A bcc lattice is made up of hollow spheres of `B`. Spheres of solids `A` are present in hollow spheres of `B`. The radius of `A` is half of the radius of `B`. The ratio of total volume of spheres of `B` unoccupied by `A` in a unit cell and volume of unit cell is `A xx (pisqrt(3))/(64)`. Find the value of `A`.

Answer

7

Explanation

Let radius of hollow sphere `B = r`
`:.` Edge length `(a) = 4r//sqrt(3)`
Volume of unit cell `= a^(3) = (4r//sqrt(3))^(2)`
Volume of `B` unoccupied by `A` (having radius `= r//2`
in unit cell `= 2 xx [4/3 pir^(3) - 4/3pi(r/2)^(3)]`
`:. ("Volume of B unoccupied by A in unit cell")/("Volume of unit cell") = (4/3pixx(7r^(3))/(8)xx2)/((4r)/(sqrt(3)))^(3)`
`= (7pisqrt(3))/(64)`
`A xx (7sqrt(3))/(64) = (7pisqrt(3))/(64)`
`:. A = 7`