Question

The unit cell of the crystal of an element has Z atoms and the edge length equal to a. If the molecular mass of the element is M, calculate the density of the crystal.

Answer 1

Given: Edge length of the unit cell = a 

Number of atoms present per unit cell = Z

Atomic mass of the element = M

For a cubic unit cell,

Volume of the unit cell = a³    ...(i)

Density of the unit cell (ρ') = `"Mass of unit cell"/"Volume of unit cell"`    ...(ii)

Mass of unit cell = Number of atoms per unit cell × Mass of one atom

Mass of unit cell = Z × m    ...(iii)

where m is the mass of a single atom. This is given by

m = `"Atomic mass"/"Avagadros number"`

m = `M/N_A`    ...(iv)

Substituting the value of m in eq. (iii), we have

Mass of unit cell = `Z xx M/N_A`

Substituting the corresponding values in eq. (ii), the density of the unit cell (p') is given by

ρ' = `"Mass of unit cell"/"Volume of unit cell"`

ρ' = `(Z xx M/N_A)/a^3`

or, ρ' = `(Z xx M)/(a^3 xx N_A)`    ...(v)

The density of a crystal (ρ) is the same as the density of its unit cell (ρ'), i.e., ρ' = ρ. Therefore, the density ρ of a crystal is given by

ρ = `(Z xx M)/(a^3 xx N_A)`