Physical Chemistry , 1st ed.

above the atoms in the first layer, as shown in Figure 21.11a. Building the

crystal can be continued by repeating each layer’s position accordingly. If the

first layer is labeled A and the second layer B, we can indicate the building of

this crystal as alternating layers of A, B, A, B,....

On the other hand, the third layer of atoms could be placed in positions that

are different from layer A or layer B; this is shown in Figure 21.11b and the

layer is usually labeled as layer C. In this crystal, the alternating layers can be

listed as A, B, C, A, B, C,....

Both arrangements are the most space-efficient ways of packing atoms, ions,

or molecules. Figure 21.12a shows that the ABAB form of crystal has a hexag-

onal Bravais lattice. The ABCABC ...form ofcrystal has a face-centered cu-

bic Bravais lattice (see Figure 21.12b). Both of these crystal lattices represent

the most space-efficient form of crystal; over 50 of the elements themselves,

from noble gases to metals, have either hexagonal or face-centered cubic crys-

tal lattices in their solid form. Because of the efficiency of the hexagonal

crystal lattice, it is sometimes called the hexagonal close-packed(or hcp) lattice.

These space-efficient crystal structures also show up in the real world of

macroscopic objects. For example, stacks of golf balls, basketballs, or baseballs

mimic an hcp or face-centered cubic arrangement. At grocery stores, fruits or

vegetables that are roughly spherical (oranges, apples, pears, citrus fruits) can

be stacked in a close-packed arrangement. Figure 21.13 shows an example of

this. This arrangement is used for its stability and, again, its efficient use of

space.

21.4 Densities

Knowledge of the crystal lattice designation of a crystal implies that we know

how many molecules are in the unit cell. If we have the unit cell parameters

(that is, the three distance parameters a,b, and c, and the three angular para-

meters , , and ) we can calculate the density of the compound. Comparison

of the calculated density with an experimentally determined density should

yield the same value. (In fact, agreement between calculated densities and ex-

perimental densities was perhaps a final—though unnecessary—supporting

argument for the atomic theory of matter.)

Recall that density is defined as mass per unit volume:

density 

vo

m

lu

a

m

ss

e

 (21.1)

Typically, the density of solids is given in units of grams/milliliter, abbreviated

g/mL; since a milliliter is equal to a cubic centimeter, densities are also com-

monly expressed in units of g/cm^3. A unit cell, however, is very small: typically

on the order of angstroms on a side (where 1 Å  1  10 ^10 m). Also, in a

unit cell we are considering (usually) a small number of atoms or molecules

per unit cell. The total mass of any one unit cell is therefore very small in an

absolute sense, on the order of 10^26 to 10^27 kg. In fact, it is typical to use

the unit “atomic mass unit,” or amu, to describe masses of individual atoms

and molecules. The amu is defined as

1 amu 1.6605  10 ^27 kg (21.2)

Thus, we have for a single unit cell

density 

vo

m

lu

a

m

ss

e

 in units of

a

Å

m

3

u



738 CHAPTER 21 The Solid State: Crystals

A

B

A

(a)

(b)

C

B

A

Figure 21.12 (a) The hexagonal unit cell for

the ABAB... pattern of close-packing. (b) The

face-centered cubic unit cell for the ABCABC...

pattern of close-packing.

Figure 21.13 Close-packed arrangements in

solids are easily mimicked in macroscopic

arrangements, like this fruit display. Can you

tell if this “crystal” would have a hexagonal or

face-centered cubic unit cell?

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