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 big drop is 10/x. The value of x is ________.

Solution

Physics sir · 3 steps

Volume stays the same when drops merge, so the big drop's radius is only 10 times the small one's.

1. Conserve volume. Let each droplet have radius r and the big drop radius R. The water does not change, so 1000 × 4/3π r³ = 4/3π R³. This gives R³ = 1000 r³, so R = 10r.
2. Surface energies. Surface energy is surface tension times surface area, E = T × 4π r². The 1000 droplets have E₁ = 1000 × 4π r² T. The big drop has E₂ = 4π (10r)² T = 100 × 4π r² T.

   E₁/E₂ = (1000 × 4π r² T)/(100 × 4π r² T) = 10

3. So. 10/x = 10, which gives x = 1.

The common slip: 100. Writing the ratio upside down, big drop to droplets, gives 1/10 = 10/x and so x = 100. The question asks for droplets to big drop.

The idea to keep. When n drops merge, volume is conserved, so R = n^(1/3) r. The total area, and so the surface energy, drops by a factor of n^(1/3).

Answer from the NTA final answer key. Solution by PrepWiser.