Question

A big drop is formed by coalescing 1000 small droplets of water. The surface energy will become:

• A. \( \frac{1}{100} \)th
• B. \( \frac{1}{10} \)th
• C. \( 100 \) times
• D. \( 10 \) times

Hint:

When droplets merge, the total volume remains constant, but the surface area (and thus the surface energy) changes significantly, as it is proportional to the square of the radius.

Answer 1

The Correct Option is B

Using the conservation of volume, the volume of the big drop is equal to the sum of the volumes of the 1000 smaller drops: \[ \frac{4}{3}\pi R^3 = 1000 \cdot \frac{4}{3}\pi r^3, \] where \( R \) is the radius of the large drop and \( r \) is the radius of each smaller drop. Simplify: \[ R^3 = 1000 \cdot r^3 \implies R = 10r. \] The surface energy of a sphere is proportional to its surface area. The initial energy (for 1000 small drops) is: \[ E_i = 1000 \cdot 4\pi r^2. \] The final energy (for the large drop) is: \[ E_f = 4\pi R^2 = 4\pi (10r)^2 = 100 \cdot 4\pi r^2. \] Thus, the ratio of the final to initial energy is: \[ \frac{E_f}{E_i} = \frac{4\pi (10r)^2}{1000 \cdot 4\pi r^2} = \frac{1}{10}. \] Final Answer: \[ \boxed{\frac{1}{10} \, \text{th}}. \]