answer:

suppose the edge length of the unit cell = a

&

radius of the sphere = r

then,since the sphere are each other along the edge,therefore a = 2r

now there are 8 spheres at the corners of the cube & each sphere at the corner is shared by 8 unit cells & the contribution per unit cell is 1/8 so that

number of spheres per unit cell is 8 x 1/8 = 1

volume of sphere =4/3πr3 & volume of cube = a3 = (2r)3 = 8r3

now packing efficiency = (volume of one sphere / total volume of cubic unit cell) x 100

or

(4/3 πr3 / 8r3) x 100 = 52.4%

therefore the volume occupied in simple cubic arrangement = 52.4%

(ii) body centered cubic:

let us suppose the edge leght = a & radius of each sphere = r then there are 8 spheres at the corners & 1 in the body of unit cell

therefore number of spheres per unit cell = (8 x1/8) + 1 = 2

now volume of unit cell = a3 = (4r / √3)3

and volume of a sphere = 4 / 3πr3

total volume of two spheres = 2 x 4/3πr3

packing efficiency = (volume of two spheres in unit cell/total volume of unit cell ) x 100

= (2 x 4/3πr3 / (4r/√3)3 ) x 100 = 68%