Iron exhibits `bcc` structure at roomj temperature. Above `9000^(@)C`, it transformers to `fcc` structure. The ratio of density of iron at room temperature to that at `900^(@)C` (assuming molar mass and atomic radius of iron remains constant with temperature) is

Options

A. `(sqrt3)/(sqrt2)`
B. `(4sqrt(3))/(3sqrt(2))`
C. `(4sqrt(3))/(4sqrt(2))`
D. `1/2`

Answer

C

Explanation

For bcc structure : `4 r = sqrt(3) a`
or `" " a = (4r)/(sqrt3)`
`Z = 2`
For fcc structure : `4r = sqrt(2)a`
`a = (4r)/(sqrt2)`
`Z = 4`.
`(d("bcc"))/(d("fcc"))= ((Z("bcc") xx M)/(a^3 xx N_A))/((Z("fcc") xx M)/(a^3 xx N_A)) = ((2xxM)/(((4r)/(sqrt(3)))^3 xx N_A))/((4 xxM)/(((4r)/(sqrt(3)))^(3)xx N_A))`
`(2 xx M)/(((4r)/(sqrt(3)))^(3)xx N_A) xx (((4r)/(sqrt(2)))^(3)xx N_A)/(4 xx M)`
`= 1/2 (3/2 xx (sqrt(3))/(sqrt(2))) = (3sqrt(3))/(4sqrt(2))`