Question

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 solid 2 than in 1. What is the approximate packing efficiency in solid 2?

• A. 45%
• B. 65%
• C. 90%
• D. 75%

Answer 1

The correct option is C 90%

Given,
$r_B=2r_A$
where,
$r_A\text{is radius of atom A}$
$r_B\text{is radius of atom B}$

Packing fraction for Solid 2 is:
$P.F.=\frac{{(Z_{eff}\times \frac{4}{3}\pi r_A^3)}_A+{(Z_{eff}\times \frac{4}{3}\pi r_B^3)}_B}{a^3}$
where,
$a^3\text{is volume of the unit cell of solid B}$
For a body centred unit cell, the atoms at corner and body centred are in contact with each other by body diagonals.
$\therefore$
$2(r_A+r_B)=\sqrt{3}a$
$\Rightarrow 2(r_A+2r_A)=\sqrt{3}a$
$\Rightarrow 6r_A=\sqrt{3}a$
$\Rightarrow 2\sqrt{3}{r}_A=a$

$\text{Voume of unit cell of solid B}=(2\sqrt{3}{r}_A{)}^3=8\times 3\sqrt{3}{r}_A^3$
$Z_{eff}=1$ for both A and B atoms.

$\Rightarrow P.F.=\frac{1\times \frac{4}{3}\pi r_A^3+\frac{4}{3}\pi \left(8r_A^3\right)}{8\times 3\sqrt{3}{r}_A^3}=\frac{9\times \frac{4}{3}\pi }{8\times 3\sqrt{3}}=\frac{\pi }{2\sqrt{3}}$
$\text{Packing efficiency}=\frac{\pi }{2\sqrt{3}}\times 100\simeq 90\%$