170
4 The Thermodynamics of Real Systems
Exercise 4.7
a. Find an expression for (∂H/∂P) T ,n for a gas obeying the truncated pressure virial equation
of state:
PV m RT + A 2 P
where A 2 is a function of T . It has been shown that the second pressure virial coefficient A 2
is equal to B 2 , the second virial coefficient.
b. Evaluate (∂H/∂P) T ,n for 1.000 mol of argon at 1.000 atm and 298.15 K. Data on B 2 and
dB 2 /dT are found in Example 4.3.
We can now obtain a useful relation between C P and C V for systems other than ideal
gases. Equation (2.5-11) is
C P C V +
∂U
∂V
T ,n
+ P
∂V
∂T
P,n
(4.3-7)
The C V term represents the energy change due to an increase in temperature that
would occur if the volume were constant. The (∂U/∂V ) (internal pressure) term represents the energy absorbed in raising the potential energy of intermolecular attraction,
and the P(∂V /∂T ) term represents work done against the pressure exerted by the
surroundings.
We now use the thermodynamic equation of state to write
C P C V +
T
∂P
∂T
V ,n
+ P − P
∂V
∂T
P,n
C V + T
∂P
∂T
V ,n
∂V
∂T
P,n
(4.3-8)
We apply the cycle rule, Eq. (B-15) of Appendix B, in the form:
∂P
∂T
V ,n
∂T
∂V
P,n
∂V
∂P
T ,n
−1
(4.3-9)
which gives
C P C V − T
∂P
∂V
T ,n
∂V
∂T
P,n
2
(4.3-10)
We obtain
C P C V − T
∂P
∂V
T ,n
∂V
∂T
P,n
2
C V +
TV α 2
κ T
(4.3-11)
where α is the coefficient of thermal expansion and κ T is the isothermal compressibility.
Since α (which is occasionally negative) is squared and since κ T is always positive, C P
is never smaller than C V .