Before You Begin

Question #43480

Difficulty: 8/10 | 🟢 Free

The Question

Consider an ideal gas in thermal equilibrium at absolute temperature $T$ . The gas consists of molecules, each of mass $m$ . Let $k$ be the Boltzmann constant. Assuming the speeds $v$ of the molecules follow the standard Maxwell-Boltzmann speed distribution, which of the following expressions correctly represents the statistically averaged value of the inverse speed, $\langle v^{-1} \rangle$ , of the gas molecules?

Options

A.
$\sqrt{\frac{2m}{\pi kT}}$
B.
$\sqrt{\frac{\pi m}{8 kT}}$
C.
$\sqrt{\frac{m}{3 kT}}$
D.
$\sqrt{\frac{m}{2 kT}}$

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

Average vs. Instantaneous Calculus|Maxwell-Boltzmann Distribution|Statistical Integration

Why This Question Is Tricky

Common Mistakes

This question targets a profound misconception in statistical mechanics and calculus: the false assumption that the average of an inverse quantity equals the inverse of its average (i.e., assuming $\langle 1/v \rangle = 1/\langle v \rangle$ ).

Students falling for this trap will incorrectly select the inverse of the mean speed.

Additionally, it tests the student's rigorous understanding of the Maxwell-Boltzmann speed distribution, specifically the $v^2$ phase-space weighting factor.

Without this $v^2$ term (which arises from the volume of a spherical shell in 3D velocity space), the integral for $\langle 1/v \rangle$ would incorrectly diverge at $v=0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Average vs. Instantaneous Calculus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free