Q: Number of droplets are combined isothermally to form a big drop, the ratio of initial and final surface energies of the system is
Sol: $\large \frac{U_i}{U_f} = \frac{n(4\pi r^2)T}{4\pi R^2 T}$
$\large \frac{U_i}{U_f} = \frac{n r^2}{R^2} = \frac{n r^2}{n^{(1/3 r)^2}} $
$\large \frac{U_i}{U_f} = \frac{n^{1/3}}{1}$