Physical Chemistry Third Edition

116 3 The Second and Third Laws of Thermodynamics: Entropy

V

V

T

(a)

(c)

T

V

(b)

T

Figure 3.5 Reversible Cycles of Isotherms and Adiabats.(a) A cycle equivalent to

three Carnot cycles. (b) A cycle equivalent to eight Carnot cycles. (c) A cycle equivalent to

a large number of Carnot cycles.

Carnot cycles, and Figure 3.5c shows a more complicated cycle that can be divided

into a large number of Carnot cycles. We conclude that the line integral ofdqrev/T

around any path consisting of reversible isotherms and adiabats vanishes.

In order to represent an arbitrary cycle we construct reversible isothermal and adi-

abatic steps that are smaller and smaller in size, until the curve of the arbitrary cycle

is more and more closely approximated by isothermal and adiabatic steps. In the limit

that the sizes of the steps approach zero, any curve is exactly represented and the line

integral ofdqrev/Tvanishes for any cycle. The differentialdSdqrev/Tis therefore

exact andSis a state function. For a simple system containing one substance and one

phase,Smust be a function of three state variables. We can write

SS(T,V,n) (3.2-5)

or

SS(U,V,n) (3.2-6)

and so on. Because we have defined only the differential of the entropy, any constant can

be added to the value of the entropy without any significant change, just as a constant

can be added to the value ofUwithout any significant change.