Physical Chemistry Third Edition

28.2 Crystal Vibrations 1169

6

4

Pb


D
 90.3 K

Ag


D
 213 K

AI


D
 389 K

C (diamond)

(^2)
D
1890 K
1.0 2.0
log 10 (T/ 1 K)
3.0
0
C
/ cal KV
^1
mol
^1
Figure 28.9 The Heat Capacity of Several Elements as a Function of Tem-
perature, with Debye Curves.From G. N. Lewis and M. Randall,Chemical Ther-
modynamics, 2nd ed., rev. by K. S. Pitzer and L. Brewer, McGraw-Hill, New York,
1961, p. 56.
The Debye temperature of diamond is equal to 1890 K, whereas its Einstein
temperature is 1320 K. Since the Einstein frequency should correspond to an average
frequency in the Debye model, the relationship between these values is reasonable.

EXAMPLE28.5

Obtain a formula for an average Debye frequency

〈v〉

∫νD

0

νg(ν)dν

∫νD

0

g(ν)dν

Solution

Sinceg(ν)∝ν^2 ,

νav

∫νD

0

ν^3 dν

∫νD

0

ν^2 dν



1

4

ν^4 D

1

3

ν^3 D



3

4

νD

νE

kBθE

h



(1. 3807 × 10 −^23 JK−^1 )(1320 K)

6. 6261 × 10 −^34 Js

 2. 75 × 1013 s−^1

νav

3

4

kBθD

h



3(1. 3807 × 10 −^23 JK−^1 )(1890 K)

4(6. 6261 × 10 −^34 Js)

 2. 95 × 1013 s−^1

The average frequency is larger than the Einstein frequency by only 7%.

For small temperatures (Tsmaller thanΘD/10),uDis large compared with values

ofuhν/kBTthat make significant contributions to the integral in Eq. (28.2-23), and