Page 98 - Modern physical chemistry

ss Entropy and the Second Law

Consider a general unifonn system A and a unifonn ideal gas system B separated by
a fixed heat conducting membrane. Thus, the systems are kept at the same temperature.
The above argument can clearly be applied to changes in each system by itself. Fur-
thennore, it can be applied to changes in the combined system.
Since the heat added to the combined system equals that added to A plus that added
to B, we have
dq =dqA +dqB' [5.13]
For a reversible step, equation (5.11) changes (5.13) to

tdS=tA dSA +tB dSB· [5.14]
Because we consider entropy to be extensive and thus additive, we also have

[5.15]
Now, expression tA is a function of the independent variables of system A, ta a func-
tion of those of system B, and t those of the combined system. But for both (5.14) and
(5.15) to hold, these must be the same. Furthennore, since the only common indepen-
dent variable is the temperature T, this function must depend only on T.

5.3 Form for the Integrating Factor
The expression for the integrating factor can be induced from the behavior of an
ideal gas.
Consider n moles of gas subject only to reversible changes with all work done work
of compression. Then

dqrev =dE-dwrev =[aE) dT+(aE) dV +PdV. [5.16]
aT v av T
Assume that the gas is ideal so that both (4.35) and (4.37) apply. Also introduce the ideal
gas equation to construct
dV
dqrev = Cv dT + nRT- [5.17]
V
whence
dqrev =C dT +nRdV. [5.18]
v
T T V
Since Cv is a function of T alone, both terms on the right can be integrated without
specifying the path. Thus, dqr..JT is an exact differential. In equation (5.12), we can sub-
stitute T for t to get
dS = dqrev . [5.19]
T
From the argument at the end of section 5.2, we assume that relationship (5.19) holds
for any unifonn system or region.

5.4 Entropy Changes of Systems

Equation (5.19) can be applied to determine the entropy difference between any two
states that can be linked by a reversible process.