Question

Metal M crystallizes in a simple cubic crystal and has a packing efficiency of 52.2%. When it combines with a halogen $X_2$ and forms MX, it forms a NaCl type crystal with a cation to anion size ratio of 1:2. Packing efficiency (%) of the MX crystal is

Answer 1

We know, $\text{packing efficiency}=\frac{\text{Volume of sphere in unit cel}l}{\text{Total volume of unit cell}}$
For a simple cubic crystal $\eta =\frac{\frac{4}{3}\pi r^3}{(2r{)}^3}\times 100=52.2\Rightarrow \frac{4}{3}\pi \times 100=52.2\times 8$
Since MX crytalillizes in NaCl type of crystal, it is FCC type of crystal.
Thus for MX, $\eta =\frac{4\times \frac{4}{3}\pi \left(\right(r_c{)}^3+\left(r_a{)}^3\right)}{2(r_c+r_a{)}^3}\times 100$ = $\frac{4\times \frac{4}{3}\pi (\frac{r_a}{2}+r_a{)}^3}{\left(2\right(r_c+r_a){)}^3}\times 100$ (Since given $\frac{r_c}{r_a}=0.5)$
$=\frac{4\left(\frac{9}{8}\right)}{27}\times 52.2\times 8=69.6\%$