Question

Packing fraction in a hexagonal arrangement is:

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

Answer 1

The correct option is A $\frac{\pi }{3\sqrt{2}}$

$packingfraction=\frac{volumeofatomsinunitcell}{volumeofunitcell}$
Total number of atoms in hcp = 6
Total volume of atoms in unit cell = $6\times \frac{4}{3}\pi r^3$
The base area of the unit cell, a regular hexagon made up of six equilateral triangles with side lengths = $2r$ would be
= $6\left[\frac{1}{2}\right(2r\left)\right(\sqrt{3}r\left)\right]=6\sqrt{3}{r}^2$
The height of the unit cell = $[\frac{2}{3}{]}^{\frac{1}{2}}.4r$
Volume of unit cell = $\left(6\sqrt{3}{r}^2\right)\left(\right[\frac{2}{3}{]}^{\frac{1}{2}}.4r)$
$\therefore packingfraction=\frac{6\times \frac{4}{3}\pi r^3}{\left(6\sqrt{3}{r}^2\right)\left(\right[\frac{2}{3}{]}^{\frac{1}{2}}.4r)}=\frac{\pi }{3\sqrt{(}2)}$