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Question #42616

Difficulty: 8/10 | 🟢 Free

The Question

A classical non-ideal gas of molar mass $M$ is in thermal equilibrium at temperature $T$ inside a spherical container of radius $R_0$ . The gas is subjected to an external central potential $U(r) = \alpha r^2$ (where $\alpha > 0$ ). A small sensor on the inner wall of the container records the instantaneous velocity of every molecule that collides with it. What is the root-mean-square (RMS) speed of the molecules recorded by the sensor? (Assume classical statistical mechanics applies and $R$ is the universal gas constant).

Options

A.
$\sqrt{\frac{3RT}{M}}$
B.
$\sqrt{\frac{4RT}{M}}$
C.
$\sqrt{\frac{3RT}{M} + \frac{2\alpha R_0^2}{M}}$
D.
$\sqrt{\frac{8RT}{\pi M}}$

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

Mean vs. RMS Values|Temperature and Kinetic Energy|Ideal Gas Assumptions

Why This Question Is Tricky

Common Mistakes

This question tests two major conceptual traps in classical statistical mechanics.

First, students often incorrectly assume that an external potential or gas non-ideality alters the kinetic energy distribution; however, because the classical Hamiltonian separates, the local velocity distribution remains strictly Maxwellian.

Second, students routinely conflate the volume distribution of velocities with the flux-weighted distribution of molecules striking a surface.

The collision rate introduces a factor of $v_z$ , which skews the distribution of the recorded molecules toward higher speeds, shifting the RMS speed from $\sqrt{3RT/M}$ to $\sqrt{4RT/M}$ .

This forces a re-integration of the moments rather than blind formula application.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Mean vs. RMS Values
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free