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Question #42177

Difficulty: 8/10 | 🟢 Free

The Question

A closed cavity of variable volume $V$ contains one mole of an ideal monoatomic gas. The cavity also contains an electromagnetic radiation field in thermal equilibrium with the gas at temperature $T$ . Due to the presence of the gas, the effective wave impedance of the cavity is $Z = \beta \sqrt{V}$ , where $\beta$ is a positive constant. The radiation energy density is given by $u = \frac{\alpha T^4}{Z}$ , where $\alpha$ is a positive constant. The radiation exerts a pressure $P_{rad} = \frac{u}{6}$ on the cavity walls. If the system undergoes a reversible adiabatic process, which of the following quantities remains constant? (Here $R$ is the universal gas constant)

Options

A.
$V T^{3/2} \exp\left( \frac{4 \alpha}{3 \beta R} T^3 V^{1/2} \right)$
B.
$V T^{3/2} \exp\left( \frac{5 \alpha}{3 \beta R} T^3 V^{1/2} \right)$
C.
$V T^{3/2} \exp\left( \frac{\alpha}{\beta R} T^3 V^{1/2} \right)$
D.
$V T^{3/2} \exp\left( \frac{\alpha}{3 \beta R} T^4 V^{1/2} \right)$

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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

Internal Energy Dependency|Energy Density Partitioning|Frequency Sensitivity of Z

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the First Law of Thermodynamics to a composite system of an ideal gas and a radiation field with a non-standard, volume-dependent impedance.

The primary trap is neglecting the radiation pressure's contribution to the work done ( $P_{rad} dV$ ) or incorrectly assuming the standard vacuum radiation pressure ( $P_{rad} = u/3$ ) instead of the given $u/6$ .

Students conditioned to treat internal energy as purely kinetic ( $U = \frac{f}{2}nRT$ ) will fail to account for the $dV$ term arising from the volume dependence of the radiation energy density, leading to an incorrect adiabatic invariant.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Internal Energy Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free