Question

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of the big drop is $\frac{10}{x}$.The value of x is _________.

Answer 1

The volume of 1000 small drops is equal to the volume of the big drop:

$1000 \times \frac{4}{3}\pi r^3 = \frac{4}{3}\pi R^3$

$R = 10r$

The surface energy (S.E.) of the 1000 drops is:

$\text{S.E. of 1000 drops} = 1000 \times (4\pi r^2)T$

The surface energy of the big drop is:

$\text{S.E. of Big drop} = 4\pi R^2T$

Substituting $R = 10r$:

$\text{S.E. of Big drop} = 4\pi (10r)^2T = 400\pi r^2T$

The ratio of the surface energy is:

$\frac{\text{S.E. of 1000 drops}}{\text{S.E. of Big drop}} = \frac{1000 \cdot 4\pi r^2T}{4\pi (10r)^2T}$

$\frac{\text{S.E. of 1000 drops}}{\text{S.E. of Big drop}} = \frac{1000}{100} = \frac{10}{x}$

$x = 1$