A body centred cubic lattice is made up of hollow sphere of B. Sphere of solid A are present in hollow sphere of B. Radius of A is half of the radius of B. What is the ratio of total volume of sphere

Question

A body centred cubic lattice is made up of hollow sphere of B. Sphere of solid A are present in hollow sphere of B. Radius of A is half of the radius of B. What is the ratio of total volume of sphere B unoccupied by A in unit cell and volume of unit cell?

TestHub · Accepted Answer

Effective number of atoms of B present in a unit cell = 2

Total volume of B unoccupied by A in a unit cell

$= 2 \times \frac{4}{3} (R^{3} - r^{3}) \times π$

$= \frac{7}{3} π R^{3}$ $(r = \frac{R}{2})$

Volume of unit cell = a³

${(\frac{4 R}{\sqrt{3}})}^{3} = \frac{64}{3 \sqrt{3}} R^{3}$ $(\sqrt{3} a = 4 R)$

Ratio of total volume of sphere B unoccupied by A in unit cell and volume of unit cell $= \frac{7 / 3 π R^{3}}{\frac{64}{3 \sqrt{3}} R^{3}} = \frac{7 π \sqrt{3}}{64}$

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