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LEE A. DuBRIDGE, Consulting Editor 


The quality of the materials used in the manufacture 
of this hook is governed by continued postwar shortages . 


G. P. Hartwell, Consulting Editor 

Bacher and Goudsmit— ATOMIC ENERGY STATES 
















Slater and Frank — ELECTROMAGNETISM 



Dr. Lee A. DuBridge was consulting editor of the series from 1939 
to 1946. 


With an Introduction to 
Statistical Mechanics 



Professor of Physics , Cornell University 

■ ' ^ v * .• 


First Edition 
Fifth Impression 


193 8 

JI/V Lib 

Copyright, 1938, by the 
McGraw-Hill Book Company, Inc. 


All rights reserved. This book , or 
parts thereof, may not be reproduced 
in any form without permission of 
the publishers. 


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 . 





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 


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 






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 




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 


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 




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 


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 


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 



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 


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 


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 




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 


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 




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 




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 




[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 


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, 



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, 



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 



[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 



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 



[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) 



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). 



[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 

Vo = 0, i o , hx - hu = Vo, - 0* (8) 

Foi the total velocity we can then write 

v - Vo + v' (9a) 


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 



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/, 


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- 



[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 



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 

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 



[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 

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 

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 



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 


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 


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 



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 



[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 



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." 



(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. 


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, 

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, 


= 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, 





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 ) 


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, 


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 


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 


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) 

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) 


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 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 



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 (unit, 
10-* g) 

R (unit, 10° 


(unit, JO 3 

(unit, UP 


2 016 

3 349 

41 25 

188 8 

174 0 


4 027 

6 880 

20 65 

133 0 

123 I 


4 002 

6 648 

20 78 

134 0 

| 123 r> 

h 2 o 

IS 016 

29 93 

4 615 

03 18 

58 19 


20 18 

33 52 

4 120 

59 08 

| 54 08 


28 02 

48 54 

2 968 

60 05 

40 07 


32 00 

53 16 

2 598 

47 39 

43 66 


36 46 

60 56 

2 280 

44 40 

40 90 


39 94 

66 34 

2 082 

42 42 

30 08 


44 00 

73 09 

1 890 

40 42 

37 24 

Krypton , 

82 9 

137 7 

1 0030 

29 45 

27 13 


130 2 

216 3 


23 50 

21 05 


200 6 

333 2 


18 03 

17 44 


(28 96) 

(48 11) 

2 871 

49 82 

46 00 


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), 



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. 


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. 




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 

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 


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, 


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. 



[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 


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). 



(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 


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 


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. 



[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 


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. 


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 



{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 


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) 



(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 


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 


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: 



— eki == dl 

Ja ^ Kt J J - /t/a) dxidw 

+ n\ dt dsy j j <pv,(FiF{ - /,/{) dx[du. 



[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 


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 


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 


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 

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 


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 

do i, 

. , 1 ,. I 00*1 

dv i* + ~ dvzx 


^; v dVtv + & dVlt 

= 0. 


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 


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 

_ „ 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 





0vi»dv i„’ 

d Off i 

dri, Stiix 


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 



(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 

dfyi _ dfyi 
dv\ x 0v 2 J 

dg i 


m_i 5(72, 
m 2 dy ; 



dvL u ’ 

(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 


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, 


/(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 ) 


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 ) 


* 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 


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 


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 


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 


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 



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: 


v i e~ p,v> dv — 


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 


( 656 ) 

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. 



((’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 


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 — ** 


f v dv 1 H — jz five-** 9 * — $((tv) 



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: 














' 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 


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) 


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 



<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 



(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% 


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 


[Chap II 


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 

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 


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 

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 



[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 

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 


J = J oe ~ 2 

J o standing for a new constant 


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). 




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 


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 ! 



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 



(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 



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 

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 


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 


[Chap IH 


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 

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 ) 


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 

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 

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) 


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), 



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) 



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 



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. 



[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) 


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 

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), 



[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 

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 



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 



[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 


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 «: 


F = — 

F, - 



F u = - 



Ft = 



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 



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 . — 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 

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 


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 . 


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. 





































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 


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 ) 


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 

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>« 


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 

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 


recross the lower one in a downward direction with v x just reversed 
(cf. Fig. 21); the maxwellian flow downward is thereby made 

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 



(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 

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 


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 


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 

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) 


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. 






[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 




N a 


















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 



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 



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 • 


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), 



[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 ) 


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 


the resulting equation ancl moving all terms but one into the left 
member, we thus obtain 



( n f ) + + V v 

<W) | .. d(nf) 

(*>/) + ± (F t nf) 


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. 



[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 

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 

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. 

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^ 


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 

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 


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- 


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 

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,] 


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 , 

P,« = Ge k ?) (GO) 

here e is the total eneigy of the molecule and tho constant C has tho 

C = kT dxdy dzdp x dp v dp,] 


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 


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 

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) 


C m being a new constant" of magnitude C'2_ f c kT Since p x ~ mv X) 


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, 



[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 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, . 


' e -V*r 



If multiple quantum states aie employed, with multiplicities u>„ and 
energies we have from (93a) or (936) for their probability 

P K = pern = 


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 


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 


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 .) 


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 



(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 

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 




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- 



[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 

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 


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 



[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 





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 


























2 3 4 


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 = ~\ 



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„ 


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 




[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, 


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) 


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 



[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- 

0i2 dt = ^7i } Sv r} dt — dt ~ £m> r dt 


20 — Relative 

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 | 



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%\ 


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: 


Vr = 1>1 + if Vl > Vi, (101a) 

v r = Vi jf tn < ti 2 . (101b) 


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- 

v 3 
0 4n.S’ 


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 


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 



[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 

= 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 



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 


$(*) = “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 


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, 



G 0 05 10 15 20 2 5 30 


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 


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) : 



- JJ dv i J 0 U ( 2 /^i + 

= j (zi3 t v\ + dvi 

-Ilf + S + w] r ** h '** 

_ M + m , i m ^ 


The first pait of the integral m the ongmal expicssion for m0 12) 
on the othei hand, has the value 


•i fh) l 



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)*) 



(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]‘ 



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 


homogeneous maxwellian gas : 

0 = -\Z2nSS, L = 

For hard elastic spheres of diameter <r these become 

0 = \/2 TrnDir 2 , L = - > 

V2ir w* 


-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 



[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 



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) 



[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 

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 *>, 


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 


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 ) 



[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 

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 



1 . 0.0 

2 alt cos ^ sm ^ 

sin 0 

= iff* 



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 , 


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 ^ 


m r 

- a :*)-» 

U(n) - 


Wg) l 

c 1 )* 4 • • ■ j 

1 w* 

— X 2 j 



or approximately 

0 = 

2 p f/Q-p) - U(n/x) 
mv 2 Jo (1 — a ,s ) M 


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 £ 



[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 ‘ 


(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 


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 


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, 




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), 



|#(b)l Sill Jr 

Fr(Or) - 



(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 


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 

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 

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 



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 



[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 

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 


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). 



[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 




Limiting angle 

0 9° 


By v 

0 9° 



Absolute temp 


1 4 ; 

2 7 

2 5 

3 0 

5 5 

4 7 


1 7 ' 

3 5 


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 





(XIO' 1 ) 


(xio- 3 ) 


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 






H 2> f(0) 






■ — 




• ■ 



















\ : 








— — 

—— 1 



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, 




[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 


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 


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. 



[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 


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 

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 


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 


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) 


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. 



[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 



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, 




[Chap IV 


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 


The factor of proportionality tj in this equation is called the coefficient 
of viscosity of the fluid or, for short, its \iscosity 



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 

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. 



[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 


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- tii mu v or u,\.srs 

ii’mc IV 


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 

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 

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 , 



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 


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 


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 

‘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 



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 

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< 


(<‘u\r IV 


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, 



xvh ' 1 "* 11 dx 



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, 



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 


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 


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 



1121 ) 


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. 


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. 


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 = 


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, 




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. 



Some Molecular Data 







(10~ 7 dyne sec/ 

(10 _(} cm) 

(10-« cm) 





871 (1)(2)(1C) 





Ill (heavy hyd.) 


871 (17) 

11.77 ■ 






1043 (3) 





CI-I 4 (methane) 


1077 (1) 





ni-i 3 


070 (4) 














3095 (3)(12) 







1734 (3) (5) 




CaH,, (ethylene) 


998 (4) 


C 2 II 0 (otliano).. 


900 (1) 


O s 


2003 (1)(0) 







1397 (16) 







2190 (3) (7) 




co 8 


1448 (3)(8)(10) 




CIIjBr (methyl 


2431 (9) 




bromide) .... 


1310 (10) 






2230 (18) 




Hg.. ! 


4700 (13) 
(21D.4°C) • 



4 , 20 . 

(219, 4°C) 







1790 (11) 






* 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 



[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 



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* 



[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 " 


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 

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 



[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 


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 



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 ) 



[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 - 


wheie I/o = 3/(4wSo), T is the absolute tempeiatuio, and C stands foi 


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)’ 


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 



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 

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 

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« 





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 






1 \r* 




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 




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). 



[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 ) 






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 





r, + %% 

^ l 

, Stf[l + Wi/MiW 
f<# „_ , 


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 




i+f»? ■ i +h.£ 

Ui 111 


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 ’ 


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). 


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 


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 



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, 


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 



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 


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 

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 



[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. 


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), 


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 : 


If wo then insert fo from (1415), the second member of (145) 


{nf o) 


. dn , dA 

A Tx + n li 

nAv 2 


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 



[Chap. IV 


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 


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 


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 


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 


denote by D%v x v 2 . The entire integrated equation can then be 

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. 




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 

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 



[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 

= (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 ; 


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- 



tion of w and l w the cosine of the angle between this direction and the 



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 . 


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 . 




[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 

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 


»‘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 




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 = - 

4 jS 7 


If we insert the mean speed v = 2/ (\/ x/S) we have, therefore, 

DPvxV* _ 16 8 v 

V 15 Vl UVSvc ~ 15 17/ 


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 


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 ' 



[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 ] 


~ V W = m2VxV *> 






I coll 

= 2nA/3 2 v x v y e-P iv * 

dv Qy 


at a point where v Qy = 0 — elsewhere v y must be replaced by ( v y — Vov) 
in this and the following equation; 


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 


5t mv 5? r 

V ~ 32 V2 S xc ~ Z2 pvLv °- 


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). 



[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. 







\ (9y - 5) 


(10“ 7 

10-3 cal 



cm sec deg 

g deg 






















H 2 







n 2 







O 2 







H a O at 100°C 







OO 2 







nh 3 


0: 0514 





CH 4 , methane 







C 2 H 4 , ethylene 




1 . 25 



CaHe, ethane 







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. 


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 ). 



[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 


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). 



[Chap. IY 


T (abs.) 


K 27 Z 0 


( K/rjcv)T 

(K /y)Cv) 27S° 































h 2 













n 2 











0 2 















- - - 




CH 4 (methane) 









C2H4 (ethylene) 




C 2 H 6 (ethane) 




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. * 


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). 



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. 



[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 

( 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. 


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 



[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 



TV _ _ 1 , r dny 

Ti ~ i ViLi w 

For the second kind of molecule we find similarly 


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, 



[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 



= Vi* w; (wi/oi) 

coll ox 

Aiv lx e~^ 




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 

fal = CiVi x e~^\ 

fa 2 = C 2 V 2 ,e^\ 

as a result of which 



2th. = niCi v^-^d , d = V 

7T* 4 JliCl 

2 /9J 

S«2x = 


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 


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), 


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. 



[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 , 

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* 


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 



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: 





7rnift 2 VM2Ch 

S d 

Sd = 2 tAPiJ* 0 J G(w,-d)(l — cos 0) sin 0 d$~^w*e~** lh ** dw (165a) 

Sa - H s ) a - c “ •> “ 1 ’• "] 

x'tr* dx. (1656) 



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 



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 


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 


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 = 


[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- 


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. 



[Chap. IV 

Du == (1 + Xu) 

3t V __ ^ rj 
16 \^nS~d^ ? 


in the first of which Sd refers as before solely to collisions between unlike 

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). 


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 








n\ H- n 2 

Oa- 1"1 2 













C0 2 -H 2 




0,75 ' 




























113. Diffusion at Various Pressures and Temperatures. The 

theoretical results indicate that at a given temperature the coefficient of 



[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 

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 

(T 7 :) X 1 "*. 



Now D, being the ratio of molecular flow over unit area per second to a 
molecular density gradient, will be changed in the ratio 



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 s 

3 8 



or like T n , where n = 

3 5 



D cc - T n , 



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 



[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. 



Diffusion Diameters 




(unit, 1( 

j ^av 

)“ 8 cm.) 




(unit, 1< 

| ^av 

)~ 8 cm.) 



3 . 15 










CH 4 -CO 2 








H 2 O-air 




H 2 -C 2 H 6 










o 2 -n 2 




H 2 -C 0 2 



Air- CO 2 




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 

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 



[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 


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’ 


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 


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 


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 


(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 



[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). 


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 

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). 



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 



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). 



[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 

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] 



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 


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 - 



[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 


tc, °C 

Pc atm 

d„ = 1/Fo, 




12 8 








H 2 Q 



JO. 4 (?) 





























C0 2 





Ethyl ether, C*HioO 





U a " critical temperature, p c - critical pressure, V c — critical volume. 

Sec. 121] 



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 


Vic - ~y> 

( 176 ) 



[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] 



«- 7.7e" 2 <-£68 

L (r/a 0 yj 

F = ~ = 0.354 e~ 2A3 »• 

X 10~ 10 erg, 

0/a 0 ) 




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 



[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] 



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 



[Chap. V 

p r = tt 2 m<r 2 


T Jo Jo Jo 

sin 0 cos 0 dtf dp 

"* W i. 


/* 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 ] 



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 ) 



[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 

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] 



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, 



[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 = 

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] 



_ 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). 


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 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. 


h 2 


n 2 


o 2 









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 



[Chap. V 

is employed in place of B, we have B p 

b — and B p should there- 

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 



N 2 

0 2 


h 2 o 

C0 2 




a/p 0 Vl 




































|7rno<r 3 









Sec. 129] 



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 


r from the center of A is Ce w dr ' by (94a) in Sec. 55, provided all 

e kTdT r , integrated for all possible 



[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 


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 

Vi — — f rn2 J 0 r 3 u'e~kf dr 


Sec. 129] 




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 



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 

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 ] 



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 


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 

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 mnlnnlo 

on anolhor al a iIihIiuioi* r, mcnmitod pu,,itivolv in a ropnh'tnn, in tho 

* 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] 



X M 


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. 









10~ 8 




10~ 8 




10~ fl 






2,35 X 10~ 11B 

2.33 X 10~« 





5 ! 

7.38 X 10- 80 

1.G8 X 10-“ 







4,38 X 10- 8 * 

1,72 X 10““ 




N a 



1.58 X 10" 72 

1.82 X 10““ 







1.04 X 10” 111 

1.13 X 10““ 



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. 



[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 

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) 


they wore reached along the more fundamental of the two lines of 

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.) 










B/RT (obs.) 







B/RT (oalo.) 









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). 



[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), 


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| 



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 

n 2 -n 2 





20 °C 









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). 



[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 ] 



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). 



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 




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. 


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 


employed to specify its state. From (188) and (187) we have then 

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 



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 ' 


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 



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 


I (?E\ p 

T\dpjr + T\dpJ T . 

+p( dv \ ' 
ap ^lT\dTj p ^ T \dT/ P 

Then, by similar reasoning, we obtain 



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), 


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 

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. 



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 


(§\ - (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 ’ 


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), 


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 



[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 


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. 



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, 


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 

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 


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, 


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 



( 197 ) 



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..; 


then, the above equation for 8 takes the form 

8 ■» <v log 7' + li log V + const., 

( 200 a) 


8 «* c p log T — R log p + const., 


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 


[Chap. VI 


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) 


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 



-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 



[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 

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 



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 



Cj)/ R 


Spherically symmetrical 





2(r - 1) 





3(r - 1) 





3r - 2 

% = 1-667 
% = 1.400 
% = 1.333 

y, = 1.286 

1+ 1 

L + 3(r - 1) 


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- 



[Chap. VI 

Specific Heats 




c v 

Cp — — Cy 





1.659 (18°) 
1.64 (19°) 
1 . 66 s 

1.252 (18°) 









1.68 (19°) 
1.66 (19°) 


h 2 




HC 1 

1.40 (20°) 




n 2 









0 2 







0. 233i 



Cl 2 





H 2 0 

1.32 (100°) 
1.32 (18°) 

0.48 (100°) 



H 2 S 



co 2 




so 2 










c 2 n 2 





CH 4 (methane) 





C 2 H 4 (ethylene) 





C 2 He (ethane) 





C 2 HoO (ethyl alcohol) 

1 . 13 (90°) 

1.08 (35°) 






C 4 H 10 O (ethyl ether) 

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 


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 


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 

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. 


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 


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 

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. 



[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) 


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) 


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] 



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 



'[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-\ 

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 

Z per oscillator 

hv 00 jhv 

Z = e = 



hv ™ i nv " j/n> 

6 = Jive 2 kT '^j e kT /Z~\~ 2 ^ ve 2kT "^ e kT /Z 

i = o 












e kT — 1 



IV molecules in a gram, each containing such an oscillator, would then 
contribute IVe to £/*, and to the specific heat Cv the amount 





R h 2 v 2 

ik 2 T 2 / 

siuh 2 


2 kf 


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 


= r(i 

12 k 2 T 2 

1_ h 2 v 2 

12 WT 2 ' 


















































—The harmonic 
e = mean energy, 

1.0 1.2 1.4 


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. 



[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 

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. 


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 

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), 



[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] 



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 


(2 J + l)e w 

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. 



[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] 



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.- 


-Total specific heat 




[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 . 


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 


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: 


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. 



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, 




[Chap. VII 

N = 3 


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 


( 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.” 



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 


(U“) *■) 

log JV - log 2 



N 2N' i 

TIkui, dropping powers of 1/iV beyond the fimt,, wo obtain 
Pur AT 4» hver 2 — ** loir f2*rt — 


t | '>»/* 

log /’ w * - ;j log N + log 2 - ^ log (2 tt) - * 2 ^ 


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 £ 



[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 = 




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. 


Sec. 155] 


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> = 




"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( 





2 Xa “d£x. 



[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 


of throe expressions such as (215?;) or 


P = 

X 3 (2jr)«' 

2X a . 


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 


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] 



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). 


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 


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 


Uu* not of irn'intniiiK Mio- I I multiple » U\ 

(' - 


1 Fur 

N t< \/k 


»< I 1 I tt* I l)(s 11 


I 1 

sr | Jitfl 

I i 

miii'n A 1 ' -« «»»; lii-itri* /*,,i » /', nt'MinlitiR it* »• 
ur iut 


II I * ? o 

f -- f 

i l 

Now for n iimximtiiii nl i> * v n wr mu >i lmv<< 

I’m I t 


«K I 


« 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) 



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 


hence in 

Av(v - vY - ^(r - vYP v = ^P v - 2-v^vP, + P 

by (220) 


{< 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 * 


.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* >* «* 


* 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 



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. 


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] 



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 

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 ] 



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 

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 


kinetic rummy of a i ses 


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 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 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] 



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) * 


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 « 


(AirkTUt)* 5 

4 kTUt * 



this makes Iff” dx dy ds 

1 ancl also 



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 



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] 



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. 



[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 


with respect to a: and noting that 


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 


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 

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] 



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 



(a— itQ* 

, e " 


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 




I'lo, 00. — Drift plus Brownllin motion. 




(« — 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- 


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\ 



(’Itr nt )'« 

(if h 

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 


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 







<»' tltx 





mV 1 

t*t «iMl 


Tho mean time of traiiKit from the origin to thr> plane Is * 



« r * tit 
V-lr/^/w Vt 


4 (H 




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 */ 




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 


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, 


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 ) 


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) 



V — ► 



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. 


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 




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, 


- JL f“ 
“ RTjo ■ 

Znrv dr 


kin uric r it unity nr uasfs 


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 


v a 4 
K nltT 




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) 



tv + 2f 


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), 




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 


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 



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 


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 



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. 






Air or COj 011 rnficMnotl brass or old shellac 



Air on mormity 1 



Air on oil , , , , T , * 



COa on oil . 



Hydrogen on oil 



Air on glass 



Helium on oil 1 

87. 4 


Air on fresh shellac 



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). 



[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 



(At rest) 

Fig 09 — Viscoua drag on a 

* Blanicbnstidw, Phya Rev , 22, 582 (1923) 



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 


lh = 


A + ft — AA 


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 


~ jTi +/,”!7. (2 «RT)» 


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 


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- 



[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- 



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 


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 

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 


, , 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_ ( 


x l + s 2 

-i tou- 

i r 

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, 



[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, 


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> 



J f>a r'(a*~x l )M 

sdS — I dx I [(a 2 — s 2 )W — y] dy = 

J-a J — (o s — 


Hence s dS ^ 167ra 3 /3, and (233a) becomes, for a circular 



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), 



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 

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- 



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 = 

1 + * {l/a) 

(pi - ?Ja). 


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. 



[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 + 


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). 



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 



[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 


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 

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). 



At high pressures, ( L/a « 1), on the other hand, the formula 

becomes F = — — — and comparison of this equation with one 

1 + — 

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. 


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). 



[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) 


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 

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 


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, 


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 



(Chap VIII 

*• - * - K % 

E r representing the energy carried away by the reflected stieam 
From these equations together with (235) we find 




CvfiT K ” Tw) 

(2t t RT)» y 

whence, by (234), 

g = 2—2 (SfcrB20» 



(7 + l)cvV 


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 


4 Wk 


2 — a 4c K £ 

CL 7 + 1 V C Y 


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 



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 


d + </i T (7a 


(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 





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. 


[Chap VIII 


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 

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). 


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 


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)' 


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 - 



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 



[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 

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 

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 



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 


«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 “’ 





[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) 


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 



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) 


a function of the temperature. With these changes (2396) can be 

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: 











so 2 





0.24, 0.18 















(3) Hi on glass (rocalo, by (7) Ho on clean fresh tungsten 0.07 



Ho on clean long-heated 

( 4 ) Ho on glass, 130°C 




(6) On bright platinum: 

IIo on gas-filmed fresh t 




Ho on gas- filmed long-heated 





On Pt-blftckonoci Pt; 

Ho on clean fresh tungsten, 







— 78°0, 0,040; — 104°C 


(0) II, on oloan tungsten 

Ho on nickel 




(8) IIo on clean tungsten (old?) 0.17 

Ha on tungsten: 

lie on gas-filmed tungsten 

Ha film 

0 fe 0,14 



Oa or oxido film 

O.l to 0.2 

Ar on oloan tungston (old?) 0.63 

Ar on gas-filmed t (old?),, , 


(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 


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 

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) 



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), 



(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) 


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.* 


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). 



[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 



_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 


3 n T' . A 

- -ssr-o ~m Sill 0 
IT ppW 2 

3 »| RdT 

ip dy’ 


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 



(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 

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?) 



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 


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, 

■k a 4 p 
8 t)R T 

dp . 3ff pa? dT 
dx ' 4 T dx * 


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. 



[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 





(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 



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, 



omit 4 and choose for the mean pressure p — ^ (pi + p 2 ) — Pt — 


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). 



[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 



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 

* and Sente, A cits, Physik, 78, 43, 418 (1932); 79, 700 (1982); 8t, 
418 (1933). 

f Czerny and Hhttnuu, Hails. Physik, 80, 268 (1024), 



[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 


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 = 




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 



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 . 



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 


Sec. 191) 



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. 


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 



[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] 



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 

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 



[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] 



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: 


(i = 1, 2 • • • s); 


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,* 



[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 



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 


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 ] 



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) 

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 


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. 


The new integral occurring here is independent of E] let it bo denoted 
byCV.* Then 


<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)| 



[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 “ 

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 


approximately, in analogy with the expression last written for c r(7V). 

I . ^(E - | mv 2 ) dK = dK, 

(Cf Fig, 84 ) 


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) 



SE = 

3(iV - 1) 


— 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 


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.- 

-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 


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. 



[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| 



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 

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) 


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 


JVitiV 2 ! • • •’ 


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!). 



[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] 



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| 


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, 



[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 


2 « 


the last sum extending ovei all cells, and this second form of tho 
expiession lesultmg from substitution m the equation ^JV, = N. 


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] 



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 

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. 



(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 


- 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 = — — -> 



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 


coefficient of every dN'i, dN" • • - , we obtain 

log N\ + a> + P e 'i ~ 0) log -|- a" + j8e" = 0, * ' ‘ > 


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. 



[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 ] 



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) 


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 


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, 


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) 


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 



[Chap IX 

result is pdQ = ^(log 0 — log Ni) d$„ from which the term in C can 


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] 



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 


The quantity 2\e kT is what we called in Sec. 147 the state sum; 


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- 



(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 

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, 


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] 



Now consider that part of the ergodic layer in which A lies in Sg A . 
The volume of this part is 



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. 



[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 


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 

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. 


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«., 


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 



do) A do>B 


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] 



(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 

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 



[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 

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 

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 


Sec, 210 ) 


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), 


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. 


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 


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 

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| 



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 ) = 

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 



[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 

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, 



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 


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 * 


; <)'I' - 0; 


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). 



[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) 


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 

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 

H = Ho + Hr, 



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 



[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 


with suitable values of the constants at, a 2 
will be lepresented, as in (259), by 

2th E n t 


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 

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 


Sac. 217] 



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, 



[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 

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 



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 


ff/u h (l) • • - u w (N) = «/,(!) * • • «/(,-■>(* ' • • 1 



[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- 


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 

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 


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 


numbei of molecules lepiesented by t ]/„ as being in state 3 , repre- 
sents molecules as being m this state 


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] 



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 


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 


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 

= N, - E, 

Sec. 222] 



*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+“ 


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- 

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 


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 

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 



[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) 


The leveisiblo heat ab&oi bed duiing the piooess is then, in analogy 
with (261o), dQ = dE + dW = + dW, or 


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, 


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] 



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&). 



(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 


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 


^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) 


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] 




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) 


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 

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 


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, 


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 


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 



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 


first system is proportional to e kT and can be written 


-Ei 6 ~Sr 

Pi = Ce ** = -2— g. (272) 



the value of G being fixed by tho condition that — 1. 


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 




[Chap IX 

E lepresenting the mean eneigy, ^EJP X) and the additive constant 


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 

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 



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. 




[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] 



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 

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 


C sin uiir r sin /*2 tt 4 - sin mv ■ 

h h ( 


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 


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 



[Ciur X 


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 

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 


* 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 ] 



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- 




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 

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), 


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] 



X /(&) - ^h~»(2m)»vf £« /(£) d(. (281 b) 


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 


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, 


p 0-TV f 
it n Jo 

’ x^dx 
Be* ± 1 

= 1. 


The total energy E is similarly given by V\ “ntfdt - E or, since 
V = W/n, 



Nk Q TV f 
n Jo 

" x^dx 
Be * ± 1 



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. 


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 

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, 



3 V’ 


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 


Sec. 232] 


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 


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 , 




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 


13 = ^, 



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) 




[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 


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 


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] 



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 


dy i- 

dyw = 

3 N 

* 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, 



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^ 


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 


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] 



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), 


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 


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 



(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 

The mtegial by itself can be written 

(krfl f dx = {kT)%(% n] = (kT)™ LM + 1 ) 

JO \ £ / 3iY/2 

Sec. 236] 



(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 



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, 


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 



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 

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 _ 


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" ' 


affl dx 
Bo* H - 



(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., loc, til. 
t E, Ftmiui, Zeita. Phyaik, 36, 002 (1926). 



[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 ' 


» _ 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 

Cr = (dTNm) v “ 2 Nik { 1 ~ ' )’ 

Sue. 230] 



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 


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 


1 fdQ\ _ 6 „ , / 
cp ~ Nm\dT/p “ 2 Nlk \ 

y = SS = 5 (\ 

y c v 3 V 1 




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 + 


— ATfc^log - 

- JVfcjlog [ 

+ H + 


2 1 WQTft 


(WW 4 1 

nh 3 



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 



|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 




* + 1' 

y = 

r zM 

kT ' 



- kT=f o /)6 




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 


» a 0 } there is & tail of 

wellian form, since for large y or (f - approximately, 


^ f Hg-cr-fi >/Ar 


n t = 


The situation is illustrated graphically m Fig 89 

DeJon S ln § t0 1 = °t we have at this temperature 

Sec. 237] 



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 


in order to harmonize with the discontinuous n t curve. The equation 
then becomes 


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. 



[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. 


2Q mi C w (y + 

V * 

dy « n 

f° k + ^ 

J-xt e v 4 - 1 


(si - yW 

e~ v + 1 


(1/ + 8Si) H 

, e» + l 

dy = (a - y)» dy 


+ v)» - (xx - 



Jx, e” + 1 


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 


yjv_ = _ r l iog£ Az _ 

1 Jo 1 

e v + 

+ 2 



Hence the original equation can be written 



*i ?4 + 12 

' )- 

n t 

* Of Peirce, “Short Table of Integrals,” no 510 , 


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, 


£i = kTxt - fo 


In a similar way on