Physical Chemistry , 1st ed.

Example 21.4

Solid nitrogen, N 2 , has a density of 1.026 g/cm^3 at about 20 K. If solid nitro-

gen is known to have some cubic lattice with a length of 5.66 Å, what kind

of cubic lattice does solid N 2 have?

Solution

From the length of the lattice, we can determine the volume of the unit cell:

volume (5.66 Å)^3 181 Å

From the known density (and a few unit conversions), we can now determine

the amount of mass in each unit cell:

740 CHAPTER 21 The Solid State: Crystals

massdensity volume

1.026

cm

g

 3 181 Å

(^3) 
1.6605

1



am

10

u

 (^27) kg 1

1

02

c

4

m

Å

3

 3  1

1

00

k

0

g

g



mass of unit cell 112 amu

Each nitrogen molecule, with two nitrogen atoms each, has a mass of 28 amu.

If there are 112 amu per cell, then there are



1

2

1

8

2

a

a

m

m

u

u

p

p

e

e

r

r

N

ce

2

ll



1

2

1

8

2

a

a

m

m

u

u

/

/

N

ce

2

ll



4

ce

N

ll

^2

or 4 nitrogen molecules per unit cell. (Note how the units work out alge-

braically.) The type of cubic Bravais lattice that has four species per unit cell

is the face-centered cubic lattice, which has ^18 of a species at each of eight cor-

ners (1 molecule overall) and ^12 of a species at each of six faces (3 molecules

overall) for a total of 4 molecules per unit cell. We therefore predict that solid

N 2 has a face-centered cubic lattice.

The two previous examples used cubic crystals, which are easier to visual-

ize because of their perpendicular sides. Noncubic crystals are treated similarly.

21.5 Determination of Crystal Structures

How do we know what the crystal lattices are for elements, molecules, and

compounds? The answer is an interesting historical story.

In the late 1800s, scientists were finally realizing that what we call light was

not just visible light but included a lot of invisible light, too. Light could be

thought of as a wave that had varying frequency and wavelength. On the very

short-wavelength and high-energy side of “light” (now better referred to as

electromagnetic radiation) are the X rays, originally named by the German

physicist and discoverer Wilhelm Roentgen in 1895 because of their unknown

nature. X rays were ultimately shown to be a form of electromagnetic radia-

tion having very short wavelength, generally considered to be between 0.01 and

100 Å.

At the same time, it was known to physical science that a series of equally

spaced parallel grooves in a mirrored surface, called a grating,could diffract

light just like a prism could. This effect was especially pronounced if the grooves

were spaced on the order of the wavelength of the light. (Indeed, most modern

dispersive spectrometers use a grating as the dispersive element.Details of in-

strumentation can be found in analytical chemistry or instrumental analysis

Figure 21.14 Max von Laue (1879–1960) was

a German physicist who originally proposed that

crystals, if they were indeed made up of atoms,

should act as gratings to diffract X rays. He was

right, and he won the 1914 Nobel Prize in physics

as a result.

Photograph by Francis Simon, Courtesy AIP Emilio Segre Visual Archives