172 4 The Thermodynamics of Real Systems
From the appendix, the value ofCP, mfor liquid benzene is 135.31 J K−^1 mol−^1. The molar
volume is
Vm
(
78 .11 g mol−^1
0 .8765 g cm−^3
)(
1m^3
106 cm^3
)
8. 912 × 10 −^5 m^3 mol−^1
CV, m 135 .31JK−^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 .96JK−^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 constant-
volume 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),
CP
(
∂H
∂T
)
P,n
(4.3-12)
CV
(
∂U
∂T
)
V,n
(4.3-13)
We take Eq. (4.2-11) for a closed system and convert it to a derivative relation, speci-
fying thatPis fixed:
(
∂H
∂T
)
P,n
T
(
∂S
∂T
)
P,n
+V
(
∂P
∂T
)
P,n
(4.3-14)
The derivative ofPwith respect to anything at constantPis equal to zero, so that
CP
(
∂H
∂T
)
P,n
T
(
∂S
∂T
)
P,n
(4.3-15)
Similarly,
CV
(
∂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
CP
CV
κT
κS
(4.3-17)
whereκSis theadiabatic compressibility,
κS−
1
V
(
∂V
∂P
)
S,n
(4.3-18)