Chemistry - Solid State Question with Solution | TestHub

In a bcc unit cell, the atoms touch along the body diagonal.

Length of body diagonal = $\sqrt{3}a$

Since the body diagonal comprises the radii of two corner atoms and the diameter of the body-centered atom, we have $4r=\sqrt{3}a$, which gives $r=\frac{\sqrt{3}a}{4}$.

The number of effective atoms in a bcc unit cell is $Z=2$.

Total volume occupied by atoms is $V=Z\times \frac{4}{3}\pi r^3$.

Substituting the value of $r$:

$V=2\times \frac{4}{3}\pi {\left(\frac{\sqrt{3}a}{4}\right)}^3$

$V=\frac{8}{3}\pi \left(\frac{3\sqrt{3}{a}^3}{64}\right)$

$V=\frac{\sqrt{3}\pi a^3}{8}$

Answer: $\mathrm{V}=\frac{\sqrt{3}\pi {\mathrm{a}}^3}{8}$