Two soap bubbles coalesce to form a single bubble. If $V$ is the subsequent change in volume of contained air and $S$ the change in total surface area, $T$ is the surface tension and $P$ atmospheric pressure, which of the following relation is correct ?

Question

When two soap bubbles of radii $ a $ and $ b $ ($ b>a $) coalesce, the radius of curvature of the common surface is:

Answer 1

To solve this problem, we begin by understanding the physical principles involved in the coalescence of soap bubbles.

When two soap bubbles merge into one, there is a change in the volume of enclosed air and the surface area of the bubble. The problem involves understanding the thermodynamic relationship between these quantities and the forces acting on the bubble, such as surface tension and pressure.

When the bubbles combine, the pressure inside the bubbles, which is due to both the atmospheric pressure $P$ and the Laplace pressure due to surface tension $T$, changes.

The equation that relates the change in volume $V$, change in surface area $S$, surface tension $T$, and atmospheric pressure $P$ is derived from the balance of pressure forces before and after coalescence.

By considering the equilibrium state and rearranging the contributions of these forces, we arrive at the relationship:

3PV + 4ST = 0

This particular equation asserts that the sum of the work done by and against the pressure and surface tension forces is zero, indicating a state of equilibrium after the bubbles have coalesced.

Let's briefly consider other options to understand why they are incorrect:

4PV + 3ST = 0: This equation mistakenly gives a higher importance to atmospheric pressure changes than surface tension changes.
2PV + 3ST = 0: This equation underestimates the relative effect of the surface tension change after coalescence.
3PV + 2ST = 0: Similarly, this option too underestimates the significance of the surface tension in the equilibrium equation.

Therefore, the correct equation relating the change in volume, surface area, surface tension, and atmospheric pressure during this process is 3PV + 4ST = 0. This captures the balance of work and describes how the mechanical and surface forces reach equilibrium in the system.

Answer 2