Lectures on Gas Theory, Vol. 2

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Section III.

Propositions of General Mechanics Useful for Gas Theory . . . 62
§ 25. Conception of molecules as mechanical systems characterized
by generalized coordinates Coordinates that describe the position and orientation of a complex system using the minimum number of variables. . . . . . 62
§ 26. Liouville's Theorem A fundamental law in statistical mechanics regarding the constancy of phase-space volume. . . . . . . . . . . . . . . . . . . . . . . 66
§ 27. On the introduction of new variables into products of
differentials . . . . . . . . . . . . . . . . . . . . . . . . . . 69
§ 28. Application to the formulas of § 26 . . . . . . . . . . . . . 74
§ 29. Second proof of Liouville's Theorem . . . . . . . . . . . . . 77
§ 30. Jacobi's Theorem of the Last Multiplier . . . . . . . . . . . 82
§ 31. Introduction of the energy differential . . . . . . . . . . . 86
§ 32. Ergods original: "Ergoden"; a term used by Boltzmann to describe a specific type of ensemble or system path that covers all possible states. . . . . . . . . . . . . . . . . . . . . . . 89
§ 33. Concept of the momentoid A term coined by Boltzmann for a generalized momentum variable. . . . . . . . . . . . . . . . . . . . . 93
§ 34. Expressions for probability; mean values . . . . . . . . . . . 96
§ 35. General relationship to temperature equilibrium . . . . . . . 102

Section IV.

Gases with Compound Molecules . . . . . . . . . . . . . . 105
§ 36. Special consideration of compound gas molecules . . . . . . . 105
§ 37. Application of Kirchhoff's method to gases with
compound molecules . . . . . . . . . . . . . . . . . . . . . . 108
§ 38. On the possibility that for a very large number of
molecules, the variables determining their state
lie within very narrow limits . . . . . . . . . . . . . . . . . 110
§ 39. Consideration of collisions between two molecules . . . . . . 112
§ 40. Proof that the state distribution assumed in § 37
is not disturbed by collisions . . . . . . . . . . . . . . . . 117
§ 41. Generalizations . . . . . . . . . . . . . . . . . . . . . . . 120
§ 42. Mean value of the kinetic energy original: "lebendige Kraft"; literally "living force," the historical term for kinetic energy. corresponding to a momentoid . . . . . . . . . . . . . . . . . . . . . 122
§ 43. The ratio $\varkappa$ of specific heats . . . . . . . . . . . . . . . 127
§ 44. Values of $\varkappa$ for special cases . . . . . . . . . . . . . . . 128
§ 45. Comparison with experience i.e., experimental data. . . . . . . . . . . . . . . . . . 130
§ 46. Other mean values . . . . . . . . . . . . . . . . . . . . . . 133
§ 47. Consideration of molecules currently in the process
of interaction . . . . . . . . . . . . . . . . . . . . . . . . 135

Section V.

Derivation of van der Waals' Equation by means of the Virial
Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
§ 48. Clarification of the points where van der Waals' reasoning
requires supplementation . . . . . . . . . . . . . . . . . . . 138
§ 49. General concept of the virial A theorem relating the average kinetic energy of a system to its potential energy. . . . . . . . . . . . . . . . 139