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Question #42707

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated spherical soap bubble of initial radius $r_0$ contains $n$ moles of an ideal monatomic gas at an initial temperature $T_0$ . The bubble is given a uniformly distributed total charge $Q$ . The system is placed in a vacuum and allowed to expand quasi-statically to a new radius $2r_0$ . Assuming the capacitance of the bubble is that of a thin spherical shell and the surface tension of the soap solution is $S$ , what is the final temperature $T_f$ of the gas? (Let $R$ be the universal gas constant and $\epsilon_0$ be the vacuum permittivity).

Options

A.
$T_0 + \frac{Q^2}{24\pi \epsilon_0 n R r_0} - \frac{16\pi S r_0^2}{n R}$
B.
$T_0 + \frac{Q^2}{24\pi \epsilon_0 n R r_0} - \frac{8\pi S r_0^2}{n R}$
C.
$T_0 - \frac{Q^2}{24\pi \epsilon_0 n R r_0} - \frac{16\pi S r_0^2}{n R}$
D.
$T_0 + \frac{Q^2}{12\pi \epsilon_0 n R r_0} - \frac{16\pi S r_0^2}{n R}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Self-Energy of Charged Sphere|Internal Energy Dependency|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the thermodynamic equivalence of electrostatic self-energy and mechanical work in deformable bodies.

Students often fall into the trap of treating the electrostatic pressure as an external opposing force (wrong sign) or failing to recognize that the decrease in electrostatic self-energy ( $\Delta U_{\text{elec}} < 0$ ) provides energy that drives the expansion.

Additionally, it tests the classic trap of forgetting that a soap bubble has two liquid-gas interfaces, which doubles the surface energy contribution.

By requiring global energy conservation ( $\Delta U_{\text{gas}} + \Delta U_{\text{elec}} + \Delta U_{\text{surface}} = 0$ ), the problem forces students to correctly assign signs to the changes in internal, electrostatic, and surface energies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Self-Energy of Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free