Question

\[8000\] identical water drops are combined to form a big drop then the ratio to the final surface energy to the initial surface energy, if all the drops together is
\[\begin{align}
  & A.1:10 \\
 & B.1:15 \\
 & C.1:20 \\
 & D.1:25 \\
\end{align}\]

Answer 1

Hint: Surface tension is the tendency of liquids to occupy minimum surface area. It is defined as the force acting per unit length of the liquid. It can also be defined as the energy needed by the liquid to reduce its surface area. This is observed at the liquid- air interface of the liquid, where the force of attraction among the liquid molecules, is greater than that of the liquid-air force.

Formula used: $T=\dfrac{E}{A}$

Complete step by step answer:
We know that the surface tension $T$ is also defined as the force $F$ experienced due to unit length
$L$ of the tube it is filled with. Mathematically, it is given as
$T=\dfrac{F}{L}$ or $T=\dfrac{E}{A}$, where $E$ is the energy of the liquid and $A$ is the surface area of the liquid.
Here, let us assume that the surface tension of the water droplets to be $T$.
Since $n=8000$ small droplets form one big droplet, we can say that the volume of the small droplet of radius $r$ is equal to the volume of the big droplet$R$.
Then, we can express the same mathematically as $\dfrac{4}{3}\pi r^{3}\times 8000=\dfrac{4}{3}\pi R^{3}$
$\implies 8000r^{3}=R^{3}$
$\implies 20r=R$
From,$T=\dfrac{E}{A}$, we can say $E=TA$
Then the surface energy of the small droplets is given as $s=8000(4\pi r^{2}T)$
And the surface energy of the big droplets is given as $S=4\pi R^{2}T$
Taking the ratio, we get $\dfrac{S}{s}=\dfrac{4\pi R^{2}T}{8000(4\pi r^{2}T)}$
$\implies \dfrac{S}{s}=\dfrac{R^{2}}{8000r^{2}}$
$\implies \dfrac{S}{s}=\dfrac{(20r)^{2}}{8000r^{2}}=\dfrac{400r^{2}}{8000r^{2}}$
$\implies \dfrac{S}{s}=\dfrac{1}{20}$

So, the correct answer is “Option C”.

Additional Information: At the surface of the liquid there are to forces, one is the liquid-liquid interaction which draws the liquid molecule inwards, and the other is the tangential force parallel to the liquid surface. This is responsible for the spherical shape of drops and meniscus formation of liquids when filled in tubes. These are also responsible for the separation of oil and water in the oil-water mixtures.

Note: Surface tension is expressed in different forms and units. It is due to unbalanced forces acting on the molecules. Remember that surface tension can be reduced by adding surfactants. It is suggested to remember that $1dyne=10^{-5}N$ and $1cm=10^{-2}m$