Chemistry - Solid State Question with Solution | TestHub

In a body-centered cubic (BCC) unit cell, the atoms touch each other along the body diagonal.

Let the edge length of the unit cell be $a$ and the radius of the atom be $r$.

The length of the body diagonal is $\sqrt{3}a$.

Since the atoms touch along the body diagonal, we have:

$\sqrt{3}a=4r\Rightarrow a=\frac{4r}{\sqrt{3}}$

The number of atoms per unit cell in a BCC structure is $Z=2$.

Volume of the unit cell is $V=a^3={\left(\frac{4r}{\sqrt{3}}\right)}^3=\frac{64r^3}{3\sqrt{3}}$

Volume occupied by the atoms is $v=2\times \frac{4}{3}\pi r^3=\frac{8}{3}\pi r^3$

Packing efficiency is given by:

$\text{Packing efficiency}=\frac{v}{V}\times 100$

$\text{Packing efficiency}=\frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}}\times 100=\frac{\sqrt{3}\pi }{8}\times 100$

Substituting the values of $\pi$ and $\sqrt{3}$:

$\text{Packing efficiency}\approx 68\%$

Answer: $68\%$