The packing efficiency of the two-dimensional square unit ce... | Chemistry | SolveeIt

Question

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

$39.27 \%$

$68.02 \%$

$74.05 \%$

$78.54 \%$

Answer

$78.54 \%$

Explanation

Solution

Contribution of circle from comer of square $= \frac{1}{4}$
$\Rightarrow$ Effective number of circle per square
$= \frac{1}{4} \times 4 + 1(at\, \, centre) = 2$
$\Rightarrow$ Area occupied by circle $= 2 \pi r^2 , r = radius.$
Also, diagonal of square $4r = \sqrt 2 L,$ where L = side of square.
$\Rightarrow$ Packing fraction $= \frac{Area\, \, occupied\, \, by\, \, circles}{Area\, \, of\, \, square}$
$= \frac{2 \pi r^2}{L^2} = \frac{2 \pi r^2}{8r^2} = \frac{\pi}{4} = 0.785$
$\Rightarrow \% Packing\, \, efficiency = 78.5 \%$

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