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Question #80463

Difficulty: 5/10 | 🟢 Free

The Question

A thermally insulated cylinder equipped with a movable piston contains $N_0$ molecules of an ideal monatomic gas at an initial temperature $T_0$ and volume $V_0$ . A small hole is punctured in the cylinder, allowing the gas to slowly effuse into a surrounding vacuum. Simultaneously, the piston is moved outward quasi-statically such that the volume of the gas increases. When the number of gas molecules remaining in the cylinder drops to $\frac{N_0}{8}$ and the volume becomes $8V_0$ , what is the final temperature $T$ of the gas remaining in the cylinder? (Assume the average kinetic energy carried away by each effusing monatomic molecule is $2kT$ , where $k$ is the Boltzmann constant).

Options

A.
$\frac{T_0}{8}$
B.
$\frac{T_0}{4}$
C.
$\frac{T_0}{16}$
D.
$\frac{T_0}{2}$

Check Your Answer

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

Effusion and Diffusion|First Law Application|State Function vs Path Function|Adiabatic Process Constraint

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 application of the First Law of Thermodynamics to an open system with simultaneous boundary work and mass loss.

A common student trap is to treat the process as a standard closed-system adiabatic expansion ( $TV^{\gamma-1} = \text{constant}$ ), ignoring the energy carried away by the effusing particles, which leads to $T_0/4$ .

Another trap is assuming the effusing gas only carries away the average internal energy ( $\frac{3}{2}kT$ ), which mathematically results in the same incorrect $T_0/4$ .

Students must recognize that effusion selectively removes faster molecules (carrying $2kT$ per molecule), acting as an additional cooling mechanism.

This requires setting up the differential First Law $dU = -PdV + e_{eff}dN$ and integrating the coupled variables.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Effusion and Diffusion
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free