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Question #42586

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal diatomic gas is heated to a high temperature such that its vibrational modes become partially active. The internal energy of the gas is modeled empirically as $U(T) = \frac{5}{2}RT + \frac{RT^2}{2T_0}$ , where $R$ is the universal gas constant and $T_0$ is a characteristic constant temperature. If the gas undergoes a reversible adiabatic expansion, which of the following relationships between temperature $T$ and volume $V$ is valid throughout the process?

Options

A.
$T^{5/2} V e^{T/T_0} = \text{constant}$
B.
$T^{5/2} V e^{-T/T_0} = \text{constant}$
C.
$T^{7/2} V e^{T/T_0} = \text{constant}$
D.
$T^{5/2} V = \text{constant}$

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

Pressure-Kinetic Energy Relation|Energy of Vibrational Modes|Adiabatic Process Constraint|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

Standard intuition assumes the adiabatic index $\gamma$ is a constant, leading students to blindly apply the standard relation $TV^{\gamma-1} = \text{constant}$ .

This question introduces a temperature-dependent internal energy model, simulating the gradual activation of vibrational degrees of freedom at high temperatures.

To solve it, students must abandon the constant- $\gamma$ shortcut and derive the $T-V$ relationship from first principles using the First Law of Thermodynamics ( $dU = -P dV$ ).

The primary trap is Option D, which assumes vibrational modes are negligible (constant $f=5$ ), and Option B, which tests for sign convention errors during integration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Pressure-Kinetic Energy Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free