Question

One large soap bubble of diameter 'D' breaks into \(64\) bubbles having surface tension 'T'. The change in surface energy is

• A. \(2πTD^2\)
• B. \(4πTD^2\)
• C. \(6πTD^2\)
• D. \(8πTD^2\)

Hint:

A soap bubble has two surfaces; find the new radius from volume conservation.

Answer 1

The Correct Option is C

Step 1: Radius of the Small Bubbles:
Volume is conserved: \(64\cdot\dfrac43\pi r^3=\dfrac43\pi R^3\), so \(r=\dfrac R4\), where \(R=\dfrac D2\).

Step 2: Surface Area and Energy:
A soap bubble has two free surfaces (inside and outside), so the surface energy of a bubble of radius \(\rho\) is \(2\times4\pi\rho^2T=8\pi\rho^2T\).
Initial: \(8\pi R^2T=8\pi\dfrac{D^2}{4}T=2\pi TD^2\).

Step 3: Final Energy:
Final: \(64\times8\pi r^2T=64\times8\pi\dfrac{R^2}{16}T=32\pi R^2T=8\pi TD^2\).

Step 4: Change:
\[ \Delta E=8\pi TD^2-2\pi TD^2=6\pi TD^2 \]
This is option (C).

Final Answer:
The change in surface energy is \(6\pi TD^2\), option (C). \[ \boxed{\text{(C) } 6\pi TD^2} \]