Physical Chemistry Third Edition

1164 28 The Structure of Solids, Liquids, and Polymers

We can now write formulas for the thermodynamic functions of the Einstein model

using equations from Chapter 27:

UkBT^2

(

∂ln(Z)

∂T

)

V

U 0 + 3 NkBT^2

(

∂ln(z)

∂T

)

V

U 0 +

3 Nhν

ehν/kBT− 1

(28.2-7a)

CV

(

∂U

∂T

)

V,N

 3 NkB

(

hν

kBT

) 2

ehν/kBT

(ehν/kBT−1)^2

(28.2-7b)

S

U

T

+kBln(Z)

U 0

T

+

3 Nhν

T(ehν/kBT)− 1

−

U 0

kBT

+ 3 NkBln(z)



3 Nhν

T(ehν/kBT−1)

− 3 NkBln

(

1 −e−hν/kBT

)

(28.2-7c)

A−kBTln(Z)U 0 + 3 NkBTln

(

1 −e−hν/kBT

)

(28.2-7d)

PkBT

(

∂ln(Z)

∂V

)

T

 3 NkBT

(

ln(z)

∂V

)

T

(28.2-7e)

μ

(

∂A

∂N

)

T,V

≈

AN−AN− 1

1



U 0

N

+ 3 kBTln

(

1 −e−hν/kBT

)

(28.2-7f )

We use the chemical potential per molecule as in Chapter 26, not the chemical potential

per mole. We have replaced a derivative by a finite difference and have assumed thatU 0

for a crystal ofN−1 atoms is equal toU 0 for a crystal ofNatoms times (N−1)/N.

There is a difficulty with the pressure of the Einstein crystal model. The model does

not include any simple way to evaluate the derivative in Eq. (28.2-7e). We might try to

evaluate the pressure by finding the difference betweenGandA, sinceGA+PV.

For a one-component system,Gis given Eq. (26.1-29) as

GNμU 0 − 3 NkBTln(z)

U 0 + 3 NkBTln(1−e−hv/kBT) (28.2-7g)

so thatGA, which leads toPV0. This result is a shortcoming of a crude model,

but for a crystal the numerical value ofPVis small compared withGandA, so that

we can use the formulas forGandAto an adequate approximation.

The value of the frequencyνis determined by fitting the heat capacity formula to

experimental data. The formulas for the thermodynamic functions can be restated in

terms of the parameter

ΘE

hv

kB

(definition) (28.2-8)

The parameterΘEhas the dimensions of temperature and is called theEinstein

temperatureor thecharacteristic temperature. Figure 28.7 shows the heat capacity

of diamond as a function of temperature as well as the heat capacity of the Einstein

crystal model with an Einstein temperature of 1320 K, which gives the best fit to the

experimental data.

0

0.2

T/1320 K 5 T/QE

0.4 0.6 0.8 1.0

5

10

C

/V

J mol

21

K

21

15

20 Classical

Einstein

25

Figure 28.7 The Heat Capacity of
Diamond Fit to the Einstein Crys-
tal Model Result.The horizontal line
corresponds to the law of Dulong and
Petit. From J. S. Blakemore,SolidState
Physics, 2nd ed., W. B. Saunders,
Philadelphia, 1974, p. 121.

In the limit of high temperature

lim

T→∞

CV 3 NkB 3 nR (28.2-9)