Physical Chemistry Third Edition

28.2 Crystal Vibrations 1165

This formula agrees with the empiricallaw of Dulong and Petit, which states that the

molar heat capacity of atomic crystals is approximately equal to 3R. Most metals have

a sufficiently small Einstein temperature that the law of Dulong and Petit applies quite

well near room temperature, but the Einstein frequency of diamond is sufficiently high

that this law does not apply to it at room temperature.

EXAMPLE28.2

Show that Eq. (28.2-9) is correct.

Solution

Equation (28.2-7b) is

CV

(

∂U

∂T

)

V,N

 3 NkB

(

hν

kBT

) 2

ehν/kBT

(ehν/kBT−1)^2

lim

T→∞

CV lim

T→∞

(

3 NkB

(

hν

kBT

) 2

1 −hν/kBT

(1−hν/kBT−1)^2

)

 3 NkB

(

hν

kBT

) 2

1

(hν/kBT)^2

 3 NkB 3 nR

Exercise 28.4

Using the Einstein theory, calculate the molar heat capacity of a diamond crystal at 298.15 K,

at 500.0 K, and at 1320 K. Compare each value with 3R. The experimental value at 298.15 K is

6.113 J K−^1 mol−^1.

The Debye Crystal Model

This model is a physically motivated improvement over the Einstein crystal model.

Debye sought a realistic way to assign different frequencies to the vibrational normal

modes. He assumed that the normal modes could be represented by standing waves

that vanish at the surfaces of the crystal. The quanta of energy of these waves are called

phononssince the waves are essentially sound waves. Consider a cubic crystal with

sideL. The amplitude of a standing wave that vanishes at the boundaries of the cube

is represented by

AmplitudeBsin

(s

xπx

L

)

sin

(s

yπy

L

)

sin

(s

zπy

L

)

(28.2-10)

whereBis a constant and wheresx,sy, andszare positive integers. We let

s

√

s^2 x+s^2 y+s^2 z (28.2-11)

The wavelength is given by

λ

2 L

s

(28.2-12)

and the frequency of the wave is given by

ν

c

λ



cs

2 L

(28.2-13)

wherecis the speed of propagation of the waves (the speed of sound in the solid).