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

Options

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

Answer

D

Explanation

in the two-dimensional packing
Packing fraction
`=("Area occupiedby circles within the square")/("Area of the square ")`
`=(2xxpir^(2))/(a^(2))`
`AC=sqrt(AB^(2)+BC^(2))=sqrt(a^(2)+a^(2))=sqrt(2)a`
But AC = 4 r (r = radius of each sphere )
`:.sqrt(2)a=4r`
or `a=(4)/(sqrt(2))r=2sqrt(2)r`
`:.` Packing fraction `(2xxpir^(2))/((2sqrt(2)r)^(2))=(pi)/(4)=(3.143)/(4)=0.785`
or Packing efficiency = 78.5%