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

Answer: (C)

% packing efficiency $= \frac{\text{Vol.occupied by atom}}{\text{Vol. of unit cell}}\times100=\frac{\frac{4}{3}\pi r^{3}}{a^{3}}\times100$
Let radius of corner atom is r and radius of central atom is 2r
So, $\sqrt{3}a=2\left(2r\right)+2r=6r$
$a=\frac{6r}{\sqrt{3}}=2\sqrt{3}r$
Now
% P.E $=\frac{\frac{4}{3}\pi r^{3}+\frac{4}{3}\pi\left(2r\right)^{3}}{\left(2\sqrt{3}r\right)^{3}}\times100$
$=\frac{\frac{4}{3}\pi\left(r^{3}+8r^{3}\right)}{8\times3\sqrt{3}r^{3}}\times100$
$=\frac{4\pi\times9r^{3}}{3\times8\times3\sqrt{3}r^{3}}\times100=90.6\%\approx90\%$