Page 138 - Physical Chemistry

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Volume Dependence of U Section 4.4
We want (
U/
V) , which was discussed at the end of Sec. 2.6. The Gibbs equa- Thermodynamic Relations for a
T System in Equilibrium
tion (4.33) gives dU TdS PdV. The partial derivative (
U/
V) corresponds to
T
an isothermal process. For an isothermal process, the equation dU TdS PdV
becomes
dU T dS P dV T (4.46)
T
T
where the T subscripts indicate that the infinitesimal changes dU, dS, and dV are for a
constant-T process. Since (
U/
V) is wanted, we divide (4.46) by dV , the infinitesi-
T T
mal volume change at constant T, to give
dU T dS T
T P
dV T dV T
From the definition of a partial derivative, the quantity dU /dV is the partial derivative
T T
(
U/
V) , and we have
T
0U 0S
a b T a b P
0V T 0V T
Application of the Euler reciprocity relation (4.43) to the Gibbs equation dA
SdT PdV [Eq. (4.35)] gives the Maxwell relation (
S/
V) (
P/
T)
T V
[Eq. (4.45)]. Therefore
0U 0P aT
a b T a b P P (4.47)
0V T 0T V k
where (
P/
T ) a/k [Eq. (1.45)] was used. Equation (4.47) is the desired expression
V
for (
U/
V) in terms of easily measured properties.
T
Temperature Dependence of U
The basic equation (4.29) is the desired relation: (
U/
T) C .
V V
Temperature Dependence of H
The basic equation (4.30) is the desired relation: (
H/
T) C .
P P
Pressure Dependence of H
We want (
H/
P) . Starting with the Gibbs equation dH TdS VdP [Eq. (4.34)],
T
imposing the condition of constant T, and dividing by dP , we get dH /dP
T T T
TdS /dP V or
T T
0H 0S
a b T a b V
0P T 0P T
Application of the Euler reciprocity relation to dG SdT VdP gives (
S/
P)
T
(
V/
T ) [Eq. (4.45)]. Therefore
P
0H 0V
a b T a b V TVa V (4.48)
0P T 0T P

Temperature Dependence of S
The basic equation (4.31) for C is the desired relation:
P
0S C P
a b (4.49)
0T P T