Physical Chemistry , 1st ed.

were greater than (^) D,g( ) 0.
The frequency distribution function in equation 18.68 can be substituted
into the statistical thermodynamic expressions for the various state functions,
and various thermodynamic properties determined for crystals. We are inter-
ested in the expression for the heat capacity. It is (omitting the details of the
derivation):
CV 9 Nk
h
k
T
D

3
h D/kT
0
k
h
T

5
(eh^
e
/k
h
T
/

kT
1)^2
 4 d
This is a complicated expression that demands some simplification. First, we
substitute for the expression h /kTby defining x h /kT. Second, we define
the Debye temperature as
D 
h
k

D (18.69)

The expression for the heat capacity becomes

CV 9 Nk



T

D



3

D/T

0



(ex

x



(^4) ex
1)^2
dx (18.70)
The integral in equation 18.70 cannot be solved analytically, but its value can
be determined numerically. Just like the Einstein treatment of heat capacities
of crystals, the Debye temperature Dis selected so that the numerical evalua-
tion of equation 18.70 agrees as closely as possible with experimental data.
Figure 18.6 shows curve fits of experimental data, and Table 18.7 lists some val-
ues ofD.
Applying limits to equation 18.70 shows that
Tlim→ 0 CV

12

5

4

Nk



T

D



3

This expression shows that the low-temperature heat capacity varies with the

cube of the absolute temperature. This is what is seen experimentally (remem-

ber that a major failing of the Einstein treatment was that it didn’t predict the

proper low-temperature behavior ofCV), so the Debye treatment of the heat

18.10 Crystals 647

25

0

0

T/D

1.6

Heat capacity [J/(mol

• K

)]^20

15

10

5

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 18.6 The Debye theory of heat capacity of crystals agrees better with experimental

values of heat capacity at low temperatures.

Table 18.7 Debye temperatures of
various crystals
Material D(K)a
Al 390
C (dia) 1860
Pb 88
Na 150
Ag 215
Au 170
Fe 420
Pt 225
Gd 169
Sc 345

aIt can be shown that E (^34) D.