Expression for Work Done in Forming a Soap Bubble :

When forming a soap bubble, the work done is essentially the energy required to create the new surface area. A soap bubble has two surfaces (an inner and an outer surface). Hence, if the bubble has radius $ R $, then :

The surface area of one spherical surface is $ 4\pi R^2 $.

Since there are two surfaces, the total surface area is $ 2 \times 4\pi R^2 = 8\pi R^2 $.

The energy (or work) required to create this bubble from scratch can be approximated as:

$ W = \text{(Surface Tension)} \times \text{(Total Surface Area)} = T \cdot 8\pi R^2. $

Thus, for a bubble of radius $ R $ at room temperature with surface tension $ T_1 $:

$ W_1 = 8\pi R^2 T_1. $

When the Radius is Doubled :

If we were to form a bubble of radius $ 2R $ at the same (initial) surface tension $ T_1 $, then:

$ W' = 8\pi (2R)^2 T_1 = 8\pi (4R^2) T_1 = 32\pi R^2 T_1. $

Comparing $ W' $ and $ W_1 $:

$ \frac{W'}{W_1} = \frac{32\pi R^2 T_1}{8\pi R^2 T_1} = 4. $

If the surface tension remained the same, the work required to form a bubble of twice the radius would be $ 4W_1 $.

Effect of Heating on Surface Tension :

The problem states that the solution is heated. Generally, increasing the temperature reduces the surface tension of a liquid. Let the new surface tension at the elevated temperature be $ T_2 $, with $ T_2 < T_1 $.

Thus, when blowing a bubble of radius $ 2R $ from the heated solution:

$ W_2 = 8\pi (2R)^2 T_2 = 32\pi R^2 T_2. $

Since $ T_2 < T_1 $, we have:

$ W_2 = 32\pi R^2 T_2 < 32\pi R^2 T_1. $

But $ 32\pi R^2 T_1 = 4W_1 $. Therefore:

$ W_2 < 4W_1. $

Conclusion :

Due to the reduction in surface tension upon heating, the work required to form a bubble of twice the radius is less than four times the original work at room temperature.

Correct Answer :

$ \boxed{W_2 < 4W_1} \quad \text{(Option C)} $

Fluid Mechanics

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