3. Maxwell's proof of the velocity distribution law; frequency of collisions

Kapitel in diesem Buch

1. Frontmatter I
2. CONTENTS V
3. Translator's Introduction 1
4. PART I THEORY OF GASES WITH MONATOMIC MOLECULES, WHOSE DIMENSIONS ABE NEGLIGIBLE COMPARED TO THE MEAN FREE PATH
5. NOTE ON LITERATURE CITATIONS 20
6. Foreword 21
7. Introduction 23
8. 1. Mechanical analogy for the behavior of a gas 23
9. 2. Calculation of the pressure of a gas 30
10. CHAPTER I THE MOLECULES ARE ELASTIC SPHERES. EXTERNAL FORCES AND VISIBLE MASS MOTION ABE ABSENT 36
11. 3. Maxwell's proof of the velocity distribution law; frequency of collisions 36
12. 4. Continuation; values of the variables after the collision; collisions of the opposite kind 43
13. 5. Proof that Maxwell's velocity distribution is the only possible one 49
14. 6. Mathematical meaning of the quantity H 55
15. 7. The Boyle-Charles-Avogadro law. Expression for the heat supplied 62
16. 8. Specific heat. Physical meaning of the quantity H 68
17. 9. Number of collisions 75
18. 10. Mean free path 82
19. 11. Basic equation for the transport of any quantity by the molecular motion 87
20. 12. Electrical conduction and viscosity of the gas 91
21. 13. Heat conduction and diffusion of the gas 98
22. 14. Two kinds of approximations; diffusion of two different gases 104
23. CHAPTER II THE MOLECULES ABE CENTERS OF FORCE. CONSIDERATION OF EXTERNAL FORCES AND VISIBLE MOTIONS OF THE GAS 110
24. 15. Development of partial differential equations for f and F 110
25. 16. Continuation. Discussion of the effects of collisions 114
26. 17. Time-derivatives of sums over all molecules in a region 123
27. 18. More general proof of the entropy theorem. Treatment of the equations corresponding to the stationary state 131
28. 19. Aerostatics. Entropy of a heavy gas whose motion does not violate Equations (147) 141
29. 20. General form of the hydrodynamic equations 147
30. CHAPTER III THE MOLECULES REPEL EACH OTHER WITH A FORCE INVERSELY PROPORTIONAL TO THE FIFTH POWER OF THEIR DISTANCE 161
31. 21. Integration of the terms resulting from collisions 161
32. 22. Relaxation time. Hydrodynamic equations corrected for viscosity. Calculation of Bb using spherical functions 172
33. 23. Heat conduction. Second method of approximate calculations 182
34. 24. Entropy for the case when Equations (147) are not satisfied. Diffusion 197
35. PART II VAN DER WAALS' THEORY; GASES WITH COMPOUND MOLECULES; GAS DISSOCIATION ; CONCLUDING REMARKS
36. Foreword 215
37. CHAPTER I FOUNDATIONS OF VAN DER WAALS' THEORY 217
38. 1. General viewpoint of van der Waals 217
39. 2. External and internal pressure 220
40. 3. Number of collisions against the wall 221
41. 4. Relation between molecular extension and collision number 222
42. 5. Determination of the impulse imparted to the molecules 224
43. 6. Limits of validity of the approximations made in §4 226
44. 7. Determination of internal pressure 227
45. 8. An ideal gas as a thermometric substance 230
46. 9. Temperature-pressure coefficient. Determination of the constants of van der Waals' equation 231
47. 10. Absolute temperature. Compression coefficient 234
48. 11. Critical temperature, critical pressure, and critical volume 236
49. 12. Geometric discussion of the isotherms 240
50. 13. Special cases 243
51. CHAPTER II PHYSICAL DISCUSSION OF THE VAN DER WAALS' THEORY 246
52. 14. Stable and unstable states 246
53. 15. Undercooling. Delayed evaporation 248
54. 16. Stable coexistence of both phases 250
55. 17. Geometric representation of the states in which two phases coexist 253
56. 18. Definition of the concepts gas, vapor, and liquid 256
57. 19. Arbitrariness of the definitions of the preceding section 257
58. 20. Isopycnic changes of state 259
59. 21. Calorimetry of a substance following van der Waals' law 261
60. 22. Size of the molecule 264
61. 23. Relations to capillarity 265
62. 24. Work of separation of the molecules 268
63. CHAPTER III PRINCIPLES OF GENERAL MECHANICS NEEDED FOR GAS THEORY 271
64. 25. Conception of the molecule as a mechanical system characterized by generalized coordinates 271
65. 26. Liouville's Theorem 274
66. 27. On the introduction of new variables in a product of differentials 278
67. 28. Application to the formulas of §26 283
68. 29. Second proof of Liouville's theorem 285
69. 30. Jacobi's theorem of the last multiplier 290
70. 31. Introduction of the energy differential 294
71. 32. Ergoden 297
72. 33. Concept of the momentoid 300
73. 34. Expression for the probability; average values 304
74. 35. General relationship to temperature equilibrium 310
75. CHAPTER IV GASES WITH COMPOUND MOLECULES 313
76. 36. Special treatment of compound molecules 313
77. 37. Application of Kirchhoff's method to gases with compound molecules 315
78. 38. On the possibility that the states of a very large number of molecules can actually lie within very narrow limits 317
79. 39. Treatment of collisions of two molecules 319
80. 40. Proof that the distribution of states assumed in §37 will not be changed by collisions 323
81. 41. Generalizations 325
82. 42. Mean value of the kinetic energy corresponding to a momentoid 327
83. 43. The ratio of specific heats, K 331
84. 44. Value of k for special cases 332
85. 45. Comparison with experiment 334
86. 46. Other mean values 336
87. 47. Treatment of directly interacting molecules 338
88. CHAPTER V DERIVATION OF VAN DER WAALS' EQUATION BY MEANS OF THE VIRIAL CONCEPT 341
89. 48. Specification of the point at which van der Waals' mode of reasoning requires improvement 341
90. 49. More general concept of the virial 342
91. 50. Virial of the external pressure acting on a gas 344
92. 51. Probability of finding the centers of two molecules at a given distance 346
93. 52. Contribution to the virial resulting from the finite extension of the molecules 350
94. 53. Virial of the van der Waals cohesion force 352
95. 54. Alternatives to van der Waals' formulas 354
96. 55. Virial for any arbitrary law of repulsion of the molecules 356
97. 56. The principle of Lorentz's method 358
98. 57. Number of collisions 361
99. 58. More exact value of the mean free path. Calculation of W/ according to Lorentz's method 364
100. 59. More exact calculation of the space available for the center of a molecule 365
101. 60. Calculation of the pressure of the saturated vapor from the laws of probability 367
102. 61. Calculation of the entropy of a gas satisfying van der Waals' assumptions, using the calculus of probabilities 370
103. CHAPTER VI THEORY OF DISSOCIATION 376
104. 62. Mechanical picture of the chemical affinity of monovalent similar atoms 376
105. 63. Probability of chemical binding of an atom with a similar one 379
106. 64. Dependence of the degree of dissociation on pressure 383
107. 65. Dependence of the degree of dissociation on temperature 385
108. 66. Numerical calculations 389
109. 67. Mechanical picture of the affinity of two dissimilar monovalent atoms 393
110. 68. Dissociation of a molecule into two heterogeneous atoms 396
111. 69. Dissociation of hydrogen iodide gas 398
112. 70. Dissociation of water vapor 399
113. 71. General theory of dissociation 402
114. 72. Relation of this theory to that of Gibbs 406
115. 73. The sensitive region is uniformly distributed around the entire atom 408
116. CHAPTER VII SUPPLEMENTS TO THE LAWS OF THERMAL EQUILIBRIUM IN GASES WITH COMPOUND MOLECULES 412
117. 74. Definition of the quantity H, which measures the probabilities of states 412
118. 75. Change of the quantity H through intramolecular motion 414
119. 76. Characterization of the first special case considered 415
120. 77. Form of Liouville's theorem in the special case considered 417
121. 78. Change of the quantity if as a consequence of collisions 419
122. 79. Most general characterization of the collision of two molecules 422
123. 80. Application of Liouville's theorem to collisions of the most general kind 424
124. 81. Method of calculation with finite differences 427
125. 82. Integral expression for the most general change of H by collisions 431
126. 83. Detailed specification of the case now to be considered 432
127. 84. Solution of the equation valid for each collision 433
128. 85. Only the atoms of a single type collide with each other 435
129. 86. Determination of the probability of a particular kind of central motion 437
130. 87. Characterization of our assumption about the initial state 441
131. 88. On the return of a system to a former state 443
132. 89. Relation to the second law of thermodynamics 444
133. 90. Application to the universe 446
134. 91. Application of the probability calculus in molecular physics 448
135. 92. Derivation of thermal equilibrium by reversal of the time direction 450
136. 93. Proof for a cyclic series of a finite number of states 453
137. BIBLIOGRAPHY 454
138. INDEX 483