Physical Chemistry Third Edition

3.2 The Mathematical Statement of the Second Law: Entropy 117

Caratheodory^1 showed thatdqrev/T is an exact differential by a more formal

mathematical procedure. His argument is sketched briefly in Appendix D. It begins

with the fact that two reversible adiabats cannot cross. We now show that this is a

fact. We have already seen an example of it in the previous chapter when we derived

a formula for the reversible adiabat for an ideal gas with a constant heat capacity.

Equation (3.4-21b) is

TT 1

(

V 1

V

)nR/CV

(3.2-7)

This equation represents a family of functions ofTas a function ofV, one for each

initial state:

TT(V) (3.2-8)

No two curves in this family can intersect.

Constantin Caratheodory, 1872–1950,
was a Greek-German mathematician
who made many contributions in
mathematics in addition to his work in
thermodynamics.

To show that two reversible adiabats cannot cross for other systems we assume

the opposite of what we want to prove and then show that this assumption leads to a

contradiction with fact and therefore must be false. Assume that there are two different

reversible adiabats in the state space of a closed simple system and that the curves coin-

cide at state number 1, as depicted in Figure 3.6. We choose a state on each reversible

adiabat, labeled state number 2 and state number 3 such that the reversible process

leading from state 2 to state 3 hasq>0. Now consider a reversible cyclic process

1 → 2 → 3 →1. Since steps 1 and 3 are adiabatic,

qcycleq 2 > 0 (3.2-9)

Since∆U0 in any cyclic process,

wsurr−wcycleqcycleq 2 (3.2-10)

Heat transferred to a system undergoing a cyclic process has been converted completely

to work done on the surroundings, violating the second law of thermodynamics. The

source of this violation is the assumption that two reversible adiabats can cross. There-

fore, only one reversible adiabat passes through any given state point. The rest of

Caratheodory’s argument is summarized in Appendix D.

V

T

Two proposed

adiabats

that cross

2

3

1

Figure 3.6 Two Reversible Adiabats
That Cross (Assumption to Be Proved
False).
Entropy Changes for Adiabatic Processes

Consider a reversible adiabatic process for any kind of a system. We integrate Eq. (3.2-1)

along the curve representing the process. Sincedqrev0 for every step of the process,

∆Srev

∫

dqrev

T

0 (reversible adiabatic process) (3.2-11)

A reversible adiabatic process does not change the entropy of the system. We next

show that if a system undergoes an irreversible adiabatic process its entropy increases.

(^1) C. Caratheodory,Math. Ann., 67 , 335 (1909); J. G. Kirkwood and I. Oppenheim,Chemical Thermody-
namics, McGraw-Hill, New York, 1961, p. 31ff; J. deHeer,Phenomenological Thermodynamics, Prentice
Hall, Englewood Cliffs, NJ, 1986, p. 123ff.