Chemistry - Solid State Question with Solution | TestHub

For a simple cubic unit cell, the relation between the edge length $a$ and the radius of the atom $r$ is $a=2r$.

The packing fraction is given by the ratio of the volume occupied by the atoms to the total volume of the unit cell:

$\text{Packing fraction}=\frac{\frac{4}{3}\pi r^3}{a^3}=\frac{\frac{4}{3}\pi r^3}{(2r{)}^3}=\frac{\pi }{6}$

Taking $\pi =\frac{22}{7}$, the packing fraction is $\frac{22}{42}=\frac{11}{21}$.

The fraction of void volume in the unit cell is:

$\text{Void fraction}=1-\frac{11}{21}=\frac{10}{21}$

Given the total volume of the unit cell is $5.5\times 10^{-22}{\mathrm{cm}}^3$, the void volume is:

$\text{Void volume}=\frac{10}{21}\times 5.5\times 10^{-22}{\mathrm{cm}}^3=2.619\times 10^{-22}{\mathrm{cm}}^3$

Answer: $2.619\times 10^{-22}{\mathrm{cm}}^3$