A large number of identical small water drops, each of radius r , coalesce to form a single large drop of radius R . If the surface energy released during this process is entirely converted into heat which is retained by the water, the rise in temperature of the water is : (Given: T is the surface tension, is the density of water, and s is the specific heat capacity of water)

Step-by-step solution

Let n be the number of small drops. By conservation of volume: n 4 3 r^3 = 4 3 R^3 n = R^3 r^3 The initial surface energy of the n drops is: U_ i = n(4 r^2 T) The final surface energy of the single large drop is: U_ f = 4 R^2 T The surface energy released during coalescence is: U = U_ i - U_ f = n(4 r^2 T) - 4 R^2 T Substituting n = R^3 r^3 : U = R^3 r^3 (4 r^2 T) - 4 R^2 T = 4 R^3 T ( 1 r - 1 R ) The mass of the water is: m = Volume Density = 4 3 R^3 The heat gained by the water is: Q = m s = ( 4 3 R^3 ) s Equatin