Question

An element crystallizes in 3-D hexagonal closed packed structure. Packing fraction of the lattice is :

• A. 0.74
• B. 0.26
• C. 0.52
• D. 0.48

Answer 1

The correct option is A 0.74

Packing Efficiency of Hexagonal Close Packing :

$\text{Total no. of spheres in a hcp unit cell}=(12\times \frac{1}{6})+(2\times \frac{1}{2})+3=6$
where,
$(12\times \frac{1}{6})\text{is for corners atoms of two hexagonal layers}$
$(2\times \frac{1}{2})\text{is for the centre atoms of two hexagonal layers}$

$3\text{is for the atoms of middle layer}$
$\text{Total volume occupied}=6\times \frac{4}{3}\times \pi \times r^3=8\pi r^3$
$\text{Volume of hcp unit cell (hexagon)}=\text{Base area}\times \text{Height}$


$\text{Base area of a regular hexagon}=6\times \frac{\sqrt{3}}{4}(2r{)}^2=6\times \sqrt{3}{r}^2$

$\text{Height of Unit Cell (h)}=4r\times \sqrt{\frac{2}{3}}$

$\text{Volume of hexagon (V)}=(6\times \sqrt{3}{r}^2)\times (4r\times \sqrt{\frac{2}{3}})V=24\sqrt{2}{r}^3$
$\text{Packing fraction}=\frac{8\pi r^3}{24\sqrt{2}{r}^3}=\frac{\pi }{3\sqrt{2}}P.F\approx 0.74$