The Foundations of Chemistry

Solution

(a) One face of the face-centered cubic unit cell is shown in the left-hand drawing, with the

atoms touching. The nearest neighbor atoms are the ones along the diagonal of the face of the

cube. We may visualize the face as consisting of two right isosceles triangles sharing a common

hypotenuse, h,and having sides of length a4.086 Å. The hypotenuse is equal to twicethe

center-to-center distance. The hypotenuse can be calculated from the Pythagorean theorem,

h^2 a^2 
a^2. The length of the hypotenuse equals the square root of the sum of the squares

of the sides.

ha^2 
a^2  2 a^2  2 (4.0 8  6 Å)^2 5.778 Å

The distance between centers of adjacent silver atoms is one half of h,so

Distance2.889 Å

(b) The hypotenuse of the unit cell face is four times the radius of the silver atom.

Atom radius1.444 Å

(c) To see the number of nearest neighbors, we expand the left-hand drawing to include several

unit cells, as shown in the right-hand drawing. Suppose that this is the x–yplane. The atom

shown in red has four nearest neighbors in this plane. There are four more such neighbors in

the x–zplane (perpendicular to the x–yplane), and four additional neighbors in the y–zplane

(also perpendicular to the x–yplane). This gives a total of 12 nearest neighbors.

EXAMPLE 13-9 Density and Cell Volume

From data in Example 13-8, calculate the density of metallic silver.

Plan

We first determine the mass of a unit cell, that is, the mass of four atoms of silver. The density

of the unit cell, and therefore of silver, is its mass divided by its volume.

Solution

_?_ g Ag per unit cell

7.165 10 ^22 g Ag/unit cell

Vunit cell(4.086 Å)^3 68.22 Å^3 

3

6.822 10 ^23 cm^3 /unit cell

Density10.50 g/cm^3

You should now work Exercises 90 and 92.

Data obtained from crystal structures and observed densities give us information from

which we can calculate the value of Avogadro’s number. The next example illustrates these

calculations.

7.165 10 ^22 g Ag/unit cell



6.822 10 ^23 cm^3 /unit cell

10 ^8 cm



Å

107.87 g Ag



1 mol Ag

1 mol Ag



6.022 1023 Ag atoms

4 Ag atoms



unit cell

5.778 Å



4

5.778 Å



2

A handbook gives the density of silver
as 10.5 g/cm^3 at 20°C.

522 CHAPTER 13: Liquids and Solids