Question

How can you determine the atomic mass of an unknown metal if you know its density and the dimension of its unit cell? Explain.

Answer 1

Suppose, the 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

= Z × m    ...(iii)

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

`m = "Atomic mass"/"Avogadro’s 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 (ρ') 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)`

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

`rho = (Z xx M)/(a^3 xx N_A)`

∴ `M = (rho xx a^3 xx N_A)/(Z)`