that cell. The change in E during the process just described (ignoring its fluctuations about its mean value) can, therefore, be written dE = 2 ) Ni du + Je, dNi. (250) it Now suppose, first, that heating forces do not act. Then, during any element of time dt , whatever changes occur in the p’n and q*n of the system are governed by the Hamiltonian equations of motion and aro the same as they would be if the ot’s were constant during that interval. Accordingly, by conservation of energy, these changes, and the result- ing changes in the instantaneous values of the Ni% do not alter tho energy. The actual gain of energy during dT must, therefore, bo d'E = where d'e* is the change in that arises. during dt i from the change in the a's, By conservation of energy, since there are no heating forces, d'E is also the negative of the work done by tho system, Hence, integrating d'E throughout the small reversible proc- ess, during which the energy changes by clE = J* d'E > and assuming that the change at a uniform rate,, so that we may write fNid'a == Nj d'u = Nidet, wo have for the work during the whole process dW “ — dE or dW = -^Nidu, i de{ standing for tho whole change in €,• during the process. If wc then suppose heating forces also to act during the process, the work clone by the system, being determined by other forces, will not be affected to the first order; but the change in energy will now be given by the more general expression (250). Hence, an amount of reversible heat dQ is now given to the system of magnitude . dQ - dE + dW = ifh (251a) i In order to find an integrating factor for dQ, let us now multiply this equation through by (3, and then insert in it tho value of /3<u proper to equilibrium, as given by the second member in (249tf) above, The 360 KINETIC THEORY OE OASES [Chap IX result is pdQ = ^(log 0 — log Ni) d$„ from which the term in C can t be dropped because ^ dff t — 0, leaving pdQ~ (251 b) ' \ The expression on the right is now the differential of a single-valued function of the lf % This shows that ft is an integi ating factor for dQ, Furthermore, as shown in Sec 201, f3, and hence also any function of /9, enjoys the propeifcy of having the same value foi any two loose many-molecule systems that are m equihbuum with each other; for two such systems may always be legal ded as a single system containing two sorts of molecules Wo can, theiefore, introduce the themio- dynamic temperatuie by the defining equation T — 1 /A/0, or in which h is a universal constant of pi opoi honality. The leplacement of fi by 1/lcT in the preceding sections is thus justified That k has its usual significance as the gas constant for one molecule follows then from the identification mentioned in Sec. 199 of our le&ults with foimei expressions that included Maxwell's law as a special case 203 . Entropy. A dynamical expicssion for the entropy S of a loosely coupled many-moleculc system can now be obtained by integrat- ing (2516) Since ^ = 0, this gives 8 = = -*25#. log ff t + So, (262 a) or, after introducing log # t irom (249 d) and using « N t the total number of molecules, and = C, the total energy, % S - | - m log C + So, (252 b) « or also, since in the last membci of (249d) kT can obviously be Sec. 204] STATISTICAL MECHANICS 361 written J* e hT = kT dg, the last integral extending over the whole of molecule space, S - | + Aft log J 0 “** dp + (262c) Hero So = So — Nk log (gN) and So or S' 0 is an arbitrary integration constant. , The quantity 2\e kT is what we called in Sec. 147 the state sum; X Jc kT dg might similarly be called the state integral. In eqs. (262a, h, c) S stands for the entropy of the whole system. From the structure of the formulas it appears that, in harmony with thermodynamics, the entropy should be simply additive, provided wo add the integration constants when separate systems are coupled into a larger one. A question arises, however, when subsequent interdiffu- sion of molecules can occur, as when two masses of gas are placed in contact with each other. If the molecules in the two part systems are different in nature, then, when equilibrium has been re-established by diffusion, tho total entropy exceeds the sum of the original separate entropies; in the formulas this results from a decrease in Ni, or from an increase in tho state sums duo to enlargement of the accessible region in molecule space. If, on tho other hand, the molecules in tho two part systems arc exactly similar, their interdiffusion has no physical significance; it cannot be made to do macroscopic work, for instance; and no such increase in the total entropy occurs. Since, however, tho same changes in or in tho state sums occur as in the other case, we can prevent an increase in S only by readjusting tho integration con- stant whenever systems of similar molecules are combined into a larger one with resulting possibilities of intorcliffusion. Wo shall find that this inconvenience disappears in tho wavo-mcohanical theory. It is often avoided by using the formula only for the entropy of unit mass, i.e., for specific entropy. 204. Entropy of the Monatomic Gas. It will bo instructive to verify the agreement of the expressions just obtained with tho usual ones in the simple case of a perfect gas composed of molecules that can be treated as point masses, Let us consider a gram of gas containing N such molecules. In tho phase space for such a molecule we can tako as variables the three cartesian coordinates of its center of mass and its three com- 362 KINETIC THEORY OF GASES (Chap, IX ponents of ordinal y momentum, p x , p v , p z , the total momentum being p. Then « is the oidinaiy translatoiy kinetic energy and has the value, e = p 2 /2m Hence, taking spherical shells about the origin as elements in momentum space in place of dp x dp v dp z , wo have Je kT dg ~ J J JdxdydzJ J j'e hT dp x dp v dp z = irVj'J’e 2 " ,r; ’p 2 dp = (2tt mkT)W, since e~ x> dx ~ Vtt and ffj di dy dz = V, the volume of the vessel Furthermore, the classical eneigy of such a gas is IS = 5 NhT (cf Sec 143) Hence (252c) becomes S = Nk log (VTV) + (252 d) wheie S" = S' 0 + Nh{% + log (2wink)Yi^ This agiees with the usual thennodynamic result as expressed by eq (200a) in Sec 141, in which R = Nk and Cy — 3i?/2 for a monatomic gas. 206. The General Boltzmann Distribution Law. The combina- torial method that we have been employing stands in strong contiast to the moie duect phase-space method that was employed previously in dealing with the special case of a pci feet gas Wo shall now show that the lattei method must be applicable to the more gonoral case as well, although it may 1 un into senous mathematical difficulties By its use we shall endeavoi to lemove some of the limitations that have hitheito been laid upon the system undei consi delation. Let us select any molecule A out of a system and seek to obtain a distribution law foi it by the phase-space method. To do this, we first confine the molecule A to an element Sg A in its own 2s H -dimon- sional molecular phase space Then, 1.1 analogy with tho procedure followed in &ec 196, we can wnte for tho volume of that pail of system space, in which the total eneigy II of tho entile system' is loss than a fixed value E, <*(E) ~ ff dpdq, l!<E and for that part in which A is also confined to Sg A with energy «, 6a = a r (E - e) hg A , <j t (E - e) - J J dp' dr/, II r<E-t where dp 1 ’ dq' stands for the product of the differentials of all other Zt r P f taimng t0 A > and Hr (V, /) foi the part of tho Hamiltonian that contains just those othei variables Sue. 205] STATISTICAL MECHANICS 363 Now consider that part of the ergodic layer in which A lies in Sg A . The volume of this part is dE Scr dE = <s' r (E - «) dE Sg A , This volume divided by the volume of the entire ergodic layer is the probability that in an actual system A lies in <5 <7.1, in consequence of the principle of statistical equivalence (Sec. 195). Accordingly, we may conclude that, in ah actual system having energy E, the probability at a given moment that molecule A is in any chosen element Sg A of molecule space is simply proportional to v' r {E - e ). Now suppose that our system is of the generalized type contem- plated in Sec. 201 above, most of the molecules, but not all, forming one or more largo homogeneous groups. Then, if molecule A happens to belong to one of these largo groups, and if, furthermore, dg A lies in a cell containing many of these particles, another expression for the probability that A lies in dg A is furnished by the theory of Sec, 201; according to (249c), this probability is proportional to e -i3e Sg A = sg A , • It must be true their, under the circumstances stated, that a' t {E ~ «) - QyC-fi* where C\ is a factor that is independent of the location of A and depends only on E. This result has been obtained only for a restricted choice of tho molecule A and of tho element Sg A , Wo can safely assert, however, that the form of the function, <j' r {E — e), cannot depend appreciably upon tho particular choice of A out of many millions of molecules in the system. Ilenco, <r' t (E — <=) must have the same simple expo- nential form when any molecule of tho system is chosen. A little more generally, it follows that <j'{E) and, by integration, <r(E) itself must bo of exponential form near any value E = E j, say, , a(E) = Cie-e*, as we found it to bo for a monatomic gas (Sec. 196) ; for, an odd mole- cule with energy e could be added to it, whereupon a{E) would function as <r, in tho analysis just given, E here being represented by E— t there. 364 KINETIC THEORY OF GASES [Chap IX If the statement just made about cr r bo granted, then it follows, by leveising the aigument, that the distribution law has the form wntten above 01 m (249c) foi any molecule of the system, and whoiovci the phase-space element may be located One is then tempted to suspect that these lesults may hold likewise foi much moie geneial types of system It is, as a matter of fact, believed that accoidmg to classical theoiy c r(E) has the exponential foim Ce~^ E ) in the neighborhood of any paiticulai value of /£, for any system composed of an enormous number of moloclilos, whether these aie loosely coupled 01 not The same statement is believed to hold if a few of the generalized cooidinates of the system aic hold fixed and only the lemainder aie included in the integiation foi a , If tlioso proposi- tions aie granted, then it is easy to deduce fiom them, by the method indicated above, the following veiy general foim of the Boltzmann distribution law, the tempeiatme being introduced by a special argu- ment foi which we have no space , Suppose a few vanables, either coordinates 01 momenta, oooui only in a sepaiate term e m the Hamiltonian function //, lot tlioso vaiiables be denoted by $ 1 , If any othoi vaiiables occui in e, lot us considei only states of the system in which those vaiiables lie in given small langes Then the piobability that the lio m langcs dg = d£i di ; 2 d £* is Pdg, P = Ce-f>* « Ce~w t (253) C being a constant such that f p dg — 1 This lesult constitutes meiely an extension of the distribution law represented by (249c) above and includes the Iattei as a special case As a coiollaiy, we can diaw the important conclusion that, in a physical body, classical theoiy lequucs Maxwell's law to hold for tho velocities of the moleculai centers of mass, lcgaulloss of all other cir- cumstances, this is tiue for each sepaiate configuration, or sot of posi- tions of the molecules, and no mattei how stiong or complicated their mutual potential energy may be It must bo lemaiked, howevor, that in wave mechanics this statement has to be modified considerably when the density is high, as m a liquid oi solid 206, The Eqmpartition of Energy. Tho distribution laws that have been obtained lead to a conclusion of considei able importance l dative to the dtsinbnhon of the energy in a system in oquilibiium, Suppose the expiession for the energy (oi, more generally, the Hamiltonian function) contains a teim in which ono vaiiable occurs only in the form of a square Lot us write for this toim e s=? i Sec. 206] STATISTICAL MECHANICS 365 where the coefficient <p may contain some of the remaining variables; if it does, let these for the moment have fixed values, Then eq. (253) in the last section, or (249c) in Sec. 201, if it happens to apply, gives as the probability law for £, P = C\e 2kT , where C\ is given by the equation f*P dtj ~ 1. We find then for the mean value Since this result is independent of any other variables that may occur in tp i it will hold quito generally, Hence, whether other variables arc kopt fixed or not, the mean value of such a term is just $ kT. This is the famous principle of the equipartition of energy. The simplest example is, of course, the translatory kinetic energy rni tryA of a particle, which can bo written ^ +7^ in terms of the ’ 2m 2 m 2m cartesian momenta, p X) p V) p„. Each of the three terms in this expres- sion must average \ kT ; the whole kinetic energy, therefore, averages -$■ kT. This latter result was obtained for perfect gases in Sec. 15; hero wo have it for particles in a system of any typo. According to classical mechanics it should hold even for the molecules in a liquid or solid. Similarly, if the system is capable of small oscillations in the generalized sense, the potential energy associated with each normal generalized coordinate q< can bo got into the form. $ <piQt, and the associated kinetic energy can bo written similarly ^ in terms of the corresponding generalized momentum.* Each of those terms will then average -fr kT ; and the whole contribution of each mode of oscilla- tion to the energy will average kT. This is true for any simple har- monic mode of vibration. Finally, similar results oan bo obtained for the rotational energy of any rigid moleculo forming part of a system in equilibrium. The rotational kinetic energy of such a moleculo can bo written K = \ ^ Bo>ji + {■ Cto* in terms of the components of tho angular velocity, co n , wo, in tire directions of tho principal axes and the moments of inertia A, B, Q about those axes. In order to express this in terms of generalized * H, La Mi), “Higher Mechanics, ” 1029, Soo. 92. 306 KINETIC THEORY OF GASES [Chap IX coordinates, let us intioduce the thiee Eulcimn angles 0, f, <p; in terms of these coa = 6 sin <p — ^ sm 0 cos <p, o>d = 6 cos <p + ^ 8111 0 sin <p, wo — <p + ^ cos 0 (Of H Lamb, "Higher Mechanics, ” Soc 33, where p, Q, r are to bo identified with w A , w a, wo ) Then the concsponding momenta aie dIC j Ocoa + Bm 2jy + Cm y ot«., or pd ~ Aco a sin (p + Bon cos <p } p<p — Co) Oj = — Ao)a sin 0 cos ip + Bo>n sm 0 Bin <p + Co>c COS 0 ye »= cun y t p+ = — Ao)A sm 0 cos ip + Bo)b i Now let us compaie the molecular distribution m (oa } w*, o)c space, which we shall call o) space foi shoit, with that in p space, m which om statistical laws hold To a volume dr m the latter there conosponds a volume dr/J m oo space where J is the Jacobian dpe dpo du)A do)n A sm (p B cos <p 0 dpt dp$ —A sin 0 cos <p B sin 0 sin <p C cos 0 1 1 do) A do>B 0 0 0 ABC sin 0 (Cf Sec 24 ) Thus for given values of 6 , \j/ } <p volumes m w space aio propoitional to the couesponding volumes m momentum space, Accouhngly, the molecules will be distiibutccl m the same nmnnoi m both spaces It is easily seen that, m consequence, the piinciplo of equipaihlion will apply also to the thiee tenns in K> Each of theso toims, which can be legarded as the lunetic eneigy associated with one lotational degree of fieedom of the molecule, will again average § hT t and tho whole lotational eneigy will aveiage $ hT 207, The Canonical Distribution and Ensemble. Instead of sup- posing the system under consideration to be isolated, wc might suppose it to be connected with anothei system, Then its energy will undergo Brownian fluctuations Suppose that the connection is loose and tho other system is a veiy much huger one, Then the lattor soives to control the tempeiature of the onginal system; it lias been oallod by Fowlei a tempeiature bath The original system can now be tioatcd as a "molecule" foiming pait of the entne combined systom; and cq Sec. 200] STATISTICAL MECHANICS 367 (263) then indicates that it will spend a fraction P dg of its time in each element dg of its own phase space or system space, where P = Ce~&, (264) E being the energy of the system when in dg. A statistical distribution in phase according to the law expressed by (264) was called by Gibbs a canonical one, and an ensemble of systems so distributed, a canonical ensemble. Such a distribution or ensemble might be thought to represent better an actual physical body immersed in its surroundings than does a microcanonical ensem- ble. The macroscopic statistical properties of the two cannot differ much, however, when they correspond to the same temperature] for a canonical ensemble really consists of many microcanonical ones of different energies, and the effective relative range of these energies can be shown to be very small when the system itself contains many molecules. The canonical ensemble, or the canonical distribution in phase, furnishes a basis for the treatment of systems whose energy cannot be written as the sum of many similar terms. 208. Entropy under a Canonical Distribution. An expression for the entropy S, defined as j is easily obtained for a system that is assumed to exhibit in time a canonical distribution in phase. Tho argument runs so very closely parallel to that in Secs. 202 and 203, in fact, that wo can at once write down as the result, in analogy with (262a) and (262c), S = JV log P d<r = | + /clog J da. (265) Here the integrals extend over the whole of system space and the second one may be called the stato integral for the entire systom; the arbitrary constant of integration has been set equal to zero. Different as this expression for 8 seems to be from tho ono obtained previously, as stated in eq. (252a) or (262c), it can differ from tho latter only by a constant amount, since the entropy, boing defined as represents a definite physical quantity. In the corresponding wave-meclmnical case it will bo worth while to ascertain the actual magnitude of this constant difference. 209. The Second Law of Thermodynamics. Our demonstration that a loose many-moleculo system in equilibrium possesses a quality 368 KINETIC THEORY OF GASES [Chap IX having the piopeities of the thcimodynamic temperature constitutes a deduction of the second law 1 of theimodynanucs fiom tlio laws of mechanics, in so fai as that law icfors lo oqiuhbiium phenomena Thus the thei mo dynamics of leveisiblc piocesses is convoitod, at least foi such systems, into a subdivision of mechanics It is to be noted, howcvei, that we have consistently dealt will) the normal 01 average state of the system, ignoung the phenomenon of fluctuations Even the most unusual states should, however, sometimes occur, just as m shaking a mixkue of black and white balls it will happen once m a gicat while that all of the black ones come on top This means that occasionally the lequircmonls of the second law will not be met Such unusual occunonces aio indeed easily obseived m the Biowman motion, where a paiticlc is soon to acqune every now and then velocities much above or below the aveiage The second law of thermodynamics must, accordingly, bo regarded as statistical in nature, having lefeicnce to the normal 01 aveiage behavioi of mattei and to the macroscopic lathci than to the molecular view of phenomena So conceived, howevoi, it appeals to admit of no exception It could peihaps be imagined that fluctuation phe- nomena, such as the Biowman motion, might some day bo harnessed by means of an ingenious mechanism and utilized to effect Iho con- tinuous conveision of heat into woik It seems highly piobablo, however, that all such schemes are doomed to defeat in consequence of the inevitable fluctuations to which the mechanism itself must bo subject On the othei hand, the second law of thei mo dynamics as usually stated implies the fui tlier asseition that, in an mevcvsible passage of an isolated system fiom one equilibnum state to another, the entropy can neve / dew ease, the fact being, of course, Lhat it usually increases This statement, likewise, must be undoi stood in a statisti- cal sense as having reference to aveiage experience Not much has been accomplished towaid the theoiotical tieafcmcnt of inovorstblo piocesses, hardly anything has been added lo Boltzmann's II theorem for a point-mass gas This question is very closely 1 dated to the proof that one would like to obtain of some pnneipio equivalent to our "ergodic sunnise," 210* Entropy and Probability. Fiona the molecular standpoint the tendency of a system to move towaid equilibrium, with an accom- panying gain in entiopy, is mterpieted as lesulling from a tendency to move fiom an impiobabje towaid a more piobable state It was shown by Boltzmann in connection with his II theorom that, in some STATISTICAL MECHANICS Sec, 210 ) 309 cases at least, a quantitative connection can be traced between the increases in entropy and in probability. In an isolated many-moleculc system, for example, the probability of any distribution, as was shown in Sec. 198, is, from (247), DNl NiW t \ • • •' D being a constant of proportionality. On the other hand, from (262a) in Sec. 203 the entropy of the system, when in a state of equilibrium, is S - log N { + So. i Let us take Pi> to refer to the normal distribution, so that Ni = Ni, and lot us suppose that the Ni'a are large enough so that we can keep only the principal term in Stirling’s formula and write log (17,1) = Ni log Ni. Then log Pd = ~ log Ni + const., , t and wo see that S = k log P D + const. (266) This famous relation, first pointed out by Boltzmann, is often regarded as holding universally. It may bo illustrated concretely by the following example. Suppose a capsule containing N molecules of perfect gas in equilibrium is introduced into an evacuated vessel r times as largo in volume as the capsule, and that a hole is then made in the wall of tho capsulo. The gas thereupon escapes, ancl presently comes to thermal equilibrium in the larger vessel at its original tem- perature but with its volume increased in the ratio r; its entropy is at the samo timo increased by the amount [cf. (252d) in See. 204] Nk log r. Now, just after the capsule has been ruptured, but before any molecule has had timo to escape, the gas is in a state which, under the now conditions croated by the rupture, is a highly improbable one; as tho gas then issues and spreads throughout tho vessel, it passes into more and more probable states. Let us find the magnitude of the total resulting increase in log Pc. When the gas has again reached equilibrium, its molecules are distributed in tho molecular phase space over r times as many cells as there were at the start, but with only l/r times as many molecules per cell; lienee the now value of 370 KINETIC THEORY OF GASES [Chap IX the term — 2iV, log iV, in log Pc, written in terms of the 01 lgmal N t ’ s, is -’2f Iog f = -^'(log IV. ~ logr). This term has thus mwaased duiing the expansion by the amount SlVUog? = (log ?)2$, = iVlogi’ Compaung this with the increase in entropy, we see that oq. (256) holds for a companson of the initial and final states If, however, we dnect oui attention to the act of rupliuing tho capsule, we eneountei difficulties The act of ruptiuo converts tho existing state of the gas from a fairly piobablc to a highly impiobablo one, but without any conesponding decieaso in the entiopy (to lowoi the entropy of a body without abstiaotmg heat from it is thoimo- dynamically impossible) The difficulty pcisists in another foim if, instead of rupturing the capsule, we allow the gas to oxpand adia- batically against a piston; foi then it is impossible to compaio tho probabilities (in the ordinary sense) of tho two states of tho gas with the piston in different positions, each state being an impossible one, owing to the difference in eneigy, when the piston is in tho other position Apparently the only way to savo the Boltzmann principle m the face of external inteiventions such as these is to lay down a suitable artificial rule specifying how the sizo of tho colls in phase space is to be altered when such actions occur If this is done, how- ever, an arbitrary element is mtioduced into tho "probability” which obscures the significance of the formula Planck and otheis have endeavoicd to avoid thoso difficulties by pointing out that the quantity, — SiV, log in tho oxpiession for S is simply the vanable part of log M, wheie M is the numboi of ways of arranging the molecules in thou most piobablc distiibution, as given by (247) m Sec 198, so that we can wnto, dioppmg a constant term, S — k log M The quantity M, to which under certain conditions tho tiuo pi oil- ability is proportional, is then called the "thermodynamic prob- ability ” This, however, is meiely giving to M a now name, and it is not clear what has been gained by this pioccdure, The principal field for the Boltzmann formula appears to bo tho irreversible behavior of systems when left to themselves It is often employed to assign a value to the entiopy undci such conditions that Sec, 2I2| STATISTICAL MECHANICS 371 the ordinary thermodynamic definition of entropy cannot be applied* An alternative, however, which has much to recommend it, is to regard the entropy as a strictly macroscopic quantity, devoid of significance on the molecular scale, and possessing meaning only when the thermodynamic definition can be applied, at least to the parts of the system taken separately, 211, Relations with Boltzmann’s H. In the case of a point-mass gas there is also a close connection between the entropy S t the prob- ability P, and Boltzmann's TL For in this case we can write Ni - N(h/V)fdK = Ngf/V ; here N is the total number of molecules in the volume V, f is the ordinary distribution function for velocity as defined in Sec, 17, and g is the volume of a coll in molecule space, whose extension in velocity space is Sk and in ordinary space 5r, so that g ~ Mk. Then , log Ri = (N/V)^f log / St Sk + log ( Ng/V ) = i N Jf log / (k + N log (Ng/V), and tho expressions found above for log Pn and for S can be written log Pd - —NT1 + N log V + const., (267a) S = -Nkll + Nk log V + const. (2676) in tonns of II, as defined by eq. (70a), or II = Jf log / ch. Thus tho proof given in Sec. 32 that II decreases carries with it a proof that, if molecular clmos holds, the probability Pn will increase toward a maximum, and likewise tho entropy S in so far as (2676) is assumed to hold. These conclusions aro again only statistical in nature, however. The condition of molecular chaos is itself subject to departures in the form of fluctuations, so that tho proof refers merely to the expectation of a change in II. If the gas is far from equilibrium, a probability approaching certainty exists that II will decrease and that P n and <S will increase; but as equilibrium is approached, the fluctuations bocomo relatively more important, and finally in the state of equilib- rium II oscillates in Brownian fashion about a minimum value, and Pd and S about a maximum. 212. Entropy as a Measure of Range in Phase. If wo wish to find some means of lending greater concreteness to tho notion of entropy, we might perhaps secure it by regarding the entropy as a measure of 372 KINETIC THEORY OF GASES [Chap IX the range m phase of the system. Gieatei cntiopy goes with a greater ranging of the molecules over molecule space, as measuiocl by an increase in Si?, log (1 /A',), 01 m the avciago value of log (1/J?,); or, if we are using the canonical clistnbulion, thcie is a similai incioaso in phase range as measuied by the mcieaso in the avciago of log (1/P) [cf (255)] A non-equilibnum state is then one m which full use is not being made by the system of the phase-space langcs that aro open to it undei the conditions to which it is subject, so that its behavior exhibits less phase lange than in the state of cquilibiium 213. Relativity and Statistical Theory. Up to this point we have worked exclusively with nonielativistic or Nowtoriian mechanics, This is adequate foi all of the piactical pui poses of kinetic theory; but it is of theoretical intei est to look foi a moment at the modifications that are required by lelativity, Foi particles in a fixed force-field the only change loquired is that we must write for the mass of a paiticlo moving at speed v , m to = . „ = -) ■\/l — i> 2 /c 2 where c is the speed of light in vacuo and m the constant mass of the particle at zero speed Now, even foi liydiogon at a tempcratuio of a hundred million degiees the root-mean-squaio speed v, is only 11 X 10 8 cm/sec or 0,004 c, so that to' = 1 000007 m Cleaily, therefore, the relativistic variation of mass is quite neghgiblo for ordinary atoms undei almost every conceivable cucumstanco The same statement holds good for flee electrons moving as molecules on the earth, but at a temperatuie of 10 s degrees, for them v, — 6 7 X 10®, so that v,/c = 0 22 and the conespondmg mass is to' — 1 025 m Thus, if the tempeiatuie should go so high as that in tho interior of a star, the increase of mass of fiee elections would begin to become appreciable. Accordingly, we shall add only tho following lemarks concerning the relativistic theoiy. If as p's we take tho components of tlm momentum m'v, the Hamiltonian equations of motion still hold; everything that we have done oi might have done in tonns of momonta still holds good, therefore, including the phase-space thorny. Max- well’s law holds, provided it is stated in teims of momontum instead of velocity The scattering coefficient is slightly altoicd, both because of the variation of mass and because of tho effects of letar ela- tion of the forces, which aie piopagatecl only with tho speed of light; but this effect is no more appicciable than is the mass variation, Sec. 2X4] STATISTICAL MECHANICS 373 B* STATISTICAL WAVE MECHANICS We must now consider how the statistical investigation is to be conducted when wave mechanics is substituted for classical theory. It turns out that a decidedly different method of attack is required. 214, The Wave-mechanical Description, The fundamental phys- ical ideas of wave mechanics have been described in Secs. 75 and 76 and will bo assumed to be familiar, As a basis for our statistical work we shall now add a concise statement of the general mathematical principles, in so far as tliese will be needed for our purpose. This statement will be made essentially complete; but it must be admitted that any discussion of wave-mechanical theory can hardly be appre- ciated properly unless the reader is already somewhat familiar with the mathematics of the theory of a particle. A system of N fundamental particles (electrons, protons, etc.), which could be represented in classical theory by its 3 N cartesian coordinates qi • * * tfa.v and its 3 N momenta p\ • * * pa*, regarded as functions of the time, is represented in nonrelativistic wave mechan- ics by a wave-function, 'I'(<7i * * * <7sjy, • • • sn, t) } which contains as variables, besides the time t } the coordinates and the spin symbols • • * Sjy of all the particles, but not their momenta. dqidqt • • • dqw t summed particles, is considered to represent the probability that an accurate determination of the values of the coordinates would yield a value of the first one lying in the range dq l} of the second in dq i} etc.; for this reason 'I' is often called a probability amplitude. It follows that must bo normalized so that |'I'| 2 dqi • • * dq* N — 1, the integral it extending over all values of the q’s. As tho analogue of the ordinary equations of motion, wo have then, as a basic postulate of tho theory, the wave equation h d'l' . _h_ _d_ 2n dt + \2n dqi _h 3_, 2 ri dqw* 19 <hvi Qi * Qwl ; <)'I' - 0; (258) here h is Planck’s constant and H is the Hamiltonian operator, which contains in general the spin vectors di • • • <i,v of tho particles (if wo employ Pauli’s approximate theory of spin). 374 KINETIC THEORY OF GASES [Chap IX There aie many pioblems in which the opeiator II does not con- tain the time explicitly, this is tiue, m fact, foi any isolated dynamical system In such cases a step towaid the solution of the wave equation can at once be taken by the usual method of sopaialmg Gho vanablos As a result of the mathematical analysis it turns out that the general solution of (258) can then be written down in the form, 2*1 Ent & - ^a n e * \J/ n (qi q aN) Si - s,v), (259) n where the a n 's are aibitiary constants and the aio the various solutions of the amplitude equation H+n - finin', (260) the constant E n must be limited to such values that any T formed out of the as m (259), can be normalized to unity This condition will be assumed heie to limit £ to a discrete set 01 “spectrum” of values, only when this is tiue can the system exist in a .state of equilib- rium It is m this way, of couise, that spectioscopic energy levels mo calculated The functions \p n and the coi responding allowed values of E n mo the chat acteristic 01 pioper functions (01 eigenfunctions) and values (or eigenvalues) for the opeiator II This opoiatoi usually iopio, seats the eneigy, and foi gi eater concieteness we shall hcncefoilh assume that it does States of the system m which 'I' contains only one e g , one represented by _ 2m Ejt 'P = e h are often called quantum (or eigen) states We shall assume the ^,,’s themselves to be normalized by a piopei choice of the aibilraiy factor that enters into any solution of (260) m such a way that JV»N<7i ' dqtw = 1. When this is done, the constants in (259) can be shown to have the significance that |a„| 2 lepiesents the piobabihty that observation would reveal the system in quantum state n, with energy exactly equal to E n The formulation just described is adequate foi tho genoial tieal- ment of an isolated system If, on the other hand, wo wish to include also the influence upon it of its surroundings, we can often do this by writing H = Ho + Hr, (261) Sec. 216 ] STATISTICAL MECHANICS 375 where II o represents the Hamiltonian for the system by itself and does not contain the time, while Hi represents the influence of the sur- roundings. Then we can still write 'I' for the system in the form of a series as in (269) if we wish, the yVs being solutions of Hc'/'n = En'J'n, but the a„’s will now be functions of the time ; their changes with time, which are determined by II i, can be interpreted as representing, in a certain sense, transitions of the system from one of its own quantum states to another. This is the method by which, for example, the absorption of light is treated. 216. The Exclusion Principle. There is one other peculiar feature for which allowance must often be made, viz., exchange degeneracy. The Hamiltonian operator is always symmetric in the coordinates and spins of any two similar particles; and for this reason, if in any solution \j/„ of the amplitude equation the coordinates of any two particles are interchanged, the result is again a solution of the equation, and one corresponding to the'same energy. Thus, in the case of an A'-dimen- sional system, there may be for any particular value of E n as many as N\ independent ^,,’s. Nature is not really so prodigal as this mathematical fact suggests, however, There is always one of these solutions which is anti sym- metric in the particles, i.e., which changes sign whenever the coordi- nates of any two particles are interchanged; and for some unknown reason only solutions of this type alone can represent physically pos- sible coses. 'I' itself must, accordingly, always bo antisymmetric in the coordinates of similar particles. This requirement is commonly called the exclusion principle; it represents an extension in wavc- meohanical terms, made by Fermi and by Dirac, of a more limited principle proposed in the old quantum theory by Pauli. 216. The State of Equilibrium. From what has been said it is obvious that in statistical wave mechanics no immediate significance can be attached to the old phase space, in which half the variables were momenta. We must work primarily, not with p's and ff's, but with the quantum states. The analogue of a classical system moving on an ergodic surface with energy E seems, at first sight, to be obviously a system in a single one of its quantum states [o.g., in (259) a m = say, all other a„’s = 0]. At this point, however, wo encounter a strong contrast with ordinary atomic theory. Whereas an atom or molecule may often be regarded as oxisting in a definite quantum state, it appears from the general principles of wave mechanics that the large bodies 376 KINETIC THEORY OF GASES [Giup IX dealt with in statistical mechanics must always be legal decl as being m a state in which many a n ’8 aie diffeient fiom 0, so that the system is, as it weie, smeaied over a laige number of quantum states Tho eneigy is then fixed only within a ceitain lango A2?, which may, perhaps, be macioscopically inappieciable, but which is yet veiy laige relative to the spacing of the cneigies of the quantum states Regarding a system in equilibnum m this light, wo aie at onoo confionted by the question as to the lelative values that must bo assigned to the vanous a n 's This pioblem appeals to bo tho analogue m the new theoiy of the old ergodic question Unlike the classical problem, howevei, this one does not seem to bo answerablo oven in principle on the basis of the equations of motion alone; for tho value* of the a* s lepiesent the initial conditions with which tho system was started, and so appeal to be aibitiary To thiow light on the question, let us consul ei how wo would tieat in wave mechanics a system that is started in some special physical condition diffeient fiom equilibrium, like a gas with gradients of temperature and piessuie in it Initially the "system, assumed iso- lated, can be repiesented by n with suitable values of the constants at, a 2 will be lepresented, as in (259), by 2th E n t n the correctness of these values of the c„’s can bo verified by sub- stituting this value of "i' in (258) and using (260). We note that na time goes on, the coefficients c n do not change m absolute magnitude, since |c n | = ]a„| and so is constant, this fact, corresponds to tho classical circumstance that the systems of an ensemble remain on then respec- tive ergodic suifaces The c,/s do change, however, in phase, (i.e , in the angle of c„ as a complex number) * Ifc is, in fact, obvious that if any two E„’s aie unequal, the relative phase of the couesponchng Cn s will tiaveise at a unifoim rate all possiblo values; if tho E„’s mo also incommensurable, the phases will finally come to bo dislubuled chaotically Wo can write c n = |e B ](* , f | f boing a real number; then £ is the phase angle oi phase of c„ The “relative” phase of two c„'s is, of com so, tho tllffoicnoo of thou f 8 Thoroaftci it (262) Sac. 217] STATISTICAL MECHANICS 377 Now, from the macroscopic standpoint, what happens in the long run is that a system generally passes into a state of equilibrium. This observation suggests the hypothesis that chaotic relative phases of the c n *s constitute the essential characteristic, from the wave- mechanical viewpoint, of the equilibrium state. Accordingly, wo shall forthwith assume that a system in thermal equilibrium can bo represented by a SI', as in (262), in which an enor- mous number of c n 's occur with values differing appreciably’ 'from zero, but that the relative phases of these c n ’s are distributed chaotically. In the light of this hypothesis, our remark that in the case of incommensurable the phases must eventually become chaotic constitutes an actual proof, based on the wave equation, that equilib- rium will, in general, come into existence automatically. In this simple remark we appear to have the physical equivalent, extended now to systems in general, of Boltzmann’s II -theorem for point-mass gases in classical theory, 217, A Priori Probabilities, From the principle just laid down it can bo shown that the statistical properties of a system in equilibrium can be found simply by averaging the properties of the various quan- tum states, each one weighted in proportion to | | 2 or |a„[ 2 . For example, the average result of measuring any quantity represented by an operator Q is, according to a rule of wave mechanics, Q « JV<2'I' dq - £M 2 / dq + %X Cn * Cm f '***«*» d( t' « n m Here V* denotes the complex conjugate of ty. The integrals f 'Pn*Q'Pm dq may bo expected to have similar values for many different pairs of quantum states, but they will eventually decrease as the two states of a pair become very dissimilar; and at tho samo time the chaotic phases of the c„'s will cause c„*c,„ to take on the negative of any given value as often as it does the value itself. Hence the double series on tho right will be negligible compared to the single one, and we can write, for the equilibrium state, » Here the intogral J^n*Q^ndq represents the average of Q for tho system when it is in its nth quantum state, and, since ^|c rt | 2 = 1, 378 KINETIC THEORY OF GASES [Ciiap IX the whole expression lepiesents the weighted average of Q over all quantum states The problem of the equihbiium state thus 1 educes to tho question as to the piopei values to be assigned to the a^s, To fix these value's some new pnnciple is necessary; and in wave mechanics, in contmst with the situation m classical theoiy, it appeals that this piinnplo must take the foim of an additional fundamental postulate which is coordinate* m status with the wave equation itself, or at least with tho lules for the pkj'sical intei pi etation of 'I' Apparently tho coil cot assumption is to lequire all of the af s in the lange AE to be givcm the same value This is equivalent to lepiesentmg a given system by an ensemble containing one system in each of these quantum stales; and accordingly we can foimulate the new pnnciple in tho following way* Pnnciple of A Piion Probability The statistical piopei tics of an isolated system with maci oscopically definite eneigy E aie } in gcneial, those of an ensemble having one system m each quantum state whose energy is contained m a lange A E about JS i this lange being maci oscopi- cally small but covering a large number of quantum slates, Oi, quantum states belonging to such a range AE are to be treated, in geneial , as equally probable independent cases , the statistical featiues of tho given Hystom being obtained by avei aging ovei these states The quantum states contemplated heio and in all of tho pi needing discussion aie the fundamental states of unit multiplicity If, how- ever, some of these states have the same eneigy, wo can, if we piefor, group them into a multiple state, ancl such multiple states can then be employed as the statistical unit provided wc assign weights to thorn equal to their respective multiplicities oi numbcis of component states If then, as a special case, all of these multiple states have tho mimo multiplicity, as frequently happens when the multiplicity arises from nuclear causes, their weights can bo ignored after all and they can 1)(3 treated as if they weie simple In the historical development this piinciplo was auivod at oilg~ mally by noting that, in a ceitain sense, each quantum state of ail s-dimensionai system corresponds to a tegion of volumo h* m tho classical phase space, m which equal legions repiosentcd equal prob- abilities Some suppoit can be lent to it fiom tho wave equation by showing that the effect of an external disturbance upon the c n *H is such that a system initially in a state n is caused to pass into another slate j at the same late as one initially in j is caused to pass mto n, ho that no state appeals to be favoied ovoi any other; but it icniams essen- tially a new postulate Sue. 2181 STATISTICAL MECHANICS 379 Just as in the classical theory, the principle is subject to exceptions in special cases, for example, in the case of a gas in a smooth sphere as described above in Sec. 193. We shall assume, however, that it holds in general; and henceforth, we shall confine our attention to those cases in which it does hold. 218. The Many-molecule System without Interaction. Following the line suggested by the classical argument, let us now consider a system that is composed of a large number N of exactly similar and only slightly interacting subsystems, all of which are confined in the same way to the same region of space. An ordinary gas is an obvious example of such a system; but the subsystems may also be bare fundamental particles such as electrons, or they may have any degree of complexity. The restriction that the same region must be acces- sible to all subsystems, in order to make them exactly similar, is a modification of the classical idea that is required by wave mechanics; this restriction makes it even more appropriate here to call the sub- systems "molecules.” Let us suppose, first, that there is no interaction at all between the molecules. The problem is then so simple that we can actually construct the characteristic functions for the system as a whole by building them up out of one-molecule functions, in the following way. Each molecule now moves independently in the containing vessel, and there will exist for each of them a series of one-molecule quantum states; lot the corresponding antisymmetric characteristic functions and energies for ono molecule be Uj and e,-, the amplitude equation for «/ being Il'uj = (203) The amplitude equation for the entire system is then - EJ n , K** 1 where 11 J means IV written in terms of the nth sot of molecular coordi- nates and ML' takes the place of Ii in (200). A solution of this equation is obviously the product function* tt/,(l)t{ft(2) • • • «*?(#), each number in parenthesis meaning that the coordinates in the function aro to bo those of the molecule bearing that number; the ♦For ff/u h (l) • • - u w (N) = «/,(!) * • • «/(,-■>(* ' • • 1 380 KINETIC THEORY OF GASES [Ciiap IX conesponding energy is -Bn “ + €?a + * * 1 + (264) To obtain an antisymmetiio function foi the whole system cor- responding to eneigy E n as given by (264), wo have then only to form the usual normalized combination of such pioduct functions: ± Pu n { l)u li (2) • u w {N), (266) wheie the symbol P is to be nndei&tood as producing in the oxpiossion following it some peimutation of the older of the sets of coordinates of the fundamental paiticles, these sots of coordinates being supposed wntten in a definite oulei in the onginal pioduct function, and the paiticles of each kind being pel muted only with cacli other; the sign is to be taken plus or minus accoidmg as the peimutation loaves an even or an odd number of pairs of the sots in mveitcd ouloi, and the sum is to be extended over all such permutations, whoso numboi is denoted by v It is readily seen that a \f/ n so foimcd changes sign whenever the cooulmates of any two similar paiticles are intoi changed. As a very simple example, suppose there arc just two molecules, each containing one election and one proton; let tlio coordinates of the electrons be denoted collectively by x x and x* and those of the protons by y x and y 2} respectively, Then the initial pi oduct function is «n(® i, ffa), and the antisymmetric combination, accoidmg to (266), is i/'n = (4)~> 4 [w ?1 (aji, yi)u n (x 2> y 2 ) - u n {x 2) yi)u n (x h y *) - Un&h V* )u n {x^ y x ) + u n {x 2} y*)uu{x\ } y{)} The ordei of the u/s in any product function is immaterial; each combination of u t *s yields just one independent In tho simple example just described, for instance, u n (% i, y\)u l7 (x 2} y 2 ) and WjjG'Ti, y\)u n {x 2} y%) lead to the same This reduction m the number of independent i/^'s for a system of similar particles is a direct conscquonce of tho exclusion principle 219, Fermi-Dirac and Bose-Einstein Sets of Similar Molecules, There is another aspect of the reduction just mentioned which depends upon the number of 'particles m the molecule } and which solves to divide sets of similar molecules into two classes. Suppose that two uf s happen to be the same; i e , tho same one- molecule function is repeated two or more times in the product f uno- Sbc. 219] STATISTICAL MECHANICS 381 tion. Then, when we form an antisymmetric by combining product functions as in (265), a permutation which consists only in inter- changing the coordinates of similar particles between the two identical u/a docs not really alter the product function. Suppose, now, that each molecule contains an odd number of particles; such a system is said to be of Fermi- Dime type, which we shall abbreviate to F-D. Then any interchange between two mole- cules reverses the order of an odd number of particles; for example, if in the sequence ABODE molecules A. and E are interchanged, an odd number of particles in A paired with those in E are reversed in order, also the pairs formed by particles in A and E with those in the intervening molecules BCD , but the latter pairs are necessarily even in total number, and hence oil the whole an odd number of pairs of particles undergo reversal. Accordingly, the new term formed by tire interchange of two molecules is opposite in sign to the original. It follows that all the terms in the sum cancel each other in pairs; and honco this particular i p» is identically aero and must be discarded. If, on the other hand, the number of particles in a molecule is even, no such loss of a occurs when two or more u/a are the same. Such a system is said to be of Bose-Einstein (or B-E) type. Each formed as just described may obviously be regarded as specifying a distribution of the N molecules among the one-molecule quantum statos, N i being assigned to the first state if the correspond- ing one-molecule function u, occurs Ni times in each term of similarly to the second state if tta occurs Ni times, and so on. It is not specified, however, which molecule is in each occupied state; the molecules, like the fundamental particles, arc not completely individualized in wave mechanics. In the B-E case there is no restric- tion upon the values of the N/ s; in the F-D case, on the other hand, each Ni » 1 or 0, or "no two molecules can occupy the same quantum state." Up to this point we have assumed the molecules to be all alike. The extension to a system containing several different kinds of mole- cules, each very numerous and moving perhaps in a force-field of its own, is so easy that no equations need be written down. In the absence of interaction, the Hamiltonian operator II in eq. (260) is simply tho sum of several terms, one for each kind of molecule; and the solutions of the equation are simply the produots of the ^„a for tho separate kinds. The final wave-function must then be antisymmetric oven for interchanges of similar fundamental particles between molecules^ of different kinds. When we secure this condition, however, by building 382 KINETIC THEORY Ob' OASES [CirAi> IX up 4> n out of product functions, as m eq (265), it is now impossible for two w,’s to be identical m foim, and hence thcie can novel’ bo such a loss of in’s as we found to occui with similar molecules Each \p n foi one land of molecule, combined with each \p'„ for ovory other kind, yields one independent zeio-oidei function for the wholo system The process of antisymmetiizing between molecules of dilToiont kinds can, accordingly, be ignoicd The theoiy of similar molecules is thus to bo applied separately to each homogeneous set of molecules that may be picsent in the system In a mixed gas, foi example, some molecules may bo of DVD type while otheis aie of B-E type, those of each type will then foim a gas exhibiting the behavior chaiactenstic of that typo, just as if tho otheis were not piescnt 220. The Loosely Coupled Many-tnolecule System. Having con- stiucted in this mannei chaiactenstic functions foi the system with the molecules moving independently, let us lointroduco the slight mteiaction that was ongmally postulated to exist between them (Sec 218), so that in (200) we must wute ii - s)j/' + ii", H " repiesenting the intei action Then we can infei fiom the usual perturbation theory that the chaiactenstic functions foi the whole system will differ a little from \p n and E n as given by (265) and (264) (with the usual pioviso in case of degeneiacy), but foi a zcio-oider approximate troalmont, which is all that will be contemplated heie, this diffeience may bo neglected* Hence we may continue to woik with the zeio-ordoi \p rt J s that wo have already constructed just as if these weie the accurate characteristic functions for the actual system As a mattei of fact, they could also be used, although m a somewhat diffeicnt way, even whon tho in tor- action is large; foi each chaiactenstic function can always be expanded as a senes in teims of oui zero-oida & n% c g , wo can write for tliom = ^bi n \(/ n When this is done, it can be shown that, if jV/» is tho n numbei of molecules lepiesented by t ]/„ as being in state 3 , repre- sents molecules as being m this state n 221. Statistics of the Loose Many-molecule System, From what has been said it follows that the principle of a prion probability (See, 217) can be applied to the zeio-oulci quantum states icpioscntod by Sec. 221] STATISTICAL MECHANICS 383 our Vs as given by (265) above. Accordingly, when a system com- posed of similar molecules is in equilibrium, with energy confined to a small range A 1$, we may regard as equally probable each distribution of its molecules among their zero-order quantum states, i.e., among the u/s. Let N , denote the number of molecules in the jth. state; then each set of values of the N/s represents an equally probable case. As a statistical feature let us seek the mean number or expectation of molecules in each of these quantum states when the system is in equilibrium at a given temperature. As before, suppose first that the molecules arc all alike; let their number be N. All distributions being now equally probable, it is hopeless to look for a most probable or “normal” one as we did in the classical case. The customary and simplest way of overcoming this difficulty is to redefine the term “distribution” in such a way that all distributions will n<T not have oqual probabilities. In order to be able to do this, we must further restrict the typo of system under consideration by supposing the one- molecule states to lie so extremely Fla “ 80 ‘~ A Formi - Dirfto distribution. close together that, after dividing them into groups of v T states each, we can suppose the r T 's to be large and yet can neglect the variation of the energy within any given group of states. Let us denote the number of molecules in the rth group by n r , Wo can then ask for the probability of any particular set of values of the Vs, regarding each such set as representing a macroscopic distribution. According to wlmfc was said above, this probability will bo proportional to the number of ways in which the molecules can be parceled out among the molecular states with ni falling into the first group of states, ?h into the second, etc,, with no attention paid to the individuality of the molecules that fall into a given group. In the F-D case, in which only one molecule is allowed per state (cf. Fig, 86), this number is simply the number of ways of selecting n\ out of the first group of v\ states to hold ni molecules, n% out of the socond group of r 2 , and so on, or J'll v%\ - ni ) ! n 2 l (r 2 - ni) \ with tho convention that 01 ® 1. To find the corresponding number in the B-E case (Fig. 87a), imagine the n T molecules that are to fall into the rth group laid out in a row and v T — 1 partitions inserted to divide them among the 384 KINETIC THEORY OF OASES (Chap, XX v T cells, seveial partitions being allowed peihaps to fall together (Fig, 87 b) The number of ways of anangmg tho n r + v r — 1 molecules and paititions in ordoi m tho low is (n T + v r — 1)1; but we must divide this by nrl, the numbci of ways of permuting tho molecules among themselves, and by (v T — 1)1, the number of ways of pei muting the paititions among themselves, sineo any suoh poririu- n T = 3 7 ( for v T »5, nr “ 6 o|o|o o|o|o or- 0||0 O 0|0 0| v T = 4 6 II eh= (a) (W Fia 87 — Tho Boso-Einatom distribution* tation does not change the distiibution of tho molecules among tho states Hence m the B-E case m — ( y i dl V± H jOj ( y g ^2 — 1)1 — 1)! ^22 1(^2 — 1)1 Now let us suppose that many of tho Ur * s arc laigo, and lot us apply to them Stilling's foimula, eq (2146) in Sec 164, or log n = (n + y 2 ) log n — n + log here we will at once suppress % m companson with n, and also lump togethei terns which aic independent of tho n/s either because — N 01 othenvise Then we can wuto, with aceuiacy so far T as the large n r ’s aic concerned, in the two cases: F-D‘ log M = — ^n T log iir — — n T ) log (v T — n T ) + const , t T B-E’ log M ~ -]£n r log fir + ^(p T + fir - 1) log (v r 4* «r ~ 1) t r *f const, Proceeding then exactly as in Sec 199, wc seek thoso values of n r which make M oi log M a maximum, subject to tho two conditions that = N, - E, Sec. 222] STATISTICAL MECHANICS 386 *t being the energy of a state in the rth group and E the total energy Fox’ these values of rw, if large, we find : F-D: - log n T + log (r, - ?v) - « - 0« r = 0, — ~ 1 = <A+“ Tlr B-E: — log nr + log (r, + n r — 1) — a — 0e r = 0, ~ + 1 = &****, 7l r after dropping l/n r for the sake of consistency with previous approxi- mations. Lot us now write for the mean number of molecules per state in the neighborhood of state j t whose energy is € ; -j we can identify N j with n T /v T for the group containing this state, and its energy e,- with the value of 6 t for the group. Then the last equations give, respectively: */- -f- 1 $ i ” Qpti+Ct __ T[ (266a, 6) When the molecules are of several kinds, there being many of each kind, ono finds easily that a formula must hold for each kind which is like (266a) or (266/;) according as the number of particles is odd or even in that kind of molecule; a may vary from one land to another, but /3, as in the classical case, must bo the same for all. 222, Introduction of the Temperature. The constant /3 can now be connected with the temperature by an argument which parallels so closely the classical ono given in Sec. 202 that we need mention only the differences. The «/s and u/s arc here functions of the parameters which were there called a's, for the latter will occur in II' in the onc-molecule amplitude equation, (263). During the small thermodynamic process contemplated in Sec. 202, any system can be represented by a T 1 of the ' form given in eq. (262) above, but the a„’s themselves will now be functions of the time. The number of molecules in state j at any instant is then Nj - where N jn denotes the number that n are in that state when the system is in its quantum state n and |a*| 2 or | c „| 2 represents tho probability for the oceurrenc6 of this quantum state. If the a’s mentioned above are changed extremely slowly, it follows from a theorem of wave mechanics (the "adiabatic” theorem) that this change does not appreciably alter the an’s; changes in the a„’s and the Nj'a arise hero, therefore, only from the action of heating forces. Accordingly, if there are no heating forces, the change in the energy, which latter can be written E — is dtf t and this J I 386 KINETIC THEORY OF GASES [Chap IX is the negative of the work done by the system If, then, heating foices do act, the woik is unaffected by then picsenco to the fiist oidei Hence we can wnte foi the woik done diuing the infinitesimal piocess undei consideiation dw = (267) 3 The leveisiblo heat ab&oi bed duiing the piooess is then, in analogy with (261o), dQ = dE + dW = + dW, or 3 d Q = 2[e, dft, (268) Now, dining this piocess we assume the system to lemain close to equilibuum, so that N, is given by (266a) oi (2666) Both of these equations, like the classical one, aic special cases of the moie general foim, N, ~ F ,(0e 7 + a), the F/s being ceitain functions Whcnevci has such a form, oq, (268) multiplied through by can be wntten, since ^ (IN, =* 0, 3 P dQ ~ ^(0€, + a) dN , = ^(/3e, + «) dF ; (/3e, + a) (269) 3 3 Heie the last mcmbci is obviously the differential of a quantity which has a definite value corresponding to each state of the whole system Hence we can, as bcfoie, wntc /3 =* 1/kT in tenns of Boltzmann's constant h and the theimoclynamic tempciatuic 5f\ The final distubutiou fonnulas (266a, b) foi a loose many-molecule system composed of similai molecules can, aecoulmgly, be wntten R, - ' R» = ■» (270a, b ) Be w + 1 Be w - 1 the first of these equations refcning to the Fcimi-Diiac oi odd- paiticle case and the second to the Bose-Emstcin or even-pai tide case Ileie B takes the place of e« above and B and T aic dotoz mined by the two conditions that X N , = N, X^> " B > 3 ) Sec. 223] STATISTICAL MECHANICS 387 N is the total number of molecules in the system* and E its total energy (the slight molecular interaction being ignored) 3 and the* sums extend over all of the molecular quantum states, AvS in the classical case, the distribution represented by these formulas is to lie regarded, not only as the most probable one, but also as normal in the sense that, appreciable departures from it are rare. Strictly speaking, the formulas are limited to cells which lie within a group containing a large number rw of molecules, but this limitation is of no practical importance. 223, Case of Large Energies : Classical Theory as a Limit Form. The exact significance in any particular case of the formulas that have just been obtained will obviously depend upon the law according to which e/ varies from one molecular state to the next; and this in turn will depend upon the nature of the external force-field to which all of the molecules are subjected, The special case of the ideal gas will be taken up in the next chapter. There is one general ease, however, that of large energy, in which both formulas approximate to a simpler form that is very important. In order to make the total energy E largo, the molecules must obvi- ously move out for the most part to states of high energy. Accord- ing to (270a, &), however, is such a function of c ? - that it can never be smaller for any state of lowor energy; hence E can be largo only if the molecules spread so widely that the iV/s all become small, and this means that for all values of «/ Be& » 1 . We can, therefore, write approximately for either a F-D or a B-E systom, when E is large enough, => Ce^\ (270 c) whore 0 » l/B, If all ey > 0, 0 « 1* This very useful formula is just like the Boltzmann formula of classical theory, eq, (249d) in Sec. 201, except that here reference to a particular quantum state replaces the mention of a cell in phase space; historically, this formula was inferred many years ago as the appropriate quantum modification. Wo note, however, tlmt it represents only an approximation valid for energies sufficiently high to justify neglect of the characteristic peculiarities of the more exact wavG-meehanical formula as given in (270a) or (270&). 388 KINETIC THEORY OP GASES (Cuai> IX 224. Entropy of a Loose Many-molecule System. The entropy can be found by putting j3 = ^ in (269) and evaluating J* -jr Aftei an integiation by paits, we thus find foi it the genoial oxpi muon, -s = - k Xl dx> } J wheie In the actual cases befoio us we have, wilting B = e* again in (270a, &), ft: - fM - r the uppei sign icfeiiing to the F~D case and the lower to Iho B-K Hence here J P , (“»,) dx } = J — f ^ j = + log (1 ± er*i) H- const , ~ + log (I ± hr ) + const since e« = B Noting that ~ E t the total energy, ancl j ^F, = iV, whereas a =* log J3, \ve have, thoiefoic, if wo omit the constant of integiation, 8 - | + JV*, log J3 ± *-2 *°e (1 ± (27 la) J An equivalent expiession m tcuns of the moan densities can also be found by substituting, in the onginal expression for S, and in the expiession found foi jFtix,) dx„ (1 ± e~*0 - 1 ± (±r T lV 1 = 1 ± — - \N, J 1 q We thus obtain T Ni 1 T <5 = ^2^' lo S (j^ + x ) T ^2 log (1 T (5J716) Sec. 226] STATISTICAL MECHANICS 389 or S — -Ic^Rj log hF fc]^(l + &i) log (1 T #,-). (2716') In the classical limit discussed in the last section, where JV'y is given with sufficient accuracy by (270c), and all 19,’s are small, these expressions become approximately S = jp + NklogB + Nk = lo S ft* + Mb, (271c) J as may be verified by expanding tlio logarithms in (271a) and (2716') _ </ and keeping only first powers of N ,■ or B~ l e These expressions agree with the classical ones as given in (252a, 6) in Sec. 203, except that here So is given the special value Nk; physically, the latter difference is of no significance because it concerns only an arbitrary constant. It may be remarked again that, contrary to our previous usage in dealing with gases, S stands here for the whole entropy of the system, not for the entropy of a unit mass, 226. Statistics of Mixed Systems. The results that have been obtained for homogeneous systems are easily extended to loosely coupled systems in which there are several different kinds of molecules, many of each kind being present. In Sec. 219 wo have seen that in the zero-order approximation, in which the slight interaction is ignored, each kind of molecule can be treated as if the others were not present; interchange of coordinates between molecules of different kinds need not bo considered and the quantum states for the whole system can bo formed simply by taking all possible combinations of the quantum states for the separate groups. Accordingly, when we employ the method of ono-moleculc states, wo have to consider all distributions of the separate kinds taken independently. The mode of generalizing the calculations of See. 221 follows so closely the lines of the parallel process in classical theory, as described in Sec. 201, that few details need be given. The number of arrange- ments M is simply the product of expressions like that written in Sec. 221, one for each kind of molecule, and its logarithm is the sum of corresponding terms. There is a separate equation of the type, = N, for each kind, but a single equation expressing conservation T of the total energy E. The result is then a distribution law of type (266a) or (2666), or (270a) or (2706), for each kind of molecule, 390 KINETIC THEORY OF GASES [Ciiap IX according as it is of F-D or B-E type, the constant a 01 B vaiios fiom one kind to another, but /? oi 1 /KT docs not In biief, each land of molecule is distubuted as if it alone wcio present, except that the common tempeialuio of all is dotoi mined by their total encigy The entiopy, likewise, is easily seen to bo simply the sum of the sepaiate entiopies 226. The Canonical Distribution m Wave Mechanics. In classical statistical mechanics we dealt not only with distubution laws foi the separate molecules but also with a piobabihly distribution in phase, called canonical, of the entile system It is natiual to seek foi something similar in teims of wave mechanics Now the lack of complete individuality of similar systems which lesults from the exclusion pnnciplc puts gicat difficulties in the way of an exact imitation of the classical aigument which led us abovo to the canonical distubution, so that wo scorn hoio to bo tin own back upon a moie indirect pioceduie We shall find in the next chaptci (Sec 232) that a pei feet gas of point-mass molecules, behaving as noaily classically as desned, can bo lealiAcd at any tompeiatuio by making the density low enough If such a gas containing a huge numbei iV of molecules is used as a tempoiatuie bath in slight contact with any given system, the statistical distribution of the lattei among its quantum states can leadily be found Since the nmeioscopio behavioi of the system can scaicely depend upon whether it is coupled to a gas oi not, it seems plausible to assume that wo may adopt its distribution when so coupled as the canonical one Let the enetgies of the quantum states of the system be E„ and of the gas, E[ Then the zeio-oider quantum states of the combined system foimed by the given system and the gas will consist of each quantum state of the one combined with each state of tho other, the, total eneigy being E, + E' K Each of these states whoso energy lies m a small iange SE can then be tieated as equally piobablc (Hoc 217) The lelative probability of each state of tho original system with energy E t will accordingly be piopoitional to the numbor dp' of states of the gas with which that state of the system can combine in such a way as to make a total eneigy lying within $E; for such states tho eneigy of the gas alone lies in a lango of width $E about tho valuo E' = E - E, Now, accoiding to eq (293) m Soc 233 of the noxt 3W chaptei, the gas has v v = XE 2 states below eneigy /£, X being a coefficient independent of E; hence it has v' = X(E - Ii’,)T 391 Sec. 227 ! STATISTICAL MECHANICS states below E — Ei. Accordingly, to SE there correspond. Sp'„ = ^X(E~ Ei) 2 SE states; and the ratio of the probabilities of two states of the original system with energies E i and Ez is, therefore, Here E consists mostly of the energy IS' of the gas ancl so is vastly larger than Ei) hence we can write with sufficient accuracy E = E' « NkT in terms of the Boltzmann constant k and the absolute temperature Then, writing log (1 — a) =» — as + ■ • ■ , wo have after dropping the factor (•} N — 1)/(| N), which is indistinguishable from unity. Thus the probability of any state of energy Ei for the -Ei first system is proportional to e kT and can be written Ei -Ei 6 ~Sr Pi = Ce ** = -2— g. (272) kT i the value of G being fixed by tho condition that — 1. I This is obviously tho exact analogue of the canonical distribution in classical theory as expressed by eq. (254) in Sec. 207 above. Pre- sumably it can be taken as a basis in quantum theory for obtaining tho statistical properties of any system in equilibrium. 227. The Entropy. From (272) one obtains readily, by the method we have several times used, us the wavo-meehatiical analogue of (255) in Sec. 208, for the entropy of any system nt temperature T\ & - log Pi= *J’ + k I(, S 2 e ~^‘ i i (273) 392 KINETIC THEORY OF GASES [Chap IX E lepresenting the mean eneigy, ^EJP X) and the additive constant X being set equal to zeio The last sum heie is Planck’s state sum, The first expxession for 5 given in (273) l educes at T — 0 to S *= A, log wo, where w Q is the degiec of dcgcneiacy of the state of lowest energy, or the numbei of fundamental quantum states having that eneigy; foi at absolute zeio P x — l/w 0 foi each of these states and P, = 0 foi all otheis Thus, accoichng to (273), the entropy of any system is finite at T = 0, as it has been widely infcired to be fiom the Nernst heat theorem For any given system it could be made actually zero, if not zero alieady, by means of a diffoient and lathci unnatural choice of the arbitiary additive constant m S , At this point we are in contact with the question, so impoitant in chemistiy, whethei all changes of entropy dining chemical transforma- tions vanish at absolute zeio This question lies enthely outside the subject of gases, however, and will not be discussed hero (but cf, Sec 236), The system under consideration may, of course, be itself a loose many-moleoule one, perhaps a gas In that case statistical mechanics gives us m (273) heie and in (271a, 6, c) in Sec 223 two diffoient expiessions foi the entropy which seem even to arise physically in diffeient ways If both forms are conect, they can differ m loality only by a constant In the next chapter we shall investigate the difference in the special case of a point-mass gas (Sec 234); in that case it turns out that, in so fai as the earliei formulas (271a, 6, c) are valid at all, they happen to agiee completely with the now one, (273), just as wntten Thus the same duplicity of method and of results that aie found in classical statistical mechanics persist in the wave-mechanical formula- tion The only important difference is that in wave mechanics similm particles aie not completely independent of each other, and consequently the canonical distribution of the system can be made to yield the distnbution law directly only for dissimilar molecules, not foi a group of similar ones CHAPTER X WAVE MECHANICS OF GASES Tho most important applications of wave mechanics to the theory of gases have already been discussed in Chap. Ill (Secs. 54, 55, 75, 76, 77). In that discussion, however, tho gas was conceived of as a collection of molecules moving about in space like classical particles, and wave mechanics was applied only to their collisions with each other or to their internal heat energy, and only in an approxi- mate-form,. This method is adequate for the treatment of most problems. For tho sake of completeness, however, the theory should somewhere bo formulated from the beginning entirely in terms of wave mechanics. This will bo done in the present chapter; and we shall at the same time consider the conditions under which deviations may bo looked for from the properties predicted by classical theory. 228. The Perfect Gas In Wave Mechanics. The perfect gas con- stitutes the principal example of tho loose many-molecule type of system whose theory has already been developed in Secs. 218 to 221 of the last chapter. To avoid repetition we shall build directly upon tho results there obtained. It is necessary to add only a special assumption constituting tho mathematical definition of a perfect gas; we shall find that it is then possible to work out the theory completely. External force-fields will bo assumed to bo entirely absent except for the restraining influence of the walls of the vessel. The effect of the walls can be represented by a potential-energy term in tlie Hamil- tonian function, and wo shall assume that this can be expressed with sufficient accuracy in terms of the centers of mass of the molecules alone. Then, when tho interaction, between the molecules can Ido ignored, tho functions for the whole gas can bo built up out of func- tions each of which represents one molecule alone in the vessel; and the wave equation for one molecule can be written, as a special case of (263) in See. 218, (K + IV, + Uw)itj = in which II' m is that part of the Hamiltonian which contains the coordinates of the center of mass, IV, is a term containing the internal coordinates of the molecule (representing rotation, vibration, etc.), and Uw is the potential energy between the molecule and the wall. 303 394 KINETIC THEORY OF GASES [Ciiap X We can suppose Uw to be zeio eveiywhoie except that it uses with extieme lapidity towaicl infinity when the molecule comes extiemcly close to the wall The effect of any such teim in the Hamiltonian can be shown to consist m foicing to vanish as Uw — > «> Accoidmgly, we can also drop Uw entirely and wute as the wave equation for one molecule simply (II' + H',)u, = ,,u„ (274) provided wo add the boundaiy condition that Uj = 0 at the walls of the vessel The limitation of the translatoiy motion only by a wall of this sort, together with the absence of interaction between the molecules themselves, may be regal ded as the wave-mechanical definition of a peifect gas, As in classical theoiy, the idealization is excessive, since with no interaction whatever thcie would be nothing to bring the gas into a state of equilibrium. To repiesent an actual gas we must suppose that a slight moleculai inteiaction does in fact exist; such a gas might be called, m conti ast with the absolutely peifect type con- sidered here, a physically peifect gas The mtci action will enter into the equations as a small perturbation and, in harmony with the explanation in Sec 220, will affect the quantum states of the gas only to a negligible extent The theoiy of the absolutely peifect gas, as we shall develop it, may accoidmgly be legal ded as a good zero-order approximation to the theoiy of a physically peifect gas In eq (274) the variables can now be separated, Substituting in it u, = ¥>* (a., y, z)%y, where ^ is a function of the caitesian coordinates of the center of mass of the molecule while Xv is a function of its internal cooidinates alone, we find that u 7 so defined is a solution of (274), piovided = Ww H f iXv ~ v»x», *? = {*/* + (276a, 6, c) Here ^ and ^ are aibitrary new constants which can bo logaidecl as repiesentmg, lespectively, special values of the translatoiy kinetic and internal energies The x/s lepiesent a set of internal quantum states of the molecule, including its vanous nucleai states, and will not be considetcd in furthor detail here, the deteimination of these functions and of the allowed values of ^ is taken up as the most impoitant pioblem in treatises on wave mechanics The howevci, picsent a pioblem that belongs to kinetic theoiy Sec. 229] WAVE MECHANICS OF GASES 395 229. The Point-mass Perfect Gas, It is simplest to develop first the theory of a gas in which all the molecules are in the same internal state; for convenience in the applications, however, we shall allow this state to have a multiplicity w, due perhaps to electron spin or to nuclear causes. As a special ease, of course, it may happen that w = 1. The function x* and its associated energy t] v can then be ignored and the theory becomes the same as that for a set of point masses. For the present we shall also suppose the molecules to be all alike. Equation (276a) can bo written explicitly in the familiar form /fl* , , 87r 2 m . _ ( . (5? + dlf + a?)* " ! ~ T " °» (276) where m is the mass of a molecule, h is Planck’s constant, and the first term multiplied by { — h-/%nrhn) represents TI' m <Pn. It can be shown that nothing of physical interest is changed if we give to the vessel a special shape in order to simplify the mathematical form of the y>„’s; accordingly, wo shall assume it to have the shape of a parallelepiped of edges l h h, h, ancl we shall take axes along three of these edges. Then the boundary condition stated above requires that <p» = 0 whenever the molecule comes up to the Avail of the vessel, i.e., when- ever x «= 0 or l\, or y - 0 or h, or z — 0 or l t . The only typo of function that satisfies all of these conditions is <Pi> C sin uiir r sin /*2 tt 4 - sin mv ■ h h ( (277) in which O' is a normalization constant, while mi, M 2 i Ma may bo any three positive integers. The corresponding kinetic energy is Ji* “ £ ]t(i if 8 m \ if + tA . m|Y n + iv (278) It is easily verified by substitution that any so defined is a solution of (270). In its mathematical form this solution recalls the classical standing oscillations of a solid rectangular block, and like the latter it can bo regarded as formod by the superposition of eight trains of running waves, each of which is continually being converted into one of the other seven by reflection from the walls. These eight trains can be expressed in the form 396 KINETIC THEORY OF GABES [Ciur X where* V* == Vv =* ±M2 <}]r> V* =* ±Pa 7 j^> (279) and hence p 2 ~ vl + vl + pi — 2mf Accoiding to the theory of a single paiticle, an infinite train of waves of this charactei lcpiesents the molecule as moving in a ceitam dn action with momentum p and kinetic eneigy f Henco wc can say, speaking loughly, that any <p J{ , of the form of (277) lepiesents a molecule as moving with kinetic eneigy ^ and with an equal chance oi being found going m any ono of the eight dnections whoso cosines aie piopoitional to ±pi/h, ±p 2 /h t ipz/U As to position, however, a molecule in a state lepicsentod by a single <pn is equally likely to be found m any pait of the vessel, aside fiom the charactei istie quantum phenomenon xopresented by the iapicl fluctuation of ^* 2 over distances of the Older of the molecular wave length, X — h/p An antisymmetuc wave-function foi tho whole gas can now be built up out of the one-moleculo functions ^ as a special case of (265) in Sec, 218, viz , tn - ± P«b( 1)^(2) • vM, N being the total number of molecules, the coucspondmg kinetic energy is then “b “b * fw Such a function may be regarded as rcpiesontmg a state of the gas m which, if N, is the numbei of times that any paiticulai ^ occurs lepeated in each of the pi oduet functions out of which \p n is constructed, then tlioio aie W, molecules moving with kinetic eneigy and momen- tum p = (2 and theie is no haim in imagining that ono eighth of them are moving in each of the eight diiections that we have asso- ciated with oacli <pn We cannot say which molecules aie so moving, however; they aie not individually distinguishable Then if the gas is in a moie general state, lopiosentcd, as in eq, (262) m Sec, 216, by a wave-function « * The ± 's in p Z} p v , p £ are independent and tho ± written m f lont is the product of all three of them Seo. 231 ] WAVE MECHANICS OP' GASES 397 we can. say that ^|c«|W^ n molecules are moving with energy Jv, N Itn denoting the number so moving when the gas is in state \p tv and |c„| 2 representing the probability of this state, In the particular case of a gas in equilibrium we can .to farther and assert that these molecules are^ moving equally in all directions; for, this statement holds for each quantum state represented by a single n, and, for systems in equilibrium, according to the principle of a priori probability laid down in Sec. 217, whatever is true for all quantum states must likewise be trap for the gas in its actual state. When the gas is not in equilibrium, on the other hand, the phases of the Cn’s cannot be assumed to be chaotic, and interference between the yv 5 s belonging to different ^ n 's may then result in nonisotropic distribution of the molecular velocities. 230. The Two Types of Point-mass Gas. In accord with the general principle explained in Sec. 219 homogeneous perfect gases with point-mass molecules will now fall into two distinct classes, the Formi-Dirac (or F-D) type with molecules composed of an odd number of fundamental particles, and the Bose-Einstein (or B-E) type, in which this number is even. In the F-D type no two molecules can be in the same quantum state; this means here that, if w is the mul- tiplicity of the internal molecular state, at most w molecules can have the same kinetic energy and the same associated eight directions of motion, one being in each of the w fundamental internal states. In the B-B type there is no such restriction. It must not be concluded, however, that in a F-D gas a molecule in one corner of the vessel cannot move in a certain manner if another molecule somewhere else is doing it, or that after a multiple collision two molecules cannot move off with sensibly the same vector velocities and energies. For statements of this sort to have a meaning, ^ must consist of a series, \j/ ® ^c n \p n7 in which the c»s for many quantum states differ from zero and also do not have chaotic phases; and then it can bo shown that the indotemrination principle described in Sec. 76 results in a sufficient degree of indefmiteness in the molecular velocities to prevent us from tolling with the requisite precision whether two molecules really do have identical velocities or not. 231. The Homogeneous Point-mass Gas in Equilibrium. The molecular distribution law for either type of gas can be obtained from (270a, b) in Seo. 222 by substituting the kinetic energy for e/, changing fy to and multiplying by the multiplicity w of the internal state. We thus obtain, us the mean total number of mole- 398 KINETIC THEORY Ob' OASES IClIAF X cules in the pth state, under cqiulibi mm conditions, (280a, 6) Be*>r -f 1 Be** - 1 the Uvo formulas xefemng to the Poimi-Dirac and to tho BonoTCinstcm types, lespectively (A, ~ the Boltzmann constant, T — absolulo tempeiatuie) Usually the molecules aie widely dhtnbuted among their trans- lational quantum states, and then it is convenient to treat tho onei&y as a continuous vanable Tho numbei of one-moloculo quantum states having energies m a lange d£ will be equal to tho numbei of positive integral sets of /i i, ju 2 , Us foi which f as given by (278) hos in di To find this numbei, let us take as new vaiiables tho components of momenta “ 2V m ^ 1 >\ 111 teims of wIncl1 f =* (Pi + Vl + vl) /2m = , /A\ P 2 /2??i. Then m all piaetical cases it turns S' \\ out ^»hat l\, U, l s aro extremely largo rolalivo z'' \ \ to /{/pi, h/p 2 , /(./ p 3 (the quantity h/p ropro- — -‘-I — sentmg the molecular wave length), at least in JT.O 88 -Molecular st l 8 f are ° f sttllls ^nmportancc; plotted m p epaco, anc * accoiaingly Vh Vi\ Vz can bo treated as if they vaiied continuously Now tho number oi Ultegially spaced values of m witlnn a unit of p x is obviously 2Wh) hence, by extension of this icsult to tluoo dimensions, the number of quantum states per unit cubo of p h p s , Va Hmcn r/ 01 87 A 3 m teims of the volume of the vessel, V - IU-4 The numbei of tianslational states included in Llio tuply posrtive octant of p space up to a given numerical value of p is, there- fore (cf the two-dimensional Fig 88), (281) where X ~ h/p, the limiting wave length (cf cq (116)], and t = p*/2m. The number of states included in the lange df is thus dv { = 2ir)r*(2m)»Vpi cl£, (281a) ZLZfZFl™ ,\T ° f V,,U0S ° ! ' c “ b ° «v«rtod InUi m g a! with i aspect to i* in aecoulance with the gonouxl formula Sac. 231] WAVE MECHANICS OF GASES 399 X /(&) - ^h~»(2m)»vf £« /(£) d(. (281 b) t* To express the distribution law in terms of £, let us now multiply the expression found for dv j- by N„, as given in (280a, b), and write for the result V d£, % thus denoting the number of molecules in unit volume per unit of the energy £, We thus obtain Qt» Q = u> (2tt mfc)« (282a, 6) % ¥ the upper sign referring to the F-D type and the lower to the B-E. The condition fixing the constant B is thatj^”% df = n, the total molecular density, or, after setting £ = kTx, 2 p 0-TV f it n Jo ’ x^dx Be* ± 1 = 1. (283) The total energy E is similarly given by V\ “ntfdt - E or, since V = W/n, 2 Vx Nk Q TV f n Jo " x^dx Be * ± 1 E. (284) The two integrals in these equations define unfamiliar functions of B and in general can be evaluated only by indirect methods. The detailed discussion of these formulas can best bo done for each type of gas separately, This will be postponed, however, until several other matters have been discussed which arc common to both types. The entropy can ho found from the expression given in (271a) in See. 224 for any loose many-molecule system: 8 - | + Nk log B ± *2 8 * lo 6 (l ± ' ) In the case of the point-mass gas the sum over J becomes v> times a sum over y, and this in turn can be converted, by following (281a), into an integral over £ or over x - £/ kT . In this way, after introduc- ing Q from (2826) and V - N/n, and setting «/ = £, we obtain + Nk log B ± lo e t 1 ± dx ‘ In the present instance, however, the last term can be expressed in terms of the energy, for an integration by parts* converts it into 2/3 T * At log [1 i B^er*] ± x^B~h~* — > 0. 400 KINETIC THEORY OP OASES [Ciui*. X times the expiession given in (284) foi B, Hence we can also wrilo, for eithei type of point-mass gas, iS»JVJblogB + ||- (285) The piesaiae p can be found fiom S, but it is simplest- and moaL illuminating to find it dnectly by liisciting m its defining equation, dW = p dV, the value of dW given in oq (207) m Sec 222, which hero takes the foim, dW = * t* The determination of the lattei sum can be simplified by giving to tho vessel the special foim of a cube with edge l Then (278) abovo can bo ivntten hr h 2 & = M = 8mV^ ^ < 280 > Hence as V is altered with no change m m, g 8) ju 8) diV _2 dV f, 3 V’ and, substituting for dip m the expression just wntten for dW, Thus V 2E 3 V’ (287) or, the piessure is two thirds of the kinetic energy per unit volume just as it is for a classical gas [cf. eq (5) m Sec G] Prom (283) it is evident that, for a given kind of gas, the vn-luo of B depends only upon the tempeiatuie and the density but 1 not, ns might Lt/ TJj n a , ntlcipatccl ’ (lnectJ y u P° n the Size of tho vosaol, i , gil ^e Mattel deteimmes the molcculai quantum states Tho It Z^ 10n 0i th ? T leculcs ,n enor ^ ropiesentocl hy Z the mean eneigv pci molecule, B/N, and the cntiopy pei molecule S/N, aie likewise functions of the tempeiatuie T and density n alone' In these respects the conditions aie as in classical theory * St * nC S U1 ab0ve formulas the total entropy of any mass of gas containing tf molecules It can then easily be Jen to Tofo. To a WAVE MECHANICS OP GASES Sec. 232] 401 gram, or to a gram molecule, by assigning the appropriate meaning to E and N. 232. The Approach to Classical Behavior. In Sec, 223 it was shown that the distribution law approximates to the classical form JL whenever for all quantum states Bc kT » 1, Here, since e, = f a 0, this is equivalent to the condition that B» 1. When this is so, the term ± 1 can be dropped in all denominators. The distribution laws then take on the Boltzmann form, = wGc kT , % 2 Vwkv QCt»e «■, where C = 1/13, Both integrals in (2S3) and (284) thus become special cases of the following more general form which can easily be connected with dy ~ by the substitution x — y (cf. table of integrals at end of book) : j Q x»e-*dx = ^, J q xVe-»dx = ^£' (288a, b) Using the first of these with s ~ 1, we find from eq. (283) ^ Q r> ijj _i * V^n B 2 whence 13 = ^, n (280) approximately. Since there is no upper limit to the values of B, it follows that large values of B and of QT^/n occur together, and the condition for the validity of classical theory can be stated also in terms of tho latter quantity. An interesting alternative expression can be obtained by noting that •f IcT represents tho mean kinetic energy of a molecule at temperature T and hence can be written p%/2in in terms of the root-mean-square momentum p,; with the latter let us associate the root-mean-square molecular wave length, X, = h/p, [of, oq. (115) in Sec. 76]. Then kT - /tV3mXJ, and from (2826) _(*)*- KINETIC THEORY OF GASES 402 [Chap. X Let us also introduce 5 » (n/w)~* to lepiesont the mean linear spacing of the molecules in each internal state, Then we can write as the condition foi the approximate validity of classical theoiy* ^ > <*» This condition can be expiessed by saying that the moan spacing of the molecules must gieatly exceed then wave length, for, m the last exDiession, (2ir/3) ?5 is only of the order of 3 The condition can always be met at any given temperature by mak- ing the density low enough, &o that even at low tempeiatuios it ia possible to have a gas behaving classically, or, at any given density it can be met by raising the tcmpeiatme sufficiently At a givon density and temperatuie the appioach to classical behavioi isbettei for heavy molecules than for light ones, and it also impioves with an increase in the internal multiplicity It may be noted that if the mtegial in (284) is evaluated for a laigo value of B by moans of (288b) with s = 1, and if QT^/n is then sub- stituted for B , eq (284) gives E = -f NkT or the classical value for a gas of point masses. 233. The Number of States. At this point it may bo inteiosling to interrupt the discussion in older to consider one or two special points In Sec 231 we obtained in eq, (281) an oxpiession for the number >‘t of one-molecule states whoso eneigy is loss than a given value f By on extension of the argument theie given we can find also the total number of translational quantum states for an entile gas containing N point-mass molecules . Bach of these states is represented by a complete set of tho p’s for all of the molecules, and if we think of the concsponding p’s, defined as they are below eq (280 a, b) m Soc. 231, as coordinates pi, p% pa if in a space of 3A r dimensions, thero will be (2h/h)»(2h/kH2h/h)" or (8 V /h a ) N sets of p’s whose p’s he in a unit cube in this space. Tho 3AT t total energy is = Accordingly, extending the lango of 1 integration for each p p to ~ as well as + 00 , and then dividing by 2 3 * to correct for this extension, we seem to find for the number of quantum states of the gas corresponding to eneigy less than JE Sec. 233] WAVE MECHANICS OF GASES 403 J_ /87V f 2 Sn \ h 3 ) J 23^p a /2»i<J3 dpi dpi • • • dp tN = 2JVp a < 1 d?/3JV (291) after substituting p p = (2 The last integral here represents the volume of a unit sphere in 3iV dimensions and can be written in iterated form as ft , , /*(1 — Vi’*— I/**— j 2 jo ^ " Jo dy *Jo dvi > evaluating the integrals in succession, wo then find ultimately ’ 3V , (292) +i ) where r(n) = dx.* An error has been made, however, in allowing all /da to vary inde- pendently. In reality, all points that can be transformed into each other by permutations of the N molecular sets of p’s as wholes represent the same wave-function; for in forming an antisymmetric combination of one-molecule functions as in Sec. 229 it is immaterial in wlmt order we first write down the <p/> s. Since there are IV 1 such permutations, we must, therefore, divide the result just obtained by iVl Strictly speaking, this is still wrong as regards points for which two sots of p’s are identical; in dealing with a gas of F-D type these particular points ought actually to be omitted entirely. When E is large, however, so that the p’s are mostly largo numbers, it can be shown that the points at which two or more sets are identical form a negligible fraction of the whole, just as points regularly spaced near the lino x — y — z are negligible among all points so spaced inside a largo sphere. Hence, inserting the value given in (292) for the integral into (291) and dividing by IV t, wo have for the number of translational states of the gas whoso energy is less than B, with vanishing relative error as Jjj — -> oO j L dy i- dyw = 3 N 2 * Cf. B. O, Poirco, “Table of Integrals,” P* 02; tho result can bo worked out by means of formulas on that page, Ail equivalent form iB given as a Dirichlot integral in B. Williamson's “Intcgml Calculus,” p. 320, 404 KINETIC THEORY OP OASES [Ciiaf, X As an illustration, let us see what the older of magnitude of v q and vt are in a practical case Consider a cube 0 0001 cm on a sido containing 26 molecules of hydrogen at 15°C and a pleasure of 10" atm. Then m = 3 35 X 10" 24 g, h = 6.62 X 10' 27 , and if we lot ^ liocomo equal to the mean kinetic energy of a molecule at 15 C, so that f = |ifc? , = |Xl 38 X 10- 10 X 288 - 5 97 X 10" 14 org, wo find from (281) in Sec 231 v t = 2 5 X 10 s H h - 36 X 10 12 Again, if we take f ~ E/N, wo can write from (293) in terms of from (281) Let us take AM = (N/c) n by Stirling^ H formula and + 1^ ~ ~ (^f) noar ^ enough (since r (n + 1) - n \ foi integral n)> Then, msoitmg N =» 26 and the value just found for vt, we find SV 6JV roughly This is a tremendous number and shows how very far bolow the reach of macroscopic observation aie tho quantum states with which the theoiy operates A much quicker method of making such estimates is to rruvko a calculation m terms of wave lengths If we drop the factor iic/Z from the middle expression in (281), we have ~ T/X 3 or tho number of cubic wave lengths in the volume V Smco V — 10“ 12 cc, and at 16*C the wave length X of a hydrogen molecule moving at mean spoocl ia 1 14 X 10~ 8 cm, we find v$ = 10~~ 12 /10~ 24 - 10 12 , roughly, which is almost coriect The number of states for 26 molecules should then bo v q = vf* = 10 na , which is again very close It is worth noting that (291), the value obtained for v q before making the reduction requned by the exclusion pnnciple, is equal to <r/h m , where <r is the classical value of tho phase space up to energy as given m Sec 196 above, The same thing is likewise tiue for just ono molecule, of course, for nothing prevents us from setting N = 1 , and in that case there is no reduction to be made, eq (291) being correct. This is one way of establishing the old result that for a point moss Sec, 234] WAVE MECHANICS OF GASES 405 without internal multiplicity each quantum state corresponds to a volume A 3 in classical phase space. For a large group of N similar point masses, however, the added factor 1/N\ in (293), which is required by the exclusion principle, reduces the volume corresponding to a quantum state of the group from to h SN /N\ (approximately). 234. The Zero-point Entropy. It is interesting to compare the expression furnished by quantum theory for the entropy of a gas, under conditions such that classical theory should hold, with the expression furnished by the classical theory itself. According to eq. (285) in Sec. 231 the entropy of a perfect gas composed of N point-mass molecules, with or without internal multiplicity, can be written 8 - NklogB + ||. In the classical limit this differs from the classical expression itself as given in (252 b) in See. 203 or - Nk log C, in which for a point-mass gas C — l/B, only in the integration con- stant; for in tho limit E = | NkT and the two expressions thus differ merely by Nk. If wo insert in tho first of these expressions both E = ■ J NkT and a suitable value of B, using (289) and (290), we obtain for tiro classical limit the alternative form, given long ago by Planck (for w = 1), (294a) Hero in — molecular mass, k — Boltzmann’s constant, I T = absolute temperature, n - density in molecules per unit volume, h = Planck’s constant, 'w = multiplicity of the internal molecular state. For a gram (or a gram moleculo) of gas in volume V i, containing N i or nV i molecules, we can write the last formula in the two forms 7'W 8 m N\k log (V,2'«) + St = Nik log — + (2946, c) these differ from tho usual phenomenological formulas [cf. (252d) in Sec. 204] only in that hero tho integration constants are assigned tho definite values 406 KINETIC THEORY OF GASES (Ciiai*. X g. - H* log [» < 2M "> g£ = Nik log m l^’o) _ (A-c) j = ^ -|- Ti log li, (3940) where i2 = Nik The mtegiation constant is, or was, of particular interns l m con- nection with the Nernst heat theoiem and its applications to chonnslt y This theorem is concerned with tcmpeiatmos neai absoluto ssoui. Now at such low temperatuies the formulas that wo havo hitherto obtained become open to doubt because tho discroto spacing of Iho quantum states then assumes importance; tho whole basis of our calculation for the point-mass gas, in fact, disappears. Thoto iw another method, howevei, that of the canonical ensemble or dish ibuhon as descnbed in Sec 226, which is open to no such restriction. This latter method furnishes in eq (273) ot See 227 tho following expression . S ~ + klog^e-& ( 205 ) > where E, is the energy of the whole gas in its ?th quantum slalo and tho sum extends ovei all states This formula lias the advantage tluvt it should hold all the way down to absolute zero, the lowoi quantum Hinton being those of the liquid or solid phase. Lot us soo what hum it assumes in the classical limit Paralleling the proceduie used for one molecule in Soc 231, wc can replace the sum in (295) by w^J ( dvJdE ) dE, wheio v q is tho number of quantum states having enoigy below E, as given m oq. (293) above; the factor w N aiises from the fact that each of the w N ways of auangmg the N molecules in thcii w internal states, combined with each tiaiiH- lational state, yields a separate quantum stato of the gas Wo thus obtain The mtegial by itself can be written (krfl f dx = {kT)%(% n] = (kT)™ LM + 1 ) JO \ £ / 3iY/2 Sec. 236] WAVE MECHANICS OF OASES 407 (cf, Peirce, “Table of Integrals/ 1 p, 62) . Let us also suppose JN to be so huge that N\ can be replaced again by the ultimately important factor in Stirling’s formula, (N/c ) Then tlie value of the state sum becomes where n stands for N/V) and for the entropy we have, after inserting as usual IS » | NkT } S = ~ Nlc + Nk log [w ■ It is interesting that this expression agrees exactly with that in (294a) above. Since the original formula as given in eq. (285) can differ from that given in cq. (295) only by a constant, it follows that these two formulas must give the same value for S, not only in the classical limit, but whenever validity can be claimed for both. Thus the choice of the integration constant that was made in obtain- ing our general formulas for the entropy of a loose many-moloculo system had the advantage that the resulting formulas, at least when applied to a perfect gas, agree throughout their range of validity with another formula that holds down through the liquid and solid states to the absolute zero of temperature. The values of the constants as given in (294d, e) are, consequently, consistent with the assumption that the entropy at T — 0 is k log Wo, where w [, is the multiplicity of the state of the substance at that temperature, and is zero if this state is simple (cf. Sec. 227). 236. Chemical Constant and Vapor Pressure. The entropy of gases is of groat importance in chemistry because of its bearing on gaseous reactions. It follows from thermodynamics that at constant temperature any system tends to settle into its state of lowest free onergy. Now in the expression for the free energy, F — U — TS, the change in the onergy U involved in a reaction is simply the heat of reaction and so can bo measured or calculated from other data; if, therefore, the change in entropy that is involved in a reaction can bo ascertained, it is possible to calculate the temperature or other condi- tions under which the reaction will take place. A direct experimental determination of the chango in entropy may be impracticable, how- ever, because to effect it the reaction must be carried out reversibly. Now Nornst’s heat theorem asserts that, at absolute zero, not only is the entropy always finite (provided the volume is kept finite), but 408 KINETIC THEORY OF GAti/S# fCllA* X also the change in entiopy dunng any chemical reaction vaninhea,* If this principle xe true, we have only to find explosions for Mio entropies of the substances concerned in a inaction, with Clio intcgnx- tion constants so chosen that all of them vanish at T = 0, ami thou the cliffeienee between these expressions at any other tempo* aUuo will give the change in enti opy at that tempeiatiu e Voi a gas such a vnhu* of the entropy can be obtained by evaluating J ~ with the help of specific-heat data and the heat of sublimation 01 vapmmition at on o tempei atuie In studying such a determination theoretically wo aio led to a loin** tionship between the entiopy constant and the vapor prossuio. Tho change m entiopy during vaponzation is L/T , where L is fche heat of vapomatLon, as T —> 0, L approaches a limiting value Lc, honco Ilia gam in entiopy becomes infinite and relatively equal to L 0 / r J\ Now the vapor can certainly be tieated as a classical perfect gas, for the vapor pressure falls so veiy fast with decreasing T that the density diniininhaH rapidly and Q actually increases without limit, the molecules can bci treated also as point masses, for they will all be in their lovmsl Intel mil state, simple or multiple Hence, wilting S M for tho entropy of il\o solid phase at T = 0 and using (294c) above, wo have for tho entropy of the saturated vapor near absolute zcio & - Y + s, = R log H -f So, Jr approximately, from which log p ~RT 2 log ^ + h * " It 8 .). 2 any \ eas ° n f* " °’ 1 “ W R B °<^o of this connection with the constant * has commonly been called the "chemical constant." The expiession thus obtained foi the vapor pressure p has an intoi- e ‘ tS f possible vakie of integration constant 8' 0 is fu t ( co e) above A VaIue of s > tlwt consistent with this TtJ o V IT/ 1 H1 SeC P 7 ’ aS We found in the la ^ soation; it is, , ' * io S w », wheie we can writo w a » m loims of the arCT’S'' °f “° h th ° N ‘ mol ' !culs ’ composing mzsin Si aud s - ‘ nd b - n * ' in i "° der ™7.°' N T "*? T *» " nwtal— Fowler, “Statistical Mechanics, ’’*1 od , ioTo^Seo yf”” 011 (1980); ok lllf5 ° R Jr Sec. 236j WAVE MECHANICS OF OASES 409 V - to (%r m)& w 8 h 3 (keT)»e w. Hero and in the last two equations R and La may refer to either a gram or a mole, If w s = 1, there is nothing in this equation referring to the internal structure of the solid phase at all; in that case the solid appears to become, so to speak, a physical point without internal features. On the chemical side, however, the practical significance of these relationships promises to be less than was formerly expected. It is probable that, even if NcrnstV heat theorem is universally true, its practical usefulness will sometimes be limited by changes in entropy which occur at temperatures extremely close to absolute zero, as the solid sinks into its final simple state, and which are thus beyond the reach of experiment.* On tho other hand, wave mechanics holds out the prospect of obtaining from molecular theory a consistent set of theoretical values of the constant for various substances, which will give the chemist what ho needs. A simple example of the method is furnished by the' case of dissociation .that is treated in a later section, 236, The Fermi-Dirac Gas of Point Masses, The two types of point-mass gas will now bo taken up for a more detailed discussion, beginning with the Fermi-Dirac type.f The distribution law for a F-D gas of N point-mass molecules is, according to (282a, b) in See. 231, % _ 2 _ Qf» Be** + 1 n (2 tt (296a, b) £ being tho kinetic energy of n molecule and rq d£, the number in unit volume having energy in the range d£. B and the total energy I? are determined by tho equations 2 9. nm f * dx _ i Vfft Jo Be* + 1" ' AL&.NhT* affl dx Bo* H - 1 13, (297a, b ) in which x represents f/fcT and n is tho total number of molecules per unit voltune. For B > 1, the integrals in these equations are readily integrated in terms of series; for then wo can write (Be* + l)~ l = B~ l e~*(l - BrHr* + • • • ). * Cf. R. II. Fowi.ua, loc, til. t E, Ftmiui, Zeita. Phyaik, 36, 002 (1926). 410 KINETIC THEORY OF OASES [Chap X Substituting this senes m (297a) and then carrying out the integiation term by term with the help of formulas (288a, 6) in Sec. 232, we obtain nB \ 2«f ? T ) l ' whence » _ QT» A _ n \ S " n \ 2VQTW + ) With this value of B the distribution formula becomes, after using the senes again with £/kT substituted for x, n * = v?(^ry» ( 1 + Wqt% ) r> * e “f 1 ~ qtk 6 kT ' ’ ) Comparison of this equation with (616) m See 28 shows that here tho two series represent the departure from classical behavior as tho temperature sinks or the density rises We note a lelative increase ill the number of fast molecules The energy can be found by integrating cq. (2976) in a similar way, which gives + ).a, or, after substituting the value found for B t + ) The series represents the mciease in energy over the classical value duo to the increase in fast molecules, The pressure p is increased in the same ratio, since according to (287) in Sec, 231 it is (%){E/V) as in classical theory, since N/V ** n, we have p *= nkT^l + ^ KQT& ) * The specific heats, on the other hand, are decreased At constant volume Cr = (dTNm) v “ 2 Nik { 1 ~ ' )’ Sue. 230] WAVE MECHANICS OF OASES 411 Ni being the number of molecules in a gram and NJc representing the classical value. At constant pressure, E/V is constant, hence dE/E = dV/V and dQ = dE + v dV = cJ£ + |~d7 = ~dE. From the preceding equations " - 1 »*( l - mye* ■ ■ ■ ) dT + 1 Nk wm and dp ~ nk dT kT'dn to zero order in n, which is sufficient for our purpose, so that if dp — 0, dn =* — n dT/T ) and Hence 1 fdQ\ _ 6 „ , / cp ~ Nm\dT/p “ 2 Nlk \ y = SS = 5 (\ y c v 3 V 1 3?i 2WQF* )' For completeness we may add also the series for the entropy, ns found similarly from (286) in Sec. 231 with the help of log (1 + y) - y - 4 2/ s + 8 — ATfc^log - - JVfcjlog [ + H + ft 2 1 WQTft H- (WW 4 1 nh 3 ri- ft J T &QT* These equations exhibit at a glance the manner in which the various magnitudes pertaining to the gas begin to depart from their classical values as QT^/n decreases from infinity* In a later section we shall consider whether there is any hope of detecting such effects experi- mentally) but it is move convenient first to complete the theoretical discussion of other cases. The complete series all converge down to QT**/n “ 1.38, at which B m h Below this point, however, we must resort to other methods 8U ch as numerical integration. The theoretical treatment becomes 412 KINETIC THEORY OF GASES |Chai>. X simple again only when QT^/n is actually small, then tho gas exhibits a maximum degiee of depaituie from classical behavior and is said to be "degeneiate ” To this case we shall now turn, 237. The Degenerate Fermi-Dlrac Gas. Suppose that „ TK (2imk)K TV ^ , (2 98) this means that the moleculai wave length is large lolativo to tho linctir spacing of the molecules m any one of the w internal stales [of. cq. (290)] Then the integial in eq (297a) must bo laigo in oidor to make that equation tiue, and this can happen only if II is small; as QT^/n —* 0, it is necessary that B — » 0, n r as given by (296a) must now vaiy as for small {, but ulli* mately for laige f as e kT . Between these two extremes lies a transition . . Il region containing that value f x that makes Be hr - 1 This values £* constitutes a sort of turning point j we can write „ - 2 $ n t B = e r** fi w * + 1' y = r zM kT ' (290) _ - kT=f o /)6 ( _i V- Fla 89 fiom proportionality with will occui relatively abruptly, tho curve leaving the neighborhood of the curvo and dropping lapidly almost to zero as f varies ovoi a range that is smalt as com- pared to ft itself. At T = 0 tho curve must actually bo cut off it , Q r, . f * nmov tbV'OUCUljr UO UUb Oil T /o rm r r t r km 0 uconorgr at f = tli but for onorgy » a 0 } there is & tail of wellian form, since for large y or (f - approximately, 20 ^ f Hg-cr-fi >/Ar max- n t = VrkY- The situation is illustrated graphically m Fig 89 DeJon S ln § t0 1 = °t we have at this temperature Sec. 237] WAVE MECHANICS OF GASES 413 2 Q for 0 < f < to and % = 0 for t > f 0 , as shown in Fig. 89. To find f 0 , we can first keep T finite but very small and then in the integral in eq. (297a) drop Be x in the denominator and change the upper limit to *0 in order to harmonize with the discontinuous n t curve. The equation then becomes whence This means that the limiting molecule has a wave length or only (4ir/3) w times the linear spacing of the molecules in a par- ticular internal state, 5 - (w/n)M; for the corresponding momentum is po ~ /i/Xo [of. (116) in Sec. 76] and the value of X 0 as stated makes the kinetic energy, pl/2m, equal to fo. Division of (2976) by (297a) and tho evaluation of both integrals, similarly treated, then gives for the total energy E o NkT X | | Wo = 3 m? 10 m Thus at T - 0 the energy of tho gas is by no means zero, as it would be according to classical theory ; this is called the zero-point energy and is a very characteristic feature of quantum theory. The mean energy of a molecule amounts to % of the maximum, {•<>. Thoro will also be a zero-point pressure, equal to two thirds of the energy per unit volume (cf. Sec. 231) or of magnitude (since N/V — n) varying, therefore, as tho % power of the density. 414 KINETIC THEORY OF GASES [Chap X This pressure is an immediately observable quantity, but the signif- icance of the zero-point eneigy is less certain, since in ordinary physical observations only changes in energy are detected Of course, if energy and mass are umvci sally proportional, as is required by rela- tivity, then the zero-point eneigy would be evidenced by a slight inez ease in mass The initial depa? ture from complete degeneracy as T rises from 0 can be found by a calculation that is a bit intimate but straightforward. The equation df - n, the number of mo’lecules pei unit volume, can be written, in terms of x . _ Ji 1 hT and as given in (299), since f = KTy + = kT(y + *0, thus. Now 2Q mi C w (y + V * dy « n f° k + ^ J-xt e v 4 - 1 dy (si - yW e~ v + 1 J- (1/ + 8Si) H , e» + l dy = (a - y)» dy + + v)» - (xx - + nsLk& Jx, e” + 1 dy. Here the last mtegial is negligible when kT/fo « 1 or x } » 1, because of the huge size of c 1 ' -f 1 ; and for the same leason, after expanding the indicate in the next to the last integral, we can extend its upper limit to infinity and wnte for it Now by the substitution* yi—~ log z i yjv_ = _ r l iog£ Az _ 1 Jo 1 e v + + 2 .2 12 Hence the original equation can be written 2g_ V? *i ?4 + 12 ' )- n t * Of Peirce, “Short Table of Integrals,” no 510 , Sec. 237] WAVS MECHANICS OF OASES from which _ f3\Arn 7r 2 „ u \* _ \ f h\* 71,2 -w 1 3:1 " \4 QT» 8 * l / ” [\kTj 8 ®* ' ' J and, after expanding and inserting Xi = $a/kT to zero order, 415 £i = kTxt - fo 01) In a similar way on