Chemistry - Solid State Question with Solution | TestHub

In a body-centered cubic (BCC) structure, the particles touch each other along the body diagonal of the cube.

The length of the body diagonal is $\sqrt{3}a$, where $a$ is the edge length of the unit cell.

Since the body diagonal comprises the radii of two corner atoms and the diameter of the body-centered atom, its length is equal to $4r$, where $r$ is the radius of one particle.

$\sqrt{3}a=4r\Rightarrow r=\frac{\sqrt{3}a}{4}$

Assuming the particle to be spherical, the volume of one particle is given by:

$V=\frac{4}{3}\pi r^3$

Substituting the value of $r$:

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

$V=\frac{4}{3}\pi \left(\frac{3\sqrt{3}{a}^3}{64}\right)$

$V=\frac{\sqrt{3}\pi a^3}{16}$

Answer: $\frac{\sqrt{3}\pi {\mathrm{a}}^3}{16}$