8.2 Thermodynamics of Polarizable Media
417
and
G
(T , P , E 0 ) ≡ U
(S, V , D) + P V − T S + E 0 D.
(8.2.10c)
These thermodynamic potentials play much the same roles in the thermodynamics
of dielectric systems as do their counterparts in the thermodynamics of nondielectric media.
Thermodynamic relationships associated with a dielectric medium have many
analogies with those obtained in terms of pressure and volume, such as expression (2.4.10) relating the partial derivative of the internal energy with respect to
system volume to the system pressure and temperature, and expression (2.5.1c) for
the difference C P − C V between the constant pressure and constant-volume heat
capacities for non-dielectric media.
Example 8.1 Dielectric medium equivalent of Eq. (2.4.10).
We shall begin with the extended version of the combined 1st and 2nd Laws
expression for a uniform constant-volume thermodynamic system. If we treat the
entropy as a function of T , and D, its total differential dS is given by
dS =
∂S
∂T
V ,D
dT +
∂S
∂D
T ,V
dD.
Upon substituting this equation for dS into Eq. (2.3.9a) and collecting terms, we
obtain dU as
dU
= T
∂S
∂T
V ,D
dT +
T
∂S
∂D
T ,V
− E 0
dD
≡
∂U
∂T
V ,D
dT +
∂U
∂D
T ,V
dD.
It is perhaps worth noting that we are now considering the internal energy as a
function of T and D, rather than of S and D. Comparison of the first expression with
the formal mathematical definition of the total differential dU gives, in particular,
the expression
∂U
∂D
T ,V
= T
∂S
∂D
T ,V
− E 0 .
To replace the entropy partial derivative in this expression, we shall employ
a Maxwell-type relation derivable from the Helmholtz energy A elec (T , D) ≡
U elec (S, E 0 ) − T S − E 0 D obtained from U elec (S, E 0 ), namely
∂S
∂D
T ,V
=
∂E 0
∂T
V ,D
,