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

Options

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

Answer

D

Explanation

Diagonal `= 4r`
Diagonal `= sqrt(L^(2) + L^(2)) = sqrt(2)L`
or `4r = sqrt(2)L`
or `L = (4r)/(sqrt(2))`
Total area `= L^(2)`
or `((4r)/(sqrt(2)))^(2) rArr 8r^(2)`
Number of spheres inside the square is
`1+4((1)/(4)) = 2`
Area of each sphere `= pir^(2)`
Total area of spheres `= 2 xx pir^(2)`
Packing fraction `= ("Total area of spheres")/("Total area")`
`= (2 xx pir^(2))(8r^(2)) = (pi)/(4) approx 0.785`