Page 135 - Physical Chemistry

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Chapter 4 The heat capacities C and C have alternative expressions that are also basic equa-
V
P
Material Equilibrium tions. Consider a reversible flow of heat accompanied by a temperature change dT. By
definition, C dq /dT, where X is the variable (P or V) held constant. But dq rev
X
X
TdS, and we have C TdS/dT, where dS/dT is for constant X. Putting X equal to V
X
and P, we have
0S 0S
C T a b , C T a b closed syst. in equilib. (4.31)*
V
P
0T V 0T P
The heat capacities C and C are key properties since they allow us to find the rates
V
P
of change of U, H, and S with respect to temperature [Eqs. (4.29) to (4.31)].
The relation dU TdS PdV in (4.25) applies to a reversible process in a closed
system. Let us consider processes that change the system’s composition. There are two
ways the composition can change. First, one can add or remove one or more sub-
stances. However, the requirement of a closed system (dU dq dw for an open sys-
tem) rules out addition or removal of matter. Second, the composition can change by
chemical reactions or by transport of matter from one phase to another in the system.
The usual way of carrying out a chemical reaction is to mix the chemicals and allow
them to reach equilibrium. This spontaneous chemical reaction is irreversible, since
the system passes through nonequilibrium states. The requirement of reversibility
(dq TdS for an irreversible chemical change) rules out a chemical reaction as ordi-
narily conducted. Likewise, if we put several phases together and allow them to reach
equilibrium, we have an irreversible composition change. For example, if we throw a
handful of salt into water, the solution process goes through nonequilibrium states and
is irreversible. The equation dU TdS PdV does not apply to such irreversible
composition changes in a closed system.
We can, if we like, carry out a composition change reversibly in a closed system.
If we start with a system that is initially in material equilibrium and reversibly vary the
temperature or pressure, we generally get a shift in the equilibrium position, and this
shift is reversible. For example, if we have an equilibrium mixture of N , H , and NH 3
2
2
(together with a catalyst) and we slowly and reversibly vary T or P, the position of
chemical-reaction equilibrium shifts. This composition change is reversible, since the
closed system passes through equilibrium states only. For such a reversible composi-
tion change, dU TdS PdV does apply.
This section deals only with reversible processes in closed systems. Most com-
monly, the system’s composition is fixed, but the equations of this section also apply
to processes where the composition of the closed system changes reversibly, with the
system passing through equilibrium states only.

The Gibbs Equations
We now derive expressions for dH, dA, and dG that correspond to dU TdS PdV
[Eq. (4.25)] for dU. From H U PV and dU TdS PdV, we have
dH d1U PV 2 dU d1PV 2 dU P dV V dP
1T dS P dV2 P dV V dP
dH T dS V dP (4.32)
Similarly,
dA d1U TS2 dU T dS S dT T dS P dV T dS S dT
S dT P dV
dG d1H TS2 dH T dS S dT T dS V dP T dS S dT
S dT V dP
where (4.32) was used.