Question

Two mercury drops (each of radius 'r') merge to form a bigger drop. The surface energy of the bigger drop, if T is the surface tension, is:
A. $4 \pi r^2 T$
B. $2 \pi r^2 T$
C. $2^{8/3} \pi r^2 T$
D. $2^{5/2} \pi r^2 T$

Answer 1

In the forming of a bigger drop of radius R from two small drops of radius r, the total volume will remain the same. The surface area will change though, and surface energy is the additional energy possessed by surface molecules over the molecules in the interior of drop.

Formula used:
Surface energy = surface tension $\times$ Area.
where in our case, we require the surface area of the sphere.

Complete step by step answer:
If we assume the drops to be spherical then, the total volume of two drops of radius r will be used in making a bigger drop of radius R with same volume, so we may write:
$\dfrac{4}{3} \pi R^3 = 2 \times \dfrac{4}{3} \pi r^3$,
which will give us:
$R = 2^{1/3} r$
to be the relation between small drop radius and larger drop radius.
Now, the formula for surface energy is given as:
U = surface area $\times$ Tension .
We know, the surface area of a sphere is $4 \pi R^2$. Substituting this in the formula and also keeping the value of R in terms of r, we get:
$U = 4 \pi (2^{2/3} r^2) T$.
We can write 4 as $2^2$, so we get:
$U = 2^{(2+6)/3} \pi r^2 T$
or,
$U = 2^{8/3} \pi r^2 T$
Therefore, the correct answer is option (C).

Additional information:
In a fluid, the surface behaves differently than the interior. Interior molecules are completely surrounded by fluid whereas at the surface, the atmosphere of a molecule becomes different. The surface tension is the amount of Force that the surface has per unit length. Surface energy is the excess of energy that surface has over the fluid inside the surface.

Note:
The volume for two drops remains the same but the surface area in forming the bigger drop reduces. This is the reason why energetically it is more favorable to form a drop for water molecules as it helps in reducing the surface area and energy minimization can follow as the surface area is reduced (see the expression).