Question

A big drop is formed by coalescing 1000 small identical drops of water. If E₁ be the total surface energy of 1000 small drops of water and E₂ be the surface energy of single big drop of water, the E₁ : E₂ is x : 1 where x = ________.

Answer 1

Correct Answer: 10

The surface energy of a droplet is defined as the product of its surface tension and surface area, represented by the equation: \( E = \sigma \times A \).

Here, \( \sigma \) denotes the surface tension and \( A \) represents the surface area.

Consider a small drop with radius \( r \). Its surface area is calculated as: \( A_1 = 4 \pi r^2 \).

For a collection of 1000 such small drops, the aggregate surface area is: \( A_1(\text{total}) = 1000 \times 4 \pi r^2 = 4000 \pi r^2 \).

The total surface energy of these 1000 small drops is: \( E_1 = \sigma \times A_1(\text{total}) = \sigma \times 4000 \pi r^2 \).

When these 1000 small drops merge, their total volume is conserved. The volume of a single small drop is: \( V_{\text{small}} = \frac{4}{3} \pi r^3 \).

Consequently, the combined volume of 1000 small drops is: \( V_{\text{total}} = 1000 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (1000 r^3) \).

Let \( R \) be the radius of the resultant large drop. Its volume is expressed as: \( V_{\text{large}} = \frac{4}{3} \pi R^3 \).

Equating the total volume of the small drops to the volume of the large drop yields: \( \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (1000 r^3) \).

This equation simplifies to: \( R^3 = 1000 r^3 \), from which we derive \( R = 10r \).

The surface area of the large drop is therefore: \( A_2 = 4 \pi R^2 = 4 \pi (10r)^2 = 4 \pi \times 100r^2 = 400 \pi r^2 \).

The surface energy of the large drop is: \( E_2 = \sigma \times A_2 = \sigma \times 400 \pi r^2 \).

The ratio of the total surface energy of the small drops to the surface energy of the large drop is calculated as: \( \frac{E_1}{E_2} = \frac{\sigma \times 4000 \pi r^2}{\sigma \times 400 \pi r^2} \).

Upon simplification, this ratio becomes: \( \frac{E_1}{E_2} = \frac{4000}{400} = 10 \).

Final Answer: The ratio of surface energies is: 10.