Physical Chemistry Third Edition

1178 28 The Structure of Solids, Liquids, and Polymers

EXAMPLE28.8

a.Find the electronic contribution to the heat capacity of copper at 15 K, using the same

assumptions and data as in Example 28.7.

b.Find the ratio of the electronic contribution to the classical prediction of the electronic

contribution.

c.Find the ratio of the electronic contribution to the lattice vibration contribution for copper

at 15 K, using the Debye theory result with the Debye temperature 315 K.

Solution

a.From Example 28.7,εF0 1. 129 × 10 −^18 J.

εFεF0

(

1 −

(πkBT)^2

12 ε^2 F0

)

 1. 129 × 10 −^18 J

(

1 −

[π(1. 38 × 10 −^23 JK−^1 )(15 K)]^2

12(1. 129 × 10 −^18 J)^2

)

 1. 129 × 10 −^18 J

This value is not significantly different fromεF0. From Equation (28.3-15) the electronic

contribution to the heat capacity per unit volume is

Celec

π^2 NkB^2 T

2 εF



π^2 (8. 49 × 1028 m−^3 )(1. 38 × 10 −^23 JK−^1 )^2 (15 K)

2(1. 129 × 10 −^18 J)

 1. 06 × 103 JK−^1 m−^3

Cm,elec

(1. 06 × 103 JK−^1 m−^3 )(0.063546 kg mol−^1 )

8960 kg m−^3

 0 .00753 J K−^1 mol−^1

b.Ratio

π^2 kBT

3 εF



π^2 (1. 38 × 10 −^23 JK−^1 )(15 K)

3(1. 129 × 10 −^18 J)

 0. 0060

c.At 15 K,

CDebye

9 NkB

n^3 D

∫νD

0

(

hν

kBT

) 2

ehν/kBTν^2

(

ehν/kBT− 1

) 2 dν^3 NkBD

(

θD

T

)

uD

θD

T



315 K

15 K

 21. 0

The integral must be evaluated numerically or by use of tabulated values.aThe value of the

Debye function for 15 K is

D(21.0)

3

u^3 D

∫ 21. 0

0

u^4

eu

(eu− 1 )^2

du

3

(21.0)^3

(26.0) 0. 0084

CV,m,Debye 3 RD3(8.3145 J K−^1 mol−^1 )(0.0084) 0 .21JK−^1 mol−^1