Physical Chemistry , 1st ed.

Example 21.1

Figure 21.10 shows a unit cell for NaCl, another simple ionic crystal.

a.Identify the Bravais lattice for NaCl.

d.Determine the stoichiometry of the unit cell. Is it consistent with the for-

mula of this compound?

Solution

a.First, we should confirm that the same species is present at all corners.

Checking Figure 21.10, we see that there are Clions at all corners. (This is

a point that many people do not understand when trying to define the unit

cell for a crystal!) Next, we try to identify Clions at other points in the unit

cell. Clions are also located in all six faces of the unit cell. Therefore, we

would assign a face-centered cubic Bravais lattice to NaCl.

b.The corners contribute ^18  8 1 atom of Cl, overall, to the formula of

the compound, and the faces contribute ^12  6 3 atoms of Cl. One Naion

is in the center of the unit cell, and on each edge of the unit cell an Naion

contributes ^14 of the atom to the unit cell. (Do you see this? Refer to Figure

21.10 and show that only ^14 of each Naion actually resides in the unit cell.)

The 12 edge Naions therefore contribute ^14  12 3 sodiums. Adding the

1 Nain the center, we get a total of 4 Naions in the unit cell. Considering

the Na and Cl contributions together, we have a stoichiometry of Na 4 Cl 4 ,

which in the lowest ratio reduces to NaCl: the expected formula for sodium

chloride. (Note: students who just read this and did not actually refer to

Figure 21.10 and make these observations on their own will have learned

nothing from this example!)

Example 21.2

Explain why the CsCl unit cell is considered simple cubic and not body-

centered cubic despite having an atom in the center of the unit cell. Refer to

Figure 21.7 for the unit cell of CsCl.

Solution

Identification of a unit cell requires that the samespecies—whether atom,

ion, or molecule—be present at the proper positions in the unit cell. In the

case of CsCl, there are Csions at the corners, which is the minimum re-

quirement for any unit cell, but no Csions in the center (it’s a Clion).

Therefore, the presence of the Clion does not factor into the determination

of the type of unit cell, and the Bravais lattice is identified as a simple cubic

lattice and not a body-centered cubic lattice.

Two crystal lattices deserve some special mention. Consider a monatomic

crystal composed of atoms all the same size. What is the most space-efficient

way to make them into a crystal? Figure 21.11 shows two different ways of plac-

ing atoms most efficiently. On the bottom layer of each diagram, atoms make

a nice, regular lattice. The next-to-bottom layer lies in the natural dips created

by three adjacent, triangularly spaced bottom-row atoms.

For the third layer, there is a choice. On the one hand, and perhaps the

easier choice to illustrate, the atoms in the third layer can be placed directly

21.3 Crystals and Unit Cells 737

Cl

Na

Figure 21.10 A unit cell for sodium chloride,
NaCl. See Example 21.1.

(a)

(b)
Figure 21.11 Two different ways of getting
the most efficient packing of spherical atoms. (a)
Using the bottom layer as a base, the second layer
of atoms lies in the dips made by three adjacent
base atoms. In the third layer, atoms lie in dips
created by the second layer, but they also lie di-
rectly above atoms of the first layer. This is the
ABAB... pattern of close-packing. (b) In this
case, atoms in the third layer do not lie directly
above the first layer’s atoms, but rather in differ-
ent relative positions. This is the ABCABC...
pattern of close-packing.