Question

The packing efficiency of a two-dimensional square unit cell shown is __________.

• A. 39.27%
• B. 68.02%
• C. 74.05%
• D. 78.57%

Answer 1

Answer: (D)

78.57%

The diagonal is given as $4r$, where $r$ is the radius of an atom.

Diagonal $=\sqrt{L^2+L^2}=\sqrt{2L}$

$4r$ $=\sqrt{2L}$ or $L=\frac{4r}{\sqrt{2}}$

Total area $=L^2$

or ${\left(\frac{4r}{\sqrt{2}}\right)}^2=8r^2$

Number of spheres inside the square is

$1+4\left(\frac{1}{4}\right)=2$

Area of each sphere $=\pi r^2$

Total area of spheres $=2\times \pi r^2$

Packing fraction $=\frac{\text{Total area of spheres}}{\text{Total area}}$

$=\frac{2\times \pi r^2}{8r^2}=\frac{\pi }{4}=0.785$

So, the percentage fraction is $78.5\%$.