A large number of water drops each of radius \(r\) combine to have a drop of radius \(\mathrm{R}\). If the surface tension is \(\mathrm{T}\) and the mechanical equivalent at heat is \(\mathrm{J}\) then the rise in temperature will be (A) \((2 \mathrm{~T} / \mathrm{rJ})\) (B) \((3 \mathrm{~T} / \mathrm{RJ})\) (C) \((3 \mathrm{~T} / \mathrm{J})\\{(1 / \mathrm{r})-(1 / \mathrm{R})\\}\) (D) \((2 \mathrm{~T} / \mathrm{J})\\{(1 / \mathrm{r})-(1 / \mathrm{R})\\}\)

Let n be the number of water drops, each of radius r. The total surface area of all the drops will be given by the sum of the surface area of each drop, i.e. Total surface area = n * 4πr² And the surface energy of all drops can be found by multiplying the surface area by the surface tension T: Initial Surface Energy = n * 4πr² * T

When the water drops combine to form a single drop of radius R, the surface area and surface energy of the new single drop can be calculated as follows: Surface area of the large drop = 4πR² Surface Energy of the large drop = 4πR² * T

According to the conservation of energy principle, the decrease in surface energy will be used to increase the temperature of the water drops. Therefore, we can write: Decrease in Surface Energy = Initial Surface Energy - Surface Energy of the large drop Decrease in Surface Energy = n * 4πr² * T - 4πR² * T

The decrease in surface energy will be used to increase the kinetic energy (thermal energy) of the water. We can relate the increase in thermal energy to the rise in temperature using the mechanical equivalent of heat, J: Decrease in Surface Energy = (mass of water) * (specific heat capacity of water) * (rise in temperature) * J Since mass of water is constant, we can write: n * 4πr² * T - 4πR² * T = Rise in Temperature * Ω * J where Ω is a constant proportional to the mass and specific heat capacity of water.

From the previous step, we can solve for the rise in temperature: Rise in Temperature = (n * 4πr² * T - 4πR² * T) / (Ω * J) Let's now see which option matches our derived equation for the rise in temperature: Option (A) has a constant 2 and is independent of R, so it's not correct. Option (B) has a constant 3, but the equation we derived doesn't have any constant factor in the numerator, so it's also not correct. Option (C) has a constant 3 and contains the terms 1/r and 1/R, which looks similar to our equation. However, the exact coefficients don't match, so it's not correct. Option (D) has a constant 2 and contains the terms 1/r and 1/R, which closely match our derived equation (the difference in surface energy terms can be rewritten as a difference in reciprocal radii terms). Thus, the correct answer is: (D) \((2 \mathrm{~T} / \mathrm{J})\\{(1 / \mathrm{r})-(1 / \mathrm{R})\\}\)