Question

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

• A. 39.27%
• B. 68.02%
• C. 74.05%
• D. 78.54%

Answer 1

The correct option is D 78.54%


Diagonal = 4r
Diagonal = $\sqrt{L^2+L^2}-\sqrt{2}L$
$or4r=\sqrt{2}L$
$orL=\frac{4r}{\sqrt{2}}$
Total area = $L^2$
or $(\frac{4r}{\sqrt{2}}{)}^2\Rightarrow 8r^2$
Number of sphered 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 = $\frac{Totalareaofspheres}{Totalarea}$
$=\frac{2\times \pi r^2}{8r^2}=\frac{\pi }{4}=0.785$