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Question #42328

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal, perfectly insulated cylinder is placed in a vacuum and contains $1 \text{ mol}$ of a diatomic gas at an initial temperature $T_0$ . A frictionless, adiabatic piston of mass $M$ is given an initial inward velocity $v_0$ such that $M v_0^2 = 18 R T_0$ , compressing the gas. Assuming the gas maintains internal thermal equilibrium, and due to the excitation of vibrational degrees of freedom at higher energies, its molar heat capacity at constant volume varies as $C_V(T) = \frac{5}{2}R + 2R\left(\frac{T - T_0}{T_0}\right)$ for $T \ge T_0$ . What is the maximum temperature $T_{max}$ reached by the gas?

Options

A.
$\frac{23}{5} T_0$
B.
3 T_0
C.
$\frac{25}{7} T_0$
D.
2 T_0

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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

Mechanical Energy Conservation|Molecular Degrees of Freedom|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the First Law of Thermodynamics and mechanical energy conservation in a coupled macroscopic-microscopic system.

The lethal trap lies in the assumption of a constant specific heat capacity ( $C_V = \frac{5}{2}R$ or $\frac{7}{2}R$ ).

Students often blindly use $\Delta U = n C_V \Delta T$ , failing to recognize that at high energies, the activation of vibrational degrees of freedom makes the internal energy a non-linear function of temperature.

By forcing the integration of $dU = C_V(T) dT$ , the problem evaluates the student's fundamental understanding of internal energy beyond standard rigid diatomic models.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Mechanical Energy Conservation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free