4.3 Additional Useful Thermodynamic Identities
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4.3
Additional Useful Thermodynamic Identities
We have asserted without proof that (∂U/∂V ) T ,n vanishes for an ideal gas. We can now
prove this assertion and can obtain a formula that allows evaluation of this derivative
for nonideal gases and liquids. We convert Eq. (4.2-3) to a derivative equation by
nonrigorously “dividing” by dV , converting the quotients to partial derivatives, and
specifying that T and n are held fixed. The process is mathematically indefensible, but
gives the correct derivative relation:
∂U
∂P
T ,n
T
∂S
∂V
T ,n
− P
∂V
∂V
T ,n
T
∂S
∂V
T ,n
− P
(4.3-1)
We apply the Maxwell relation of Eq. (4.2-18) to the first term to obtain
∂U
∂V
T ,n
T
∂P
∂T
V ,n
− P
(the thermodynamic
equation of state)
(4.3-2)
The relation shown in Eq. (4.3-2) is called the thermodynamic equation of state. For
an ideal gas,
T
∂P
∂T
V ,n
T
nR
V
P
(4.3-3)
so that
∂U
∂V
T ,n
P − P 0
(4.3-4)
It is now necessary only to specify that PV nRT to define an ideal gas.
E X A M P L E 4.7
Show that P is proportional to T in an ideal gas if V and n are constant, using Eqs. (4.3-1)
and (4.3-4).
Solution
For an ideal gas
T
∂P
∂T
V ,n
P
∂P
∂T
V ,n
P
T
At constant V and n,
1
P
dP
1
T
dT
ln (P) ln (T ) + ln (constant)
P
T
constant at constant V and n