Two different elements, X and Y , crystallise in body-centred cubic (bcc) and face-centred cubic (fcc) lattices, respectively. The ratio of the edge lengths of their unit cells ( a_X : a_Y ) is 1:2 , and the ratio of their densities ( _X : _Y ) is 2:3 . If equal masses of both elements are taken, the ratio of the number of atoms of X to the number of atoms of Y is ______.

Step-by-step solution

Let the mass of each element taken be W . The number of atoms N in a given mass W is given by: N = W Z a^3 where Z is the number of atoms per unit cell, is the density, and a is the edge length. For element X (bcc lattice), Z_X = 2 . For element Y (fcc lattice), Z_Y = 4 . The ratio of the number of atoms of X to Y is: N_X N_Y = W Z_X _X a_X^3 W Z_Y _Y a_Y^3 N_X N_Y = ( Z_X Z_Y ) ( _Y _X ) ( a_Y a_X )^3 Given: a_X a_Y = 1 2 a_Y a_X = 2 _X _Y = 2 3 _Y _X = 3 2 Substituting these values into the ratio expression: N_X