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Question #81035

Difficulty: 5/10 | 🟢 Free

The Question

A large vessel contains an ideal gas of molar mass $M$ at pressure $P$ and temperature $T$ . The vessel is connected to a vacuum chamber via a perfectly absorbing conical capillary. The entrance to the capillary is a small flat aperture of area $A$ located on the vessel's wall. The capillary expands outward in the shape of a cone of half-angle $\theta_0$ , with its apex centered at the entrance aperture and its axis perpendicular to the aperture. Any gas molecule that strikes the inner wall of the capillary condenses and is permanently absorbed. What is the net mass of gas that successfully exits the capillary into the vacuum per unit time? ( $R$ is the universal gas constant).

Options

A.
$P A \sin^2\theta_0 \sqrt{\frac{M}{2\pi R T}}$
B.
$P A (1 - \cos\theta_0) \sqrt{\frac{M}{2\pi R T}}$
C.
$P A \sin\theta_0 \sqrt{\frac{M}{2\pi R T}}$
D.
$P A \cos^2\theta_0 \sqrt{\frac{M}{2\pi R T}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Temperature and Kinetic Energy|Effusion and Diffusion|Vector Decomposition Ambiguity

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 understanding of the angular distribution of effusing molecules (Lambert's Cosine Law for gases).

A common misconception is that molecules effuse isotropically in all directions, leading students to incorrectly multiply the total scalar effusion rate by the fractional solid angle of the cone, $\frac{\Omega}{2\pi} = (1 - \cos\theta_0)$ .

However, the probability of a molecule finding the aperture depends on its velocity component perpendicular to the aperture plane, introducing an additional $\cos\theta$ factor.

Integrating the true flux vector field $\cos\theta \sin\theta d\theta$ yields the correct $\sin^2\theta_0$ dependence.

This prevents the blind application of the scalar effusion formula and demands rigorous solid angle differential geometry.

Quick Facts About This Question

Question Information

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