2.3 New Thermodynamic State Functions
41
C P = lim
T →0
(δQ) P
T
≡
∂H
∂T
P
(2.3.1c)
as the heat capacity at constant pressure.
The thermodynamic state function H plays much the same role for isobaric
processes that the internal energy U plays for isochoric processes, and is known
as the enthalpy. It may also be represented as the Legendre transform (see Sect. B.2)
H = U + P V
(2.3.2)
of the internal energy U(S, V ) that replaces the independent extensive variable V
with the independent intensive variable P , so that the enthalpy can be written as
H = H (S, P ).
If we form the total differential dH using Eq. (2.3.2), we obtain
dH = dU + P dV + V dP
which, upon employing expression (2.2.12) for dU , becomes
dH = T dS + V dP .
(2.3.3)
Because this expression for dH also arises from the first and second laws, we therefore say that S and P are the natural thermodynamic variables for enthalpy. It will
be clear from Eq. (2.3.3) that the enthalpy provides the appropriate thermodynamic
energy state function for the description of isentropic and isobaric thermodynamic
processes.
Comparison of Eq. (2.3.3) with the formal mathematical representation,
dH ≡
∂H
∂S
P
dS +
∂H
∂P
S
dP ,
for the total differential dH shows that the dependent variables temperature and
volume are related to enthalpy by
T =
∂H
∂S
P
and
V =
∂H
∂P
S
,
(2.3.4a)
respectively. Equality of the mixed second partial derivatives of H then gives the
second Maxwell relation
∂T
∂P
S
=
∂V
∂S
P
,
(2.3.4b)
which carries an interpretation similar to that for the first Maxwell relation, but
involving isentropic and isobaric processes.