Question

Calculate the packing fraction for unit cell of an element 'X', given that it crystallizes in a face- centred cubic unit cell.

• A. 0.62
• B. 0.38
• C. 0.74
• D. 0.26

Answer 1

The correct option is C 0.74

Face centred cubic unit cell has a ccp lattice:


Packing efficiency (P.F) of ccp:

$P.F=\frac{\text{No of spheres in one ccp unit cell}\times \text{Volume of each sphere}}{\text{Total Volume of the Unit cell}}$

ccp unit cell have FCC arrangement



Here,
$\sqrt{2}\times a=4Ra=2\sqrt{2}\times R$

$Z_{eff}\text{for FCC}=(8\times \frac{1}{8})+(6\times \frac{1}{2})=4$
So,

$\text{The total volume occupied by sphere}V=4\times \frac{4}{3}\times \pi r^3$
$\text{Volume of the cube}=a^3=(2\sqrt{2}R{)}^3$

$\text{Packing efficiency (P.F)}=\frac{4\times \frac{4}{3}\times \pi r^3}{(2\sqrt{2}R{)}^3}P.F=\frac{\pi }{3\sqrt{2}}\approx 0.74$