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Question #43487

Difficulty: 8/10 | 🟢 Free

The Question

An ideal monoatomic gas of molar mass $M$ (in $\text{kg mol}^{-1}$ ) is initially in a thermodynamic state with absolute temperature $T_0$ (in $\text{K}$ ) and volume $V_0$ (in $\text{m}^3$ ). The Maxwell-Boltzmann speed distribution function $f(v)$ for this gas exhibits two inflection points. Let $v_1$ be the speed corresponding to the higher inflection point. The gas is then compressed reversibly and adiabatically to a volume $\frac{V_0}{8}$ . Let $v_2$ be the speed corresponding to the higher inflection point of the new distribution. Which of the following expressions correctly represents $v_2$ ? (Here $R$ is the universal gas constant in $\text{J K}^{-1} \text{mol}^{-1}$ )

Options

A.
$\sqrt{\frac{RT_0}{M} (10 + 2\sqrt{17})}$
B.
$\sqrt{\frac{RT_0}{M} \left(\frac{5 + \sqrt{17}}{2}\right)}$
C.
$\sqrt{\frac{RT_0}{M} (10 + 2\sqrt{33})}$
D.
$\sqrt{\frac{RT_0}{M} (5 + \sqrt{17})}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Graphical Concavity Significance|Maxwell-Boltzmann Distribution|Adiabatic/Isothermal Processes via KTG

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the concavity of the Maxwell-Boltzmann distribution and couple it with thermodynamic processes.

A common trap is assuming the inflection points (which represent the transition in the population density of velocity groups) occur at standard characteristic speeds like the most probable or RMS speed, bypassing the necessary calculus.

Furthermore, students often fail to recognize that the distribution's shape—and thus its inflection points—is dynamically coupled to the macroscopic state variables.

During an adiabatic compression, the temperature increases, shifting the entire distribution and its inflection points.

Forgetting to apply the adiabatic relation $TV^{\gamma-1} = \text{constant}$ or miscalculating the second derivative leads directly to the distractor options.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Graphical Concavity Significance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free