Question

The percentage of empty space in a body centred cubic arrangement is________.

• A. $26$
• B. $74$
• C. $68$
• D. $32$

Answer 1

The correct option is D $32$

$\text{Calculating the packing efficiency in the BCC arrangement}$

Packing Efficiency:

$=\frac{\text{Volume occupied by atoms in a unit cell}}{\text{Total volume of the unit cell}}\times 100$

Packing efficiency $=\frac{Z\times \frac{4}{3}\pi r^3}{a^3}\times 100$

Where,

$r=$ radius on an atom
$Z=$ effective number of atoms in a unit cell
$a=$ edge length of a cubic unit cell


If above two diagrams are compared:

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

Then, the volume of a cubic unit cell is:

$a^3=\frac{4r}{\sqrt{3}a}=\frac{64r^3}{3\sqrt{3}}$


$\frac{1}{8}$ th contribution of corner atom and there are $8$ corner atoms in $BCC$ unit cell and complete one contribution of atom at body centre of $BCC$ unit cell.

Effective number of atoms in a unit cell $\left(Z\right)$:

$=(\frac{1}{8}\times 8)+1=2$

Volume occupied by the atoms:

$=2\times \frac{4}{3}\pi r^3\times 100\%$

$=\frac{2\times \frac{4}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}}\times 100\%$

$=\frac{\sqrt{3}\pi }{8}\times 100\%=67.98\%\approx 68\%$

The packing efficiency of $BCC$ unit cell $=68\%$

In the $BCC$ unit cell, the total filled space is represented by $68\%$.

$\text{Calculating the empty space in a}BC\text{unit cell}$

Therefore, the empty space in a body centered cubic arrangement: $=(100-68)\%=32\%$