Question

You are given marbles of diameter $10$ mm. They are to be placed such that their centers are laying in a square bound by four lines each of length $40$ mm. 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

CD=10 sin 60= $10\frac{\sqrt{3}}{2}$
$5\sqrt{5}$ mm
Area of square having spherical marbles in it
$=40\times {10}^{-1}\times 40\times {10}^{-1}c{m}^2=16c{m}^2$
The maximum number of sphere of 10 mm diameter in hcp packing can be seen in the figure above.
Total length covered by spheres =$5+4\times CD$
$=5+4\times 10sin{60}^o$
$=5+4\times 5.\sqrt{3}=40mm$
Maximum number of sphere(s) = $\underset{\left(full\right)}{14}+\underset{\left(Half\right)}{8}=18$
$\therefore$ Number of spheres per $c{m}^2=\frac{18}{16}=1.125$