Question

A body centred cubic lattice is made up of hollow spheres of B. Spheres of solid A are present inside the hollow spheres of B. If the radius of A is half the radius of B, what is the ratio of the total volume of spheres of B unoccupied by A in a unit cell and the volume of the unit cell?

• A. $\frac{7\pi \sqrt{3}}{64}$
• B. $\frac{7\sqrt{3}}{128}$
• C. $\frac{7\pi }{24}$
• D. None of these

Answer 1

The correct option is A $\frac{7\pi \sqrt{3}}{64}$

Volume of A spheres in the unit cell = $\frac{4\pi r^3}{3}$
Volume of B spheres in the unit cell = $\frac{4\pi r^3\times 8}{3}$
Hence, unoccupied volume = $Volum{e}_B-Volum{e}_A$ = $\frac{7\times 4\pi r^3}{3}$
For BCC, $\sqrt{3}a=4R$, here $R=2r$
Total volume of unit cell = $a^3=(\frac{4\left(2r\right)}{\sqrt{3}}{)}^3$
Hence, ratio = $\frac{unoccupiedvolume}{volumeofunitcell}$ = $\frac{7\pi \sqrt{3}}{64}$