Question

Manganese crystallizes with a body-centered cubic unit cell. The radius of a manganese atom is 127 pm. Calculate the density of solid crystalline manganese in grams per cubic centimeter.

Answer 1

Here r = 127.0 pm

r = 1.27*10^-8 cm

length of body diagonal = sqrt(a^2+a^2+a^2)

length of body diagonal = a*sqrt(3)

use:

For BCC Lattice

length of body diagonal = 4*r

a*sqrt(3) = 4*r

Given: r = 1.27*10^-8 cm

So,

a = 4*r/(sqrt(3))

a = 4*1.27*10^-8/(sqrt(3)) cm

a = 2.933*10^-8 cm

Molar mass = 54.94 g/mol

since the cubic cell is Body Centred Cubic, the value of Z=2

d = (Z*M)/(a^3*NA)

d = (2*54.94)/((2.933*10^-8)^3*(6.022*10^23))

d = 7.23 g/cm^3

Answer: 7.23 g/cm^3