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Question #84411

Difficulty: 5/10 | 🟢 Free

The Question

A rigid, thermally insulated container holds a total mass $m \text{ (in kg)}$ of an ideal gas having an adiabatic exponent $\gamma$ (dimensionless). A paddle wheel inside the container is driven by an external agent, doing non-PV work on the gas. The initial average speed of the gas molecules is $v_0 \text{ (in m/s)}$ . The stirring continues until the final root-mean-square (RMS) speed of the gas molecules becomes $v_f \text{ (in m/s)}$ . Assuming no heat exchange with the surroundings, which of the following expressions represents the total work $W \text{ (in J)}$ done by the paddle wheel?

Options

A.
$\frac{m}{\gamma - 1} \left( \frac{v_f^2}{3} - \frac{\pi v_0^2}{8} \right)$
B.
$\frac{m}{\gamma - 1} \left( \frac{\pi v_f^2}{8} - \frac{v_0^2}{3} \right)$
C.
$\frac{3m}{2} \left( \frac{v_f^2}{3} - \frac{\pi v_0^2}{8} \right)$
D.
$\frac{m}{\gamma - 1} \left( \frac{v_f^2}{3} - \frac{v_0^2}{3} \right)$

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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

RMS vs Average vs Most Probable Speed|First Law Application|Energy Partitioning|Work Done in Non-PV Diagrams

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to bridge microscopic kinetic theory with macroscopic thermodynamics via the First Law.

The primary trap lies in energy partitioning: students often incorrectly assume that non-PV work (like stirring) only increases the translational kinetic energy of the gas molecules (leading to Option C), forgetting that energy is distributed across all active degrees of freedom dictated by the adiabatic exponent $\gamma$ .

Additionally, it tests precision in distinguishing between statistical speed distributions ( $v_{rms}$ vs $v_{avg}$ ), where swapping their dependencies (Option B) or treating them as identical (Option D) leads to incorrect temperature mappings.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	RMS vs Average vs Most Probable Speed
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free