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Question #42567

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal monatomic gas is reversibly compressed from an initial temperature $T_0$ to a final temperature $2T_0$ . During this process, the total number of atomic collisions occurring per unit time within a spatial volume equal to $\lambda^3$ (where $\lambda$ is the mean free path of the gas) is maintained constant. Calculate the work done ON the gas.

Options

A.
$\frac{1}{2} R T_0$
B.
$\frac{1}{4} R T_0$
C.
$-\frac{1}{4} R T_0$
D.
R T_0

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

Work Done in Non-PV Diagrams|Mean Free Path|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the bridge between microscopic kinetic theory and macroscopic thermodynamic work.

A common trap is confusing the per-molecule collision frequency ( $Z \propto T^{1/2}/V$ ) with the total volumetric collision rate ( $Z_V \propto T^{1/2}/V^2$ ).

If a student incorrectly uses $Z \lambda^3 = \text{constant}$ , they derive $T^{1/2} V^2 = \text{constant} \implies n=5$ , leading to the trap answer of $R T_0 / 4$ .

Furthermore, students must correctly translate the microscopic constraint into a macroscopic polytropic index ( $n=3$ ) and apply the proper sign convention for work done ON the gas during compression.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Work Done in Non-PV Diagrams
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free