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The packing efficiency of a two-dimensio...

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

A

`39.27%`

B

`68.02%`

C

`74.05%`

D

`78.54%`

Text Solution

Verified by Experts

The correct Answer is:
D
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`
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