Question

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

• A. 39.27%
• B. 68.2 %
• C. 74.05%
• D. 78.54%

Answer 1

The correct option is D 78.54%

Area od square of side length $l=l^2$

Number of atoms in unit cell $=1+4\times \frac{1}{4}=2$

Let, radius of each circle be $r$

Area occupied $=\pi r^2\times 2$

Now, diagonal of square $=l\sqrt{2}$

$\Rightarrow 4r=l\sqrt{2}$

$r=\frac{l}{2\sqrt{2}}$

Now, Packing efficiency $=\frac{Areaoccupied}{Totalarea}\times 100$%

$=\frac{2\pi r^2}{l^2}\times 100$%

$=\frac{2\pi l^2}{(2\sqrt{2}{)}^2\times l^2}\times 100$%

$=\frac{100\times 2\pi }{8}$%

$\Rightarrow PackingEfficiency=78.54$%