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Question #42134

Difficulty: 8/10 | 🟢 Free

The Question

A solid isotropic medium contains a spherical cavity of volume $V_0$ . The cavity is filled with one mole of an ideal monatomic gas at an initial temperature $T_0$ and pressure $P_0$ . The solid is heated quasi-statically. The cavity volume $V$ responds to both the temperature $T$ of the solid and the internal gas pressure $P$ according to the relation: $V = V_0 \left[ 1 + \gamma (T - T_0) + \frac{P - P_0}{K} \right]$ where $\gamma = \frac{5}{2T_0}$ is the volumetric thermal expansion coefficient and $K = 2P_0$ is an effective elastic stiffness parameter. Assuming the gas and the solid are always in thermal equilibrium, what is the molar heat capacity of the gas at the beginning of the heating process ( $T = T_0$ )?

Options

A.
$\frac{7}{2}R$
B.
$4R$
C.
$\frac{11}{6}R$
D.
$\frac{3}{2}R$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Expansion of Cavities in Solids|Polytropic Process|First Law of Thermodynamics

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple the thermodynamic state of a gas with the thermo-elastic boundary conditions of its container.

The 'Lethal Trap' lies in the energy balance: students often incorrectly assume that the work done by the gas is converted entirely into the solid's strain energy, leading them to subtract a strain energy term or assume $dW=0$ initially (Option D).

Others fail to couple the effects, either ignoring the elastic pressure response to get a purely thermal expansion (Option B) or ignoring thermal expansion to get a purely elastic response (Option C).

The correct approach requires simultaneously solving the differential forms of the ideal gas law and the thermo-elastic volume constraint to find the true boundary work $P dV$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Expansion of Cavities in Solids
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free