Question

The radius of a single atom of generic element X is 145 picometers (pm) and a crystal of X has a unit cell that is body-centered cubic. Calculate the volume of the unit cell.

Answer 1

Examine the following diagram:

The triangle we will use runs differently than the triangle used in fcc calculations. d is the edge length of the unit cell, however d√2 is NOT an edge of the unit cell. It is a diagonal of a face of the unit cell. 4r is a body diagonal. Since it is a right triangle, the Pythagorean Theorem works just fine.

Using the Pythagorean Theorem, we find:

d² + (d√2)² = (4r)²

3d² = (4r)²     r - the radius of the X atom; d- edge length of the unit cell

d= sqrt[(4r)² / 3]   ; 145 pm = 1.45 x 10^-8 cm

d= sqrt[(4*1.45 x 10^-8 cm)² / 3]

d = 3.3486 x 10^-8 cm = edge length of the unit cell

d = cube root of volume of the unit cell ==> volume of the unit cell = (edge length of the unit cell)³

volume of the unit cell = (3.3486 x 10^-8 cm)³

volume of the unit cell = 3.7549 x 10^-23 cm³