Question

What will be the packing fraction in the following case?

The sphere is touching the sides of the cube exactly.

• A. pi/3
• B. pi/6
• C. pi/4
• D. pi/2

Answer 1

The correct option is B

pi/6

We know that,

Packing fraction = $\frac{Volumeoccupiedinsidetheunitcell}{Volumeofunitcell}$

Here the unit cell is the cube and volume occupied inside the unit cell will be equal to the volume of the sphere.

Let's assume the sphere is of radius r and cube is of side a.

Volume of sphere will be $\frac{4}{3}{}^{\ast }pi{}^{\ast }{r}^3$

Volume of cube will be $a^3$

Notice that the side of the cube is equal to the diameter of the sphere since it is touching all the sides.

So, a = d

a = 2r

Using the relation, we get

Packing fraction = $\frac{4}{3}{}^{\ast }pi{}^{\ast }\frac{r^3}{(2r{)}^3}$

= $\frac{4}{3}{}^{\ast }pi{}^{\ast }\frac{r^3}{8r^3}$

= $\frac{pi}{6}$ = 0.523