Question

A soap bubble of radius \(r\) is blown up to form a bubble of radius \(2r\) under isothermal conditions. If \(T\) is the surface tension of soap solution, then energy spent in blowing the bubble is

• A. \(3\pi Tr^2\)
• B. \(6\pi Tr^2\)
• C. \(12\pi Tr^2\)
• D. \(24\pi Tr^2\)

Hint:

Soap bubble has two surfaces, so total area \(= 2 \times 4\pi R^2\). Energy \(= T \times\) total area.

Answer 1

The Correct Option is D

Step 1: Understand surface energy of soap bubble.
Soap bubble has two surfaces (inner + outer).
Surface energy = Surface tension \(\times\) total surface area.
So energy:
\[ E = T \times (2 \times 4\pi R^2) = 8\pi TR^2 \]
Step 2: Initial and final energies.
Initial radius = \(r\):
\[ E_1 = 8\pi Tr^2 \]
Final radius = \(2r\):
\[ E_2 = 8\pi T(2r)^2 = 8\pi T \cdot 4r^2 = 32\pi Tr^2 \]
Step 3: Energy spent = Increase in surface energy.
\[ \Delta E = E_2 - E_1 \]
\[ \Delta E = 32\pi Tr^2 - 8\pi Tr^2 = 24\pi Tr^2 \]
Final Answer: \[ \boxed{24\pi Tr^2} \]