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Question #42151

Difficulty: 8/10 | 🟢 Free

The Question

A microscopic system of initial rest mass $M$ is initially at rest. It undergoes a phase transition where its internal rest energy decreases by $\Delta U$ . During the transition, the system expands by a volume $\Delta V$ against a constant external pressure $P$ , doing mechanical work $P\Delta V$ . The remaining available energy is emitted as a single photon, and the system recoils freely in vacuum. Which of the following gives the exact energy $E_\gamma$ of the emitted photon? (Here $c$ is the speed of light in vacuum).

Options

A.
$\frac{(\Delta U - P\Delta V)(2Mc^2 - \Delta U - P\Delta V)}{2(Mc^2 - P\Delta V)}$
B.
$\frac{(\Delta U - P\Delta V)(2Mc^2 - \Delta U + P\Delta V)}{2Mc^2}$
C.
$\frac{(\Delta U + P\Delta V)(2Mc^2 - \Delta U - P\Delta V)}{2(Mc^2 + P\Delta V)}$
D.
$\frac{(\Delta U - P\Delta V)(2Mc^2 - \Delta U)}{2(Mc^2 - \Delta U)}$

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

Latent Heat and Volume Change|Recoil of Atom|Energy Level Scaling

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous synthesis of the First Law of Thermodynamics with relativistic energy-momentum conservation.

The primary trap (Option B) represents the standard semi-classical approximation for atomic recoil $E_\gamma \approx \Delta E (1 - \Delta E / 2Mc^2)$ where $\Delta E = \Delta U - P\Delta V$ .

Students falling for this trap fail to account for the exact relativistic mass defect ( $M_f = M - \Delta U/c^2$ ) and incorrectly partition the $P\Delta V$ work, missing that the work done reduces the total invariant mass-energy available in the denominator of the exact expression.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Latent Heat and Volume Change
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free