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Question #42166

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The Question

One mole of a monatomic ideal gas of molar mass $M$ (in $\text{kg mol}^{-1}$ ) undergoes a cyclic process consisting of four steps: (I) Reversible isothermal expansion at a state where the variance of its 1D Maxwell-Boltzmann velocity distribution is $\sigma_1^2$ (in $\text{m}^2\text{s}^{-2}$ ). (II) Irreversible isochoric cooling by placing it in direct thermal contact with a heat reservoir until the velocity variance becomes $\sigma_2^2$ ( $\sigma_2 < \sigma_1$ ). (III) Reversible isothermal compression at the new state. (IV) Irreversible isochoric heating by placing it in direct thermal contact with a heat reservoir until the velocity variance returns to $\sigma_1^2$ . What is the total entropy change of the universe ( $\Delta S_{\text{univ}}$ ) per mole of the gas for one complete cycle? ( $R$ is the universal gas constant in $\text{J K}^{-1}\text{mol}^{-1}$ )

Options

A.
$\frac{3}{2}R \frac{(\sigma_1^2 - \sigma_2^2)^2}{\sigma_1^2 \sigma_2^2}$
B.
0
C.
$\frac{3}{2}R \ln\left(\frac{\sigma_1^2}{\sigma_2^2}\right)$
D.
$\frac{1}{2}R \frac{(\sigma_1^2 - \sigma_2^2)^2}{\sigma_1^2 \sigma_2^2}$

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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

State Function vs Path Function|Entropy and Reversibility|Maxwell-Boltzmann Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental distinction between the entropy change of a system and the universe in cyclic processes, specifically targeting the misconception that $\Delta S_{\text{universe}} = 0$ for any cycle.

Students often memorize that entropy is a state function and incorrectly apply $\oint dS = 0$ to the universe rather than just the system.

By embedding the temperature states within the variance of the Maxwell-Boltzmann velocity distribution ( $\sigma^2 = RT/M$ ), the problem forces students to translate microscopic statistical parameters into macroscopic thermodynamic variables.

Furthermore, it traps students who fail to recognize that irreversible heat transfer across a finite temperature difference generates entropy, requiring them to calculate the exact entropy production ( $\Delta S_{\text{surr}}$ ) at the thermal reservoirs.

Option D serves as an additional trap for students who confuse the 1D velocity distribution variance with the degrees of freedom, incorrectly using $C_v = R/2$ instead of $3R/2$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	State Function vs Path Function
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free