Physical Chemistry Third Edition

1162 28 The Structure of Solids, Liquids, and Polymers

b. How many bases are there in a unit cell?

c.The density of CsCl is 3988 kg m−^3. Find the unit cell

dimension and the distance between nearest-neighbor

Cs and Cl atoms.

28.8Barium crystallizes in the body-centered cubic

lattice.

a.Find the number of atoms per unit cell.

b. The density of barium is 3.5 g cm−^3. Find the unit

cell dimension and the apparent radius of a barium

atom.

c.Calculate the molar volume of solid barium and

the volume of empty space in 1.00 mol of solid

barium.

28.9Argon forms a face-centered cubic lattice. Assume that in

solid argon the nearest-neighbor distance is equal to the

distance at the minimum in the Lennard–Jones potential

function, 3. 82 × 10 −^10 m.

a.Find the unit cell dimension.

b.Find the density of solid argon.

28.10Iron forms a body-centered crystal with a unit cell

dimension equal to 2. 861 × 10 −^10 m. Calculate the

density of iron. Compare it with the experimental value

(look it up).

28.11 a.Find the perpendicular distance between the 110
planes in the iron crystal, which forms a
body-centered cubic lattice with a unit cell dimension
of 2. 861 × 10 −^10 m.
b. Find the value ofθfor then1 reflection of X-rays
with a wavelength of 1.5444 Å from these lattice
planes.
28.12a.Find the perpendicular distance between 111 planes
in the molybdenum carbide lattice, which is
face-centered cubic witha 4. 28 × 10 −^10 m.

b. If then1 reflection from the 111 plane gives

θ 36. 5 ◦, find the wavelength of the X-rays.

c.There are two molybdenum carbides, MoC and Mo 2 C.

Determine which formula applies for this unit cell size

and find the density of molybdenum carbide.

28.13a.The stretching of a uniform bar due to a tensile

(stretching) force is described byYoung’s modulus,E,

defined by

E

stress

strain



F/A

∆L/L

whereFis the magnitude of the tensile force,Ais the

cross-section area,Lis the length, and∆Lis the

change in the length due to the forceF. Derive an

approximate expression for Young’s modulus for a

perfect crystalline substance with a simple cubic

lattice and an intermolecular potential energy

given by

u(r)u(a)+

k

2

(r−a)^2

wherekis a constant,ris the lattice spacing, andais

its equilibrium value. Include only nearest-neighbor

interactions and assume that each unit cell stretches in

the same ratio as the entire bar.

b.The value of Young’s modulus for fused quartz is

7. 17 × 1010 Nm−^2. Estimate the force constantkfor

the Si–O bond, assuming (contrary to fact) that quartz

has a simple cubic lattice. Assume that

a 1. 5 × 10 −^10 m, the approximate Si–O bond

distance. Comment on your value in view of the fact

that force constants for most single bonds in

molecules are roughly equal to 500 N m−^1.

28.14The value of Young’s modulus for iron is 2. 8 × 107 lb in^2

(pounds per square inch). Make reasonable assumptions

and estimate the force constant for an Fe–Fe bond. See

the previous problem.

28.2 Crystal Vibrations

The atoms or ions making up a crystal can vibrate about their equilibrium positions.

A crystal is like a very large molecule and if a crystal consists ofNatoms it must

have 3N−6 vibrational normal modes, all of which correspond to collective motions

of many atoms. We will discuss two model systems that represent the vibrations of

a crystal.