Physical Chemistry Third Edition

D. Some Derivations of Formulas and Methods

This appendix expands on some of the things presented without derivation.

D.1 Caratheodory’s Theorem

Caratheodory devised a three-part proof that the mathematical statement of the second

law follows from a physical statement of the second law.^3 The first part is to establish

that in the state space of the system only one reversible adiabat passes through any

given point. This was shown in Chapter 3. The second part of the argument is to show

that this fact implies that a functionSexists whose differential vanishes along the

reversible adiabat on whichdqrevalso vanishes. This implies thatdqrevpossesses an

integrating factor, which is a functionythat produces an exact differentialdSwhen it

multiplies an inexact differential:

dSydqrev (D-1)

The third part of the proof is to show thaty 1 /Tis a valid choice for an integrating

factor.

We will give only a nonrigorous outline of Caratheodory’s proof.^4 We represent the

state of a simple closed system by a point in the state space withTon the vertical axis

andVon the horizontal axis. The main idea is that if there is a single curve in this space

along whichdqrevvanishes there is also a differential of a function,dS, which vanishes

on the same curve. Consider reversible adiabatic processes of a closed simple system

starting from a particular initial state. Since no two adiabats can cross the reversible

adiabat can be represented mathematically by a function

Tf(V) (D-2)

Equation (2.4-21b) is an example of such a function, holding for an ideal gas with

constant heat capacity, but for another system it would be whatever function applies to

that system. Equation (D-2) is the same as

0 f(V)−T (valid only on the curve) (D-3)

(^3) C. Caratheodory,Math. Ann., 67 , 335 (1909).
(^4) J. G. Kirkwood and I. Oppenheim,Chemical Thermodynamics, McGraw-Hill, New York, 1961, p. 31ff;
J. deHeer,Phenomenological Thermodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1986, p. 123ff.
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