Consider the bcc unit cells of the solids 1 and 2 with the position of atoms as shown below. The radius of atom B is twice that of atom A. the unit cell edge length is `50%` more in soild 2 than in 1 . What is the approximate packing efficiency in solid 2?

Options

A. `45%`
B. `65%`
C. `90%`
D. `75%`

Answer

C

Explanation

No. of A atoms per unit cell = 1
Volume of A atoms = `1 xx 4/3 pi r_A^3`
No. of B atoms per unit cell = 1
Volume of B atoms = `1 xx 4/3 pi r_(B)^(3)`
Packing efficiency = `(1 xx 4/3 pi r_A^3) + (1 xx 4/3 pi r_B^3)/(a^3)`
In figure, `2(r_A - r_B) = sqrt(3)a`
or `2(r_A + 2r_A) = sqrt(3)a " " ( :. r_B = 2r_A)`
`2(3r_A) = sqrt(3)a`
`2sqrt(3) r_A = a`
and `r_B = 2r_A`
` :.` Packing efficiency = `(4/3 pi r_A^3 + 4/3 pi (2r_A)^3)/((2sqrt(3)r_A)^3)`
`= (4/3 pi r_A^3 (1 + 8))/(r_A^3 xx 8 xx 3sqrt(3)) = (9 xx 4/3 pi)/(24 xx sqrt(3)) = pi/(2sqrt(3))` .