Physical Chemistry Third Edition

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)