Question

The efficiency of packing in the case of a metal crystal for face-centered cubic is

Answer 1

Packing efficiency is the percentage of total space filled by the constituent particles in the unit cell. For a face centred cubic unit cell, there are four atoms per unit cell. The edge length, $a=2\sqrt{2}r$ .
In face centred cubic, packing efficiency $=\frac{volume occupied by particles in a unit cell \times 100}{volume of a unit cell}$
$= \frac{\text{Volume occupied by 4 atoms in the unit cell} \times 1 0 0}{\text{Volume of the unit cell}}$
$=\frac{4 \times \left(4 / 3\right) \pi r ^{3} \times 100}{\left(2 \sqrt{2} r ⁡\right)^{3}}=\frac{\left(16 / 3\right) \pi r ⁡^{3} \times 100}{16 \sqrt{2} r ⁡^{3}}=74\text{\%}$