The molecules of a solid is compactly arranged. It is made up of repeating blocks called unit cells. The density of a unit cell is:
ρ = nA/VNₐ
where
n is the number of atoms per unit cell
A is the atomic weight of solid
V is the volume of unit cell
Na is Avogadro's number (6.022×10²³)
For a face-centered cubic unit cell, there are 4 atoms (n=4). The volume of the unit cell follows the formula of the cube, s³. The atomic weight of gold is 197 g. So, the solution is as follows:
ρ = (4 atoms)(197 g/mol)(1 kg/1000 g)/(407×10⁻¹² m)³(6.022×10²³ atoms/mol)
ρ = 19,408.5 kg/m³
