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Question #42011

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The Question

A velocity spectrometer with a resolution (least count) of $\Delta v = 65 \text{ m/s}$ is used to measure the speed distribution of an ideal gas of molar mass $M = 16 \text{ g/mol}$ . The instrument cannot resolve the root-mean-square speed $v_{rms}$ and the average speed $v_{avg}$ as distinct peaks if their difference is less than or equal to $\Delta v$ . Taking $\sqrt{3} \approx 1.73$ , $\sqrt{\frac{8}{\pi}} \approx 1.60$ , and the universal gas constant $R = \frac{25}{3} \text{ J mol}^{-1}\text{K}^{-1}$ , what is the maximum temperature $T$ of the gas for which these two speeds remain indistinguishable?

Options

A.
$480 \text{ K}$
B.
$240 \text{ K}$
C.
$960 \text{ K}$
D.
$720 \text{ K}$

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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

Least Count and Resolution Limits|RMS vs Average vs Most Probable Speed

Why This Question Is Tricky

Common Mistakes

This question tests the physical reality of measurement uncertainty applied to statistical mechanics.

Students often treat $v_{rms}$ , $v_{avg}$ , and $v_{mp}$ as purely theoretical, infinitely precise points on the Maxwell-Boltzmann curve.

By introducing a finite resolution (least count $\Delta v$ ), the problem forces the realization that these statistical aggregates can merge into a single indistinguishable peak if the thermal broadening (which scales with $\sqrt{T}$ ) is insufficient to separate them beyond the instrument's limits.

It traps students who fail to link abstract kinetic theory formulas with the practical constraints of experimental physics, specifically the requirement to convert molar mass to standard SI units (kg/mol) before evaluating the limit.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Least Count and Resolution Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free