Physical Chemistry , 1st ed.

Luckily, going from amu/Å^3 to g/mL is simply an exercise in unit conversions.

We can use equation 21.2 along with the fact that

1 mL 1 cm^3  1024 Å^3 (21.3)

to determine a macroscopic density in measurable units from a microscopic

density in terms of unit cell parameters.

The masses of the atoms and molecules are taken from the cumulative num-

ber of atoms or molecules in the unit cell. The volume of the unit cell is de-

termined from geometry. From geometry, the equation that gives the volume

of a six-sided object with parallel opposing faces (called a parallelepiped) is

21.4 Densities 739

volume abc(1 cos^2 cos^2 cos^2 
2 cos cos cos  ) (21.4)

Equation 21.4 is general; the user should recognize that for some crystal lat-

tices, two or three sides of the unit cell may have the same length, and two or

three of the angles may also be the same (that is, 90°). The definitions of the

angles , , and with respect to the lattice sides should also be remembered.

For a unit cell that has 90° angles, equation 21.4 reduces to

volume abc

as it should for a right-angle solid. Examples 21.3 and 21.4 show how to relate

macroscopic densities and microscopic information.

Example 21.3

Solid silver exists as a face-centered cubic crystal with a4.09 Å. What is the

density of silver? Assume that each silver atom has a mass of 108 amu.

Solution

First, we must determine the number of silver atoms in a face-centered cubic

unit cell. Each corner atom contributes ^18 of an atom to each unit cell. There

are eight corner atoms, contributing a total of^18  1 1 silver atom. Each

face contributes ^12 of an atom to each unit cell. Overall, the six faces of the cu-

bic lattice add ^12  6 3 silver atoms to each unit cell. Each cell therefore has

3  1 4 silver atoms per cell. If each silver atom has a mass of 108 amu,

then the total mass of each unit cell would be four silver atoms:

mass  4 108 amu 432 amu

The volume of the unit cell can be determined using equation 21.4, in part

by recognizing that in a cubic cell the angles are all 90°:

volume abc4.09 Å 4.09 Å 4.09 Å

volume 68.4 Å^3

Using the definition of density and converting to its more common units:

density 

vo

m

lu

a

m

ss

e





4

6

3

8

2

.4

am

Å^3

u



1.6605

1



am

10

u

 (^27) kg


1

1

02

c

4

m

Å

3

3



10

1

0

k

0

g

g



density 10.5 g/cm^3

The measured density of silver is 10.5 g/cm^3 , exactly the same (to three sig-

nificant figures) as the calculated density.