An element crystal has a density of 8570 kg/m3 packing fraction is 0.68 . Determine mass of an atom if nearest neighbor distance is 2.86 Å

Density=8570 kg/m3

packing fraction=0.68

d=2.86Å

As packing fraction=0.68 it is a BCC crystal

$∵ $ packing fraction$=\dfrac{\sqrt3\pi}{8}=0.68$

As it is BCC , number of atoms per unit cell is 2

As d=2.86Å for BCC

$d=\dfrac{\sqrt3a}{2}=2.86$

$a=2.86\times\dfrac{2}{\sqrt3}=3.302Å$

as,

$density=mass\times\dfrac{no.of atoms per unit cell}{a^3\times N}$

where N is Avogadro number

$\therefore mass=(a^3\times N)\times \dfrac{density}{(no.\ of\ atoms\ per\ unit\ cell)}$

$M=((3.302\times10^{-10} )^3\times6.023\times10^{23})\times\dfrac{8570}{2}$

M=92.90×10-3kg

$\therefore$ M=92.90g

Mass of atom per unit cell is $\dfrac{92.90}{2}=46.45$g