Question

An element with molar mass $2.7\times {10}^{-2}kgmo{l}^{-1}$forms a cubic unit cell with edge length 405 $pm$. If its density is $2.7\times {10}^{3}kg{m}^{-3}$ , what is the nature of the cubic unit cell?

Answer 1

Given,
Density of element,$d=2.7\times {10}^{3}kg{m}^{-3}$
Molar mass,$M=2.7\times {10}^{-2}kgmo{l}^{-1}$
Edge length,$a=405pm=405\times {10}^{-12}m=4.05\times {10}^{-10}m$

Value of $Z$
Let $Z$ be the number of atoms in unit cell. Number of atoms in 1 mole substance,
$N_A=6.022\times {10}^{23}$atoms

Using formula,

Density(d)$=\frac{Z\times M}{a^3\times N_A}\Rightarrow$

$Z=\frac{d\times a^3\times N_A}{M}$

$Z=\frac{2.7\times {10}^3\times (4.05\times {10}^{-10}{)}^3\times 6.022\times {10}^{23}}{2.7\times {10}^{-2}}$
$Z=4$

Nature of unit cell
Hence, there are four atoms of the element present per unit cell. So, the unit cell can be a face centred cubic (fcc) or cubic close packed (ccp) structure.

Final answer: Face centred cubic (fcc) or Cubic close packed (ccp) structure.