Page 131 - Physical Chemistry

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Chapter 4 4.3 THE GIBBS AND HELMHOLTZ ENERGIES
Material Equilibrium
We now use (4.8) to deduce conditions for material equilibrium in terms of state func-
tions of the system. We first examine material equilibrium in a system held at constant
T and V. Here dV 0 and dT 0 throughout the irreversible approach to equilibrium.
The inequality (4.8) involves dS and dV, since dw PdV for P-V work only. To
introduce dT into (4.8), we add and subtract SdT on the right. Note that SdT has the
dimensions of entropy times temperature, the same dimensions as the term TdS that
appears in (4.8), so we are allowed to add and subtract SdT. We have

dU T dS S dT S dT dw (4.9)
The differential relation d(uy) udy y du [Eq. (1.28)] gives d(TS) TdS SdT,
and Eq. (4.9) becomes
dU d1TS2 S dT dw (4.10)
The relation d(u y) du dy [Eq. (1.28)] gives dU d(TS) d(U TS), and
(4.10) becomes
d1U TS2 S dT dw (4.11)

If the system can do only P-V work, then dw PdV (we use dw rev since we are as-
suming mechanical equilibrium). We have
d1U TS2 S dT P dV (4.12)

At constant T and V, we have dT 0 dV and (4.12) becomes

const. T and V, closed syst. in
d1U TS2 0 (4.13)
therm. and mech. equilib., P-V work only
where the equality sign holds at material equilibrium.
Therefore, for a closed system held at constant T and V, the state function U TS
continually decreases during the spontaneous, irreversible processes of chemical
reaction and matter transport between phases until material equilibrium is reached. At
material equilibrium, d(U TS) equals 0, and U TS has reached a minimum. Any
spontaneous change at constant T and V away from equilibrium (in either direction)
would mean an increase in U TS, which, working back through the preceding equa-
tions from (4.13) to (4.3), would mean a decrease in S S S . This decrease
univ syst surr
would violate the second law. The approach to and achievement of material equilib-
rium is a consequence of the second law.
The condition for material equilibrium in a closed system capable of doing only
P-V work and held at constant T and V is minimization of the system’s state function
U TS. This state function is called the Helmholtz free energy, the Helmholtz
energy, the Helmholtz function, or the work function and is symbolized by A:

A U TS (4.14)*
Now consider material equilibrium for constant T and P conditions, dP 0, dT
0. To introduce dP and dT into (4.8) with dw PdV, we add and subtract SdT and
VdP:
dU T dS S dT S dT P dV V dP V dP
dU d1TS2 S dT d1PV2 V dP

d1U PV TS2 S dT V dP
d1H TS2 S dT V dP (4.15)