Question

The excess pressure inside an air bubble of radius R inside the liquid is:
(A) $\dfrac{2T}{R}$
(B) $\dfrac{4T}{R}$
(C) $\dfrac{T}{R}$
(D) $\dfrac{8T}{R}$

Answer 1

We know that a liquid drop consists of a single surface whereas in the case of an air bubble, it consists of two surfaces. We can derive a relation for the excess pressure inside an air bubble of radius R, using the concept of surface tension.

Complete step by step solution:
Let us consider an air bubble which possesses a surface tension of value T and has a radius of value R. We know that an air bubble contains two free surfaces. An expansion takes place due to the excess pressure inside the bubble.
The air bubble is inside a liquid. If the bubble expands by a radius dR then,
Work done by excess pressure is
$W=P\times 4\pi {{R}^{2}}\times dR$
In this case, the increase in potential energy would be the product of surface tension and increase in area
$\begin{aligned} &\Rightarrow T\times [2\\{4\pi {{(R+dR)}^{2}}-4\pi {{R}^{2}}\\}] \\\ &\Rightarrow 16\pi TRdR \\\ \end{aligned}$
Comparing this with work, we obtain
$\begin{aligned} & P\times 4\pi {{R}^{2}}\times dR=16\pi\times T\times RdR \\\ & \therefore P=\dfrac{4T}{R} \\\ \end{aligned}$
When we consider a liquid drop with a single surface and follow the same procedure we get the pressure as $\dfrac{2T}{R}$.The excess pressure of the air bubble inside liquid can be found out from the difference of these pressures which gives the value of $\dfrac{2T}{R}$.

Hence, option A is the correct answer.

Note: The formulas for excess pressure is to be learnt by heart by all the students as they get several application questions from this area. The excess pressure inside a soap bubble and an air bubble has the same value of $\dfrac{4T}{R}$whereas a water bubble has the value $\dfrac{2T}{R}$