Question

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.

Answer 1

In order to accommodate maximum of spheres, there should be hcp (hexagonal closed packing).
Area of square having spherical marbles `=16cm^(2)`
Maximum number of spheres = 14 (full) + 8 (half)
Number of spheres = 18
Per unit area (per `cm^(2)`) `=18/16=1.125`
Length PQ of square = 4 cm
Length PR = `5+4xx5sqrt40 mm = 4` cm

CD 10 sin `60^(@)=(10sqrt3)/(2)=5sqrt3`