Solid State - Result Question 20

####20. The packing efficiency of the two-dimensional square unit cell shown below is

(2010)

(a) $39.27 \text{\%}$

(b) $68.02 \text{\%}$

(c) $74.05 \text{\%}$

(d) $78.54 \text{\%}$

Show Answer

Answer:

Correct Answer: 20. (d)

Solution:

  1. Contribution of circle from corner of square $=\frac{1}{4}$

$\Rightarrow$ Effective number of circle per square

$$ =\frac{1}{4} \times 4+1(\text { at centre })=2 $$

$\Rightarrow$ Area occupied by circle $=2 \pi r^{2}, r=$ radius.

Also, diagonal of square $4 r=\sqrt{2} L$, where $L=$ side of square.

$\Rightarrow$ Packing fraction $=\frac{\text { Area occupied by circles }}{\text { Area of square }}$

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

$\Rightarrow \text{\%}$ packing efficiency $=78.5 \text{\%}$.



NCERT Chapter Video Solution

Dual Pane