Physical Chemistry Third Edition

28.3 The Electronic Structure of Crystalline Solids 1171

b.Find the value of the frequencyνcorresponding to the

Einstein temperature of 1320 K assigned to diamond.

c.Calculate the molar energy of a diamond crystal at

298.15 K and at 500.0 K relative to the ground-state

energyU 0.

d.Calculate the molar Gibbs energy of a diamond crystal

at 298.15 K and at 500.0 K relative to the ground-state

energyU 0.

e.Calculate the molar entropy of a diamond crystal at

298.15 K and at 500.0 K.

28.19a.Write a computer program to evaluate the Debye
function, using Simpson’s rule.^5

b.Use this program to evaluate the heat capacity of

aluminum at several temperatures, using the Debye

temperature of 428 K. Construct a graph of the heat

capacity of solid aluminum as a function of

temperature from 0 K to its melting temperature

of 933 K.

28.20Consider a modified Einstein crystal model with two

frequencies. There areNatoms in the crystal. One-third

of them oscillate in three dimensions with frequencyν,

and two-thirds of them oscillate in three dimensions with

frequency 2ν.

a.Write a formula forCV.

b.Draw a graph ofCVversusTforν 3. 94 × 1012 s−^1

(one-half of the Debye frequency for germanium).

c.If a table of the Debye function is available^6 find the

ratio of your result to the Debye result for several

values ofT.

28.3 The Electronic Structure of Crystalline Solids

If we ignore the intermolecular forces, the electronic wave function of a molecular

or ionic crystal can be approximately represented as a product of wave functions for

individual molecules or ions. For example, the wave function of a sample of solid argon

would be approximated by a product of atomic wave functions of the argon atoms. An

approximate description of the electronic structure of network covalent crystals can

include localized covalent bonding similar to that described in Chapter 21.

EXAMPLE28.6

Describe the bonding in diamond.

Solution

In diamond each carbon atom is tetrahedrally bonded to four adjacent carbon atoms. To

describe the bonding in diamond, we form the 2sp^3 hybrid orbitals on each carbon atom. The

lobes of these orbitals point in the tetrahedral directions, and each forms a localized bonding

orbital with a 2sp^3 hybrid on the adjacent atom. Each pair of 2sp^3 hybrid orbitals also forms

an antibonding orbital, which is vacant.

Exercise 28.8

Describe the bonding in quartz, in which each silicon atom is bonded to four oxygen atoms. Each

oxygen atom is bonded to two silicon atoms.

The atoms in metallic crystals are bonded to each other by delocalized covalent bonds

similar to those described in Chapter 21 for substances such as benzene. Consider gold

(^5) S. I. Grossman,Calculus, 3rd ed., Academic Press, Orlando, FL, 1984, p. 518ff, or any standard calculus text.
(^6) N. Davidson,op. cit., p. 359 (note 3).