172
4 The Thermodynamics of Real Systems
From the appendix, the value of C P,m for liquid benzene is 135.31 J K −1 mol −1 . The molar
volume is
V m
78.11 g mol −1
0.8765 g cm −3
1 m 3
10 6 cm 3
8.912 × 10 −5 m 3 mol −1
C V,m 135.31 J K −1 mol −1
−
(293.15 K)(8.912 × 10 −5 m 3 mol −1 )(1.237 × 10 −3 K −1 ) 2
9.67 × 10 −10 Pa −1
93.96 J K −1 mol −1
Exercise 4.9
The constant-pressure specific heat capacity of metallic iron at 298.15 K and 1.000 atm is equal
to 0.4498 J K −1 g −1 . The coefficient of thermal expansion is 3.55 × 10 −5 K −1 , the density
is 7.86 g cm −3 , and the isothermal compressibility is 6.06 × 10 −7 atm −1 . Find the constantvolume specific heat capacity at 298.15 K.
We can now obtain two additional relations for the heat capacities. From Eqs. (2.5-8)
and (2.4-4),
C P
∂H
∂T
P,n
(4.3-12)
C V
∂U
∂T
V ,n
(4.3-13)
We take Eq. (4.2-11) for a closed system and convert it to a derivative relation, specifying that P is fixed: ∂H
∂T
P,n
T
∂S
∂T
P,n
+ V
∂P
∂T
P,n
(4.3-14)
The derivative of P with respect to anything at constant P is equal to zero, so that
C P
∂H
∂T
P,n
T
∂S
∂T
P,n
(4.3-15)
Similarly,
C V
∂U
∂T
V ,n
T
∂S
∂T
V ,n
(4.3-16)
Exercise 4.10
Use Eq. (4.3-15), Eq. (4.3-16), and the cycle rule to show that
C P
C V
κ T
κ S
(4.3-17)
where κ S is the adiabatic compressibility,
κ S −
1
V
∂V
∂P
S,n
(4.3-18)