You are given marbles of diameter `10mm`. They are to be placed such that their centres are laying in a square bound by four lines each of length `40mm`. What will be the arrangements of marbles in a plane so that maximum number of marbles can be placed inside the area? Sketch the diagram and derive expression for the number of molecules per unit area.

Explanation

Area of square having spherical marbles in it
`=(40xx10^(-1)xx40xx10^(-1))cm^(2)=16cm^(2)`
The maximum number of spheres of `10mm` diameter in `h.c.p.` packing can be seen in Figure.
Total length covered by spheres
`=5+4xxCD=5+4xx10 sin 60^(@)` ltBRgt `=5+4xx5sqrt(3)~~40 mm`
Maximum number of spheres
`(s)=underset((Fu ll))(14)+underset((Half))(8)=18`
`:.` Number of spheres per `cm^(2)=(18)/(16)=1.125`