Physical Chemistry , 1st ed.

low temperatures, predicting a lower heat capacity than is measured experi-

mentally. In fact, using the mathematics of limits, it can be shown that equa-

tion 18.66 predicts the following:

Tlim→ 0 CV^3 Nk



T

E



2

eE/T (18.67)

This is not the T^3 -dependence, as experimental measurements suggest.

Therefore, while Einstein’s application of statistical thermodynamics to crys-

tals was useful, it has its limitations. (It might be considered similar, in some

respects, to Bohr’s attempt to describe electron energy levels by assuming

quantized angular momentum. It worked in some respects—mostly in appli-

cation to hydrogen atoms—but had its deficiencies in a more global sense.)

Peter Debye, a Dutch physical chemist after whom the Debye-Hückel theory

is partly named (see Figure 18.5), expanded on Einstein’s work. Rather than as-

sume that all atoms in a crystal had the same vibrational frequency (as Einstein

had presumed), Debye suggested that the possible vibrational motions of

the atoms in a crystal could have any frequency from zero to a certain maxi-

mum. That is, he suggested that atoms could have a range,or distribution,of

frequencies.

Using an argument similar to that used to determine the number of trans-

lational states for qtrans, Debye deduced that the equation for the distribution

of frequencies, symbolized by g( ), is

g( ) d

(

9

D

N

)^3

^2 d (18.68)

where (^) Dis the maximum frequency that the atoms in the crystal can have and
is called the Debye frequency.The distribution function g( ) is a function of the
frequencies , but is subject to the condition that the total number of vibra-
tions is 3N,where Nis the number of atoms in the crystal. The mathematical
way of expressing this restriction is
(^) D
 0
g( ) d 3 N
Equation 18.68 is therefore applicable for values of between 0 and (^) D.If
646 CHAPTER 18 More Statistical Thermodynamics
25
0
0
T/E
1.0
C
[J/(mol

• K

)]

20

15

10

5

0.90.80.70.60.50.40.30.20.1

Figure 18.4 The Einstein theory of heat capacity of crystals agrees reasonably well with ex-

perimental measurements.

Figure 18.5 Peter J. W. Debye (1884–1966)

was a Dutch-American physical chemist who

made important advances in the understanding

of ionic solutions and dipoles in molecules. He

also formulated an acceptable theory of the ther-

modynamic properties of crystals at low temper-

atures. He was awarded the 1936 Nobel Prize in

chemistry for his work.

Cornell University, courtesy AIP Emilio Sergre Visual Archives