Physical Chemistry Third Edition

28.4 Electrical Resistance in Solids 1179

The ratio of the electronic contribution to the vibrational contribution is

Ratio

0 .00753 J K−^1 mol−^1

0 .21JK−^1 mol−^1

 0. 036

aK. S. Pitzer and L. Brewer,Thermodynamics, McGraw-Hill, New York, 1961, p. 660.

PROBLEMS

Section 28.3: The Electronic Structure of Crystalline
Solids

28.21Construct an accurate graph of the fermion distribution,
Eq. (28.3-1), usingε/kBTas the independent variable
and assuming (a) thatTμ/ 10 kB, (b) thatTμ/kB,
and (c) thatT 10 μ/kB.

28.22Evaluate the Fermi level for copper at 298 K using Eq.
(28.3-12). Find the percent error of using Eq. (28.3-11).

28.23a.Evaluate the Fermi level of silver at 298.15 K using
Eq. (28.3-12).
b.Find the electronic contribution to the molar heat
capacity of silver at 15.00 K and at 298.15 K.

28.24The Fermi level for sodium is equal to 3.1 eV.

a.Find the number of mobile electrons per cubic meter

and per mole. The density of sodium is 0.971 g cm−^3.

State any assumptions.

b.Find the speed of electrons that have kinetic energy

equal to the Fermi level for sodium.

28.25In silicon, the band gap between the highest filled band

(the valence band) and the lowest vacant band (the

conduction band) is equal to 1.11 eV.

a.Assuming the Boltzmann distribution, find the ratio

of the population of the lowest conduction band

states and the highest valence band states at

300 K.

b.Assuming the fermion distribution with the Fermi

level at the top of the valence band, repeat

part a.

28.26Calculate the electronic contribution to the heat capacity

of copper at 298.15 K. Find the percent contribution to

the total heat capacity, using the law of Dulong and Petit

for the vibrational contribution.

28.4 Electrical Resistance in Solids

An electric current in a metallic conductor consists of moving electrons. The Drude

model^12 pictures a metal as consisting of mobile electrons and positively charged

“cores,” which are the ions produced when the conduction electrons are removed from

the atoms. Dilute occupation of the electron states is assumed. The electrons are pictured

as colliding with the cores, impeding their motion. We define a period of timeτby

(Collision probability per unit time)

1

τ

(28.4-1)

LetN(t) be the number of electrons per unit volume that have not yet collided with a

core at timet. The rate of change ofNis given by

dN

dt

−

1

τ

N

(^12) J. S. Blakemore,op. cit., p. 158ff (note 4); D. Tabor,Gases, Liquids and Solids, 2nd ed., Cambridge
University Press, Cambridge, England, 1979, p. 188ff.