Physical Chemistry Third Edition

(C. Jardin) #1

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)
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