Question

The fraction of total volume occupied by the atoms in a simple cube is -

• A. $\frac{\pi }{4}$
• B. $\sqrt{2}\frac{\pi }{8}$
• C. $\sqrt{2}\frac{\pi }{6}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is D $\frac{\pi }{6}$

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


In simple cube atoms at corners will be touching each other.
Therefore $a$ = $2r$
$r$ = $\frac{a}{2}$

Volume of the atom = $\frac{4}{3}\pi r^3$
=$\frac{4}{3}\pi {\left(\frac{a}{2}\right)}^3$

In simple cubic arrangement, there is effectively one atom per unit cell.

Volume of the cube = $a^3$
Packing fraction = $\frac{Volumeoccupiedbytheatom}{Totalavailablevolume}$

= $\frac{\frac{4}{3}\pi {\left(\frac{a}{2}\right)}^3}{a^3}$
=$\frac{\pi }{6}$