Physical Chemistry Third Edition

3.2 The Mathematical Statement of the Second Law: Entropy 115

for all reversible cyclic processes in a closed system. The symbol

∮

represents a

line integral around a closed curve in the state space (beginning and ending at the

same state).

We begin with a Carnot cycle. From Eqs. (3.1-13) and (3.1-23)

q 1

Th

−

q 3

Tc

(3.2-3)

SinceTis constant on the isothermal segments and sincedqrev0 on the adiabatic

segments, the line integral for a Carnot cycle is

◦

∫

dqrev

T



1

Th

∫V 2

V 1

dqrev+ 0 +

1

Tc

∫V 4

V 3

dqrev+ 0 

q 1

Th

+

q 3

Tc

 0 (3.2-4)

so that Eq. (3.2-2) is established for any Carnot cycle.

We now show that Eq. (3.2-2) is valid for the reversible cyclic process of

Figure 3.4a. Steps 1, 3, and 5 are isothermal steps, and steps 2, 4, and 6 are adia-

batic steps. Let point 7 lie on the curve from state 6 to state 1, at the same temperature

as states 3 and 4, as shown in Figure 3.4b. We now carry out the reversible cyclic

process 1→ 2 → 3 → 7 →1, which is a Carnot cycle and for which the line integral

vanishes. We next carry out the cycle 7→ 4 → 5 → 6 →7. This is also a Carnot cycle,

so the line integral around this cycle vanishes. During the second cycle the path from

state 7 to state 3 was traversed from left to right. During the first cycle, the path from

state 3 to state 7 was traversed from right to left. When the two cyclic line integrals are

added, the integrals on these two paths cancel each other, and if we leave them both

out the sum of the two line integrals is unchanged. The sum of the two line integrals

is now a vanishing line integral for the cyclic process 1→ 2 → 3 → 4 → 5 → 6 →

7 →1, which is the cycle of Figure 3.4a.

We now show that Eq. (3.2-2) holds for any cyclic process made up of isothermal and

adiabatic reversible steps. Consider the process of Figure 3.5a, which can be divided

into three Carnot cycles, just as that of Figure 3.4a was divided into two Carnot cycles.

We can do the same division into Carnot cycles for any cycle that is made up of

reversible isothermal and adiabatic steps. If each Carnot cycle is traversed once, the

integrals on all of the paths in the interior of the original cycle are traversed twice,

once in each direction, and therefore cancel out when all of the line integrals are added

together. The exterior curve is traversed once, and the integral of Eq. (3.2-2) is shown

to vanish around the cycle. For example, Figure 3.5b shows a cycle equivalent to eight

V

6 6

7

1 2

(^34)
5
12
3 4
5
(a)
T
V
(b)
T
Figure 3.4 A Reversible Cycle of Isotherms and Adiabats.(a) The original cycle.
(b) The cycle with an added process.