Question

Fraction of total volume occupied by atoms in simple cubic cell is:

• A. $\frac{\pi }{2}$
• B. $\frac{\sqrt{3}\pi }{8}$
• C. $\frac{\sqrt{2}\pi }{6}$
• D. $\frac{\pi }{6}$

Answer 1

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

We know that,
Effective number of atoms present in simple cubic structure $=\frac{1}{8}\times 8=1$
Volume occupied by an atom in simple cubic structure is $1\times \frac{4}{3}\pi r^3$
In simple cubic structure, atoms are present in corners of the cube and they will be touching each other, hence
$\text{Edge length =}2\times \text{radius}$
$a=2\times r$
Fraction of volume occupied by atoms in simple cube $=\frac{\frac{4}{3}\pi r^3}{a^3}$
Substituting, $r=\frac{a}{2}$
Fraction of volume occupied by atoms in simple cube $=\frac{\frac{4}{3}\pi (\frac{a}{2}{)}^3}{a^3}$
Fraction of volume occupied by atoms in simple cube $=\frac{\frac{4}{3}\pi \left(\frac{a^3}{8}\right)}{a^3}$
$=\frac{\pi }{6}$