A metal crystallizes in two different forms: a face-centred cubic (fcc) lattice and an unknown lattice type. The edge length of the fcc unit cell is exactly twice the edge length of the unknown unit cell. If the density of the fcc phase is one-fourth the density of the unknown phase, the effective number of atoms per unit cell for the unknown lattice is _______.

Step-by-step solution

The density of a crystal lattice is given by the formula: d = Z M N_A a^3 where Z is the effective number of atoms per unit cell, M is the molar mass, N_A is Avogadro's number, and a is the edge length of the unit cell. For the same metal, the ratio of densities of the two lattices is: d_ fcc d_ unknown = ( Z_ fcc Z_ unknown ) ( a_ unknown a_ fcc )^3 Given: Z_ fcc = 4 a_ fcc = 2 a_ unknown a_ unknown a_ fcc = 1 2 d_ fcc d_ unknown = 1 4 Substituting the values into the equation: 1 4 = ( 4 Z_ unknown ) ( 1 2 )^3 1 4