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Question #42595

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The Question

A solid block with a coefficient of linear thermal expansion $\alpha$ contains a spherical cavity of volume $V_0$ at an initial temperature $T_0$ . The cavity is filled with a mixture of $2$ moles of a monatomic ideal gas and $1$ mole of a diatomic ideal gas. At $T_0$ , the diatomic gas has one fully active vibrational mode. The total internal energy of the gas mixture at $T_0$ is $U_0$ . Assuming the solid expands isotropically and the gases obey Dalton's law of partial pressures, what is the rate of change of the total gas pressure with respect to temperature, $\left. \frac{dP}{dT} \right|_{T_0}$ , at $T = T_0$ ?

Options

A.
$\frac{6 U_0}{13 T_0 V_0} (1 - 3\alpha T_0)$
B.
$\frac{6 U_0}{11 T_0 V_0} (1 - 3\alpha T_0)$
C.
$\frac{6 U_0}{13 T_0 V_0} (1 - \alpha T_0)$
D.
$\frac{6 U_0}{13 T_0 V_0}$

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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

Dalton's Law of Partial Pressures|Energy of Vibrational Modes|Expansion of Cavities in Solids

Why This Question Is Tricky

Common Mistakes

This question tests the thermo-elastic coupling between a confined gas mixture and its expanding solid container.

Students often incorrectly assume the cavity volume remains constant (leading to Option D) or use the linear expansion coefficient instead of the volumetric one (leading to Option C).

Furthermore, it tests the equipartition theorem by requiring the inclusion of both kinetic and potential energy contributions of a fully active vibrational mode in a diatomic gas, which adds $R$ to the molar heat capacity.

Failing to account for the vibrational mode leads to an incorrect internal energy mapping (Option B).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Dalton's Law of Partial Pressures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free