density+of+unit+cells

As mentioned earlier, unit cells have //occupied volumes// and //empty space volumes//. These vary by the type of unit cell.

The density of a solid can be estimated using its unit cell information. In order to do this, a few things need to be known:
 * the type of unit cell
 * the edge length (or atomic/ionic radius)
 * the identity of the atom/ion (and therefore its atomic mass)

Recall that the density of a substance is the ratio of its mass to its volume. So in a unit cell, we need to know these things.


 * Mass of a unit cell**

To determine the mass of the unit cell, the number of atoms inside the unit cell must be known, as well as their atomic mass(es). The type of unit cell will tell us the first part: a simple cubic unit cell has 1 atom, a bcc unit cell has 2 atoms, and a fcc unit cell has 4 atoms.

Recall, however, that this would be the number of atoms/ions and needs to be converted to moles using Avogadro's Number.

Therefore:

unit cell mass = (# of atoms) x (atomic mass of atom/ion in g/mole) / (6.022 x 10 23 atoms/mole)

For example, iron occupies a bcc unit cell:


 * Mass of iron's bcc unit cell:** (2 atoms) x (55.845 g/mol) / (6.022 x 10 23 atoms/mole) = **1.85 x 10 -22 grams**


 * Volume of a unit cell**

The volume of a unit cell is simple to find if the edge length is known:

volume of unit cell = a 3

If only the atomic or ionic radius is known, it becomes a bit more work. The ratio of edge length to radius must be used:


 * simple cubic: a = 2 r**
 * bcc: a = 2.31 r**
 * fcc: a = 2.83 r**

The only concern is that the volume be reported in cubic centimeters.

Example: volume of iron's bcc unit cell.

Iron has an atomic radius of 126 picometers (pm), which is 1.26 x 10 -8 cm. Because it has a bcc unit cell, its edge length is:

a = 2.31 r = 2.31 x 1.26 x 10 -8 cm = 2.91 x 10 -8 cm

So, a 3 = 2.47 x 10 -23 cm 3

To find the unit cell density, finish by dividing mass by volume:


 * 1.85 x 10 -22 grams / 2.47 x 10 -23 cm 3 = 7.49 g/cm 3 **

Try some practice problems: