Question

The fraction of the total volume occupied by the atoms present in the simple cube is:

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{3\sqrt{2}}$
• D. $\frac{\pi }{4\sqrt{2}}$

Answer 1

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

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

Now, in simple cube, number of atoms present $=\frac{1}{8}\times 8=1$

So, volume occupied $=1\times \frac{4}{3}\pi r^3$

Now, in simple cube atoms at corners will be touching each other.

$\Rightarrow a=2r\Rightarrow r=\frac{a}{2}$

Volume of cube $=a^3$

So, fraction of total volume occupied $=\frac{\frac{4}{3}\pi r^3}{a^3}$

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

$=\frac{4}{3}\times \pi \times \frac{1}{2}$

$=\frac{\pi }{6}$