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

Correct 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:

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

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

\[ x = 1 \]