A metal M crystallizes into two lattices :- face centred cubic (fcc) and body centred cubic (bcc) with unit cell edge length of 2.0 and 2.5 Aᵒ respectively. The ratio of densities of lattices fcc to bcc for the metal M is ____________. (Nearest integer)

Solution

Chemistry ma’am · 3 steps

Same metal, same molar mass, so the density ratio depends only on Z and the edge length cubed.

1. Density of a cell. For a cubic unit cell, density is d = (Z M)/(N_A a³). Here Z is the number of atoms per cell and a is the edge length. M and N_A are the same for both lattices, so they cancel in the ratio.

   d_(fcc)/d_(bcc) = Z_(fcc)/Z_(bcc) × (a_(bcc)/a_(fcc))³

2. Put in numbers. fcc has Z = 4 and bcc has Z = 2, so the first factor is 2. The edge ratio is 2.5/2.0 = 1.25, and 1.25³ = 1.953.
3. So. The ratio is 2 × 1.953 = 3.906, which rounds to 4.

The common slip: 1. Writing the edge ratio as (2.0/2.5)³ instead of (2.5/2.0)³ gives 2 × 0.512 = 1.024, which rounds to 1, not 4.

The idea to keep. Density goes as Z/a³. Because a sits in the denominator, the cell with the smaller edge ends up with the larger cube factor in its favour. Always cube the edge ratio.

Answer from the NTA final answer key. Solution by PrepWiser.