Physical Chemistry Third Edition

1168 28 The Structure of Solids, Liquids, and Polymers

whereU 0 is the vibrational ground-state energy. Since we have a continuous distribution

of normal mode frequencies in the Debye model, we replace the sum by an integral

ln(Z)−

U 0

kBT

−

∫νD

0

ln

(

1 −e−hν/kBT

)

g(ν)dν (28.2-22)

The vibrational energy of the Debye model is given by the analogue Eq. (28.2-7a)

UU 0 +

∫νD

0

hν

ehν/kBT− 1

g(ν)dνU 0 +

9 N

ν^3 D

∫νD

0

hν^3

ehν/kBT− 1

dν

U 0 +

9 NkBT

u^3 D

∫uD

0

u^3

eu− 1

du (28.2-23)

whereuhν/kBTanduDhνD/kBT. The integral in Eq. (28.2-23) cannot be eval-

uated in closed form, and must be evaluated numerically.

Differentiation of the formula forUgives a formula forCV:

CV

9 NkB

ν^3 D

∫νD

0

(

hν

kBT

) 2

ehν/kBTν^2

(

ehν/kBT− 1

) 2 dν^3 NkBD(ΘD/T) (28.2-24)

which defines theDebye function,DD(ΘD/T). TheDebye temperature,ΘD,is

defined by

ΘD

hνD

kB

(definition) (28.2-25)

Tables of the value of the Debye functionDare available.^3 A software package such

as Mathematica can easily carry out the evaluation. The appropriate value ofΘDfor a

given crystal is chosen by fitting heat capacity data to Eq. (28.2-24).

The Helmholtz energy is given by

AU 0 −kBT

∫νD

0

ln

(

1

1 −e−hν/kBT

)

g(ν)dν

U 0 +kBT

∫νD

0

ln

(

1 −e−hν/kBT

)

g(ν)dν (28.2-26)

The entropy is given by combining the expressions forUandA:

S

U−A

T



9 N

ν^3 DT

∫νD

0

hν^3

ehν/kBT− 1

dν+kB

∫νD

0

ln

(

1

1 −e−hν/kBT

)

g(ν)dν (28.2-27)

Figure 28.9 shows the heat capacities of several elements, along with curves rep-

resenting the Debye function for the Debye temperatures given. At high temperatures,

the Debye expression conforms to the law of Dulong and Petit.

Exercise 28.6

Show that the energy expression in Eq. (28.2-23) reduces toUU 0 + 3 NkBTfor high tem-

peratures, so thatCV 3 NkB, conforming to the law of Dulong and Petit. Use the fact that for

small values ofu,eucan be approximated by 1+u.

(^3) N. Davidson,Statistical Mechanics, McGraw-Hill, New York, 1962, p. 359.