Physical Chemistry Third Edition

1170 28 The Structure of Solids, Liquids, and Polymers

the upper limit of the integral can be extended to infinity without serious error. The

integral is then equal toπ^4 /15, and for small values of the temperature

UU 0 +

3 π^4 NkBT^4

5 Θ^3 D

(28.2-28)

For small values of the temperature, the heat capacity is

CV

12 π^4

5

NkBT^3

ΘD

(28.2-29)

In electrical conductors, there is also a contribution to the heat capacity from the

electronic motion (see Section 28.3). Heat capacities are hard to measure at low tem-

peratures, and data for temperatures below 15 K are hard to find. Equation (28.2-29) is

commonly used as a substitute for experimental data between 0 K and 15 K. Modifica-

tions to the Debye theory have been devised that use a temperature-dependent Debye

temperature and give improved agreement with experiment.^4

Exercise 28.7

Verify Eq. (28.2-28).

PROBLEMS

Section 28.2: Crystal Vibrations

28.15The formulas for the thermodynamic functions of an
Einstein crystal can also be derived by slightly modifying
the approach of Chapter 25, recognizing that the normal
modes of vibration are distinguishable from each other.
Carry out this analysis.

28.16The value ofΘEthat fits the Einstein crystal model
heat-capacity formula to data for aluminum is 240 K.

a.What is the vibrational frequency corresponding to

this value of the parameter?

b.Draw a graph of the heat capacity of aluminum from

0 K to 300 K, according to the Einstein model.

c.At what temperature does the prediction of the

Einstein model for the heat capacity of aluminum

come within 5.00% of the law of Dulong and Petit?

At what temperature does it come within 1.00% of the

law of Dulong and Petit?

28.17The value ofΘEthat fits the Einstein crystal model

heat-capacity formula to data for diamond is 1320 K.

a.What is the vibrational frequency corresponding to

this value of the parameter?

b.Construct an accurate graph of the heat capacity of

diamond from 0 K to 300 K, according to the Einstein

model.

c.At what temperature does the prediction of the

Einstein model for the heat capacity of diamond come

within 5.00% of the law of Dulong and Petit? At what

temperature does it come within 1.00% of the law of

Dulong and Petit?

28.18a.Express the thermodynamic functions of an Einstein

crystal in terms of the Einstein temperature.

(^4) See J. S. Blakemore,Solid State Physics, 2nd ed., W. B. Saunders, Philadelphia, 1974, p. 128ff.