274 Chapter 9Functions of several variables
EXAMPLE 9.22Maxwell relations
It follows from equations (9.42) that
so that
This is one of four Maxwell relations of importance in thermodynamics. The other
three are derived from the differentials of the enthalpyH 1 = 1 U 1 + 1 pV, the Helmholtz
energyA 1 = 1 U 1 − 1 TS, and the Gibbs energyG 1 = 1 H 1 − 1 TS. For example, for the enthalpy
(and using (9.40) for dU),
dH 1 = 1 dU 1 + 1 pdV 1 + 1 Vdp 1 = 1 TdS 1 + 1 Vdp
The quantity dHis the total differential of Has a function of Sand p, so that
and the corresponding Maxwell relation is
Similarly,
dG 1 = 1 −SdT 1 + 1 Vdp
0 Exercise 56
S
p
V
T
Tp
→
∂
∂
=−
∂
∂
dA SdT pdV
S
V
p
T
TV
=− − →
∂
∂
=
∂
∂
Sp
T
p
V
S
∂
∂
=
∂
∂
T
H
S
V
H
p
pS
=
∂
∂
,=
∂
∂
SV
T
V
p
S
∂
∂
=−
∂
∂
SV
T
V
U
VS
p
S
U
SV
∂
∂
=
∂
∂∂
,−
∂
∂
=
∂
∂∂
22