Question

Two mercury drops (each of radius r) merge to form a bigger drop. The surface energy of bigger drop, if T is the surface tension, is
$\begin{aligned} & \text{A}\text{. }{{2}^{\dfrac{8}{3}}}\pi {{r}^{2}}T \\\ & \text{B}\text{. }{{2}^{\dfrac{5}{3}}}\pi {{r}^{2}}T \\\ & \text{C}\text{. }4\pi {{r}^{2}}T \\\ & \text{D}.\text{ }2\pi {{r}^{2}}T \\\ \end{aligned}$

Answer 1

Basically we know that two drops of mercury are merged together to form a bigger drop. It is clear that the new drop form will have volume equal to the sum of volume of two drops. We will first find the new radius after that using the formula of surface tension we will find surface tension.

Formula used:
Volume of drop $=\dfrac{4}{3}\pi {{r}^{3}}$,
$\text{Surface tension}=T4\pi {{R}^{2}}$

Complete step by step answer:
We know that,
Volume of new drop = sum of volume of two drop

$\begin{aligned} & 2\times \dfrac{4}{3}\pi {{r}^{3}}=\dfrac{4}{3}\pi {{R}^{3}} \\\ & R={{2}^{\dfrac{1}{3}}}r \\\ \end{aligned}$

So the radius of the new available drop is $R={{2}^{\dfrac{1}{3}}}r$.

Now,
Using the formula of surface energy or surface tension.
$\text{Surface tension}=T4\pi {{R}^{2}}$
$\text{Surface tension}=T\pi {{2}^{\dfrac{8}{3}}}{{r}^{2}}$
Hence, the correct answer is option A.

Additional Information
Surface tension is the attractive force found in liquids which is responsible for pulling surface molecules in the rest of the liquid. Further, it minimizes the surface area. The attractive forces we are talking about here are because of electrostatic forces. Usually, we refer to this cohesion at the gas-liquid surface; remember not liquid-solid or not liquid-liquid. You might have noticed it takes place in water often but it also happens with all other liquids to some extent too. Furthermore, we see that there are electrostatic forces. These forces attract molecules to each other. The slight dipole with water is able to make the molecules specifically attract to each other. Thus, it enables it to produce a tight-knit cohesive unit. The force is not able to make much impact when in a liquid state as each molecule pulls all directions towards the other molecules. However, on the surface of the liquid, these molecules do not pull up at all due to the fact that no molecules are there for pulling the others up. The molecules are pulled down by a strong force between them. Thus, it makes a solid obstacle at the liquid-gas interface. Therefore, a lot of force will be required to break through the liquid’s surface in comparison to going through liquid.

Note:
When two water droplets combine to form a large drop after its combination the surface area decreases. Surface tension of the new form droplet is reduced as Surface area of two droplets is greater than the large single droplet. The energy of new form droplets is also decreased due to decrease in surface tension. The new form droplet has volume equal to the sum of the volume of drop which is merged together.