# A large number of water drops,

Question:

A large number of water drops, each of radius $\mathrm{r}$, combine to have a drop of radius $\mathrm{R}$. If the surface tension is $T$ and mechanical equivalent of heat is $\mathrm{J}$, the rise in heat energy per unit volume will be :

1. (1) $\frac{2 T}{r J}$

2. (2) $\frac{3 T}{r J}$

3. (3) $\frac{2 T}{J}\left(\frac{1}{r}-\frac{1}{R}\right)$

4. (4) $\frac{3 T}{J}\left(\frac{1}{r}-\frac{1}{R}\right)$

Correct Option: , 4

Solution:

(4)

$\mathrm{R}$ is the radius of bigger drop. is the radius of $n$ water drops. Water drops are combined to make bigger drop. So,

Volume of $\mathrm{n}$ drops $=$ volume of bigger drop

$n\left(\frac{4}{3} \pi r^{3}\right)=\frac{4}{3} \pi R^{3}$

$R=r n^{1 / 3} \Rightarrow n=\left(\frac{R}{r}\right)^{3}$

$\Delta \mathrm{U}=\mathrm{T}$ (Change in surface area) $\Delta \mathrm{U}=\mathrm{T}\left(\mathrm{n} 4 \pi \mathrm{r}^{2}-4 \pi \mathrm{R}^{2}\right)$

$\Delta \mathrm{U}=4 \pi \mathrm{T}\left[\left(\frac{\mathrm{R}}{\mathrm{r}}\right)^{3} \mathrm{r}^{2}-\mathrm{R}^{2}\right] \Rightarrow \frac{4 \pi \mathrm{T}\left(\frac{\mathrm{R}^{3}}{\mathrm{r}}-\mathrm{R}^{2}\right)}{\mathrm{J}}$

$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\frac{4 \pi \mathrm{T}\left(\frac{\mathrm{R}^{3}}{\mathrm{r}}-\mathrm{R}^{2}\right)}{\mathrm{J} \times \frac{4}{3} \pi \mathrm{R}^{3}}=\frac{3 \mathrm{~T}}{\mathrm{~J}}\left[\frac{1}{\mathrm{r}}-\frac{1}{\mathrm{R}}\right]$