Page 121 - Physical chemistry eng

98 CHAPTER 5 Entropy and the Second and Third Laws of Thermodynamics

The equality holds only for a reversible process. We rewrite the Clausius inequality in
Equation (5.32) for an irreversible process in the form

dq
dS 7 (5.33)
T
For an irreversible process in an isolated system, dq = 0 . Therefore, we have again
proved that for any irreversible process in an isolated system, ¢S 7 0 .
How can the result from Equations (5.29) and (5.30) that
dU = dq - P external dV = TdS - PdV be reconciled with the fact that work and heat
are path functions? The answer is that dw Ú-PdV and dq … TdS , where the equali-
ties hold only for a reversible process. The result dq + dw = TdS - PdV states that
the amount by which the work is greater than –PdV and the amount by which the heat is
less than TdS in an irreversible process involving only PV work are exactly equal.
Therefore, the differential expression for dU in Equation (5.30) is obeyed for both
reversible and irreversible processes. In Chapter 6, the Clausius inequality is used to gen-
erate two new state functions, the Gibbs energy and the Helmholtz energy. These func-
tions allow predictions to be made about the direction of change in processes for which
the system interacts with its environment.
The Clausius inequality is next used to evaluate the cyclic integral dq>T for an
A
arbitrary process. Because dS = dq reversible >T , the value of the cyclic integral is zero
for a reversible process. Consider a process in which the transformation from state 1 to
state 2 is reversible, but the transition from state 2 back to state 1 is irreversible:
2 1
dq dq reversible dq irreversible
= + (5.34)
T T T
C 3 3
1 2
The limits of integration on the first integral can be interchanged to obtain

1 1
dq dq reversible dq irreversible
=- + (5.35)
T T T
C 3 3
2 2
Exchanging the limits as written is only valid for a state function. Because
dq reversible 7 dq irreversible
dq
… 0 (5.36)
T
C
where the equality only holds for a reversible process. Note that the cyclic integral of
an exact differential is always zero, but the integrand in Equation (5.36) is only an exact
differential for a reversible cycle.

The Change of Entropy
in the Surroundings and
5.7 ¢S total =¢S +¢S surroundings

As shown in Section 5.6, the entropy of an isolated system increases in a spontaneous
process. Is it always true that a process is spontaneous if ¢S for the system is positive?
As shown later, this statement is only true for an isolated system. In this section, a crite-
rion for spontaneity is developed that takes into account the entropy change in both the
system and the surroundings.
In general, a system interacts only with the part of the universe that is very close.
Therefore, one can think of the system and the interacting part of the surroundings as
forming an interacting composite system that is isolated from the rest of the universe.
The part of the surroundings that is relevant for entropy calculations is a thermal reser-
voir at a fixed temperature, T. The mass of the reservoir is sufficiently large that its tem-
perature is only changed by an infinitesimal amount dT when heat is transferred