Question

Inorganic Chemistry: Show that the atoms occupy only 52.4% of the total volume in a primitive or simple cubic unit cell.(By contrast, in a closest-packed structure like fcc, the atoms occupy 74% of the total volume)

Answer 1

In a primitive or simple cubic unit cell, the atoms are located at the corners of unit cell such that the adjacent atoms touch each other. Thus, radius of atom becomes (a/2).

Each atom is shared by eight cubes, hence, one eighth of each atom is in the cube. Therefore, total number of atoms in a cube = 1/8 x 8 = 1 atom.

Expression for total volume occupied by atom in a unit cell can be given as follows:

Volume of atom / volume of cell = (4/3 πr³)/ a³

Where, r = radius of atom = a/2

a = length of a side of cube

Therefore, Volume of atom / volume of cell = [4/3 π (a/2)³]/ a³

Volume of atom / volume of cell = π/6

Volume of atom / volume of cell = 3.142/6

Volume of atom / volume of cell = 0.524

Thus, if volume of cell is 100 then volume of atom will be 52.4. Alternatively, 52.4% of total volume in primitive or simple cubic unit cell is occupied by an atom.