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Question #42605

Difficulty: 8/10 | 🟢 Free

The Question

One mole of a non-ideal diatomic gas has a temperature-dependent molar heat capacity at constant volume given by $C_v(T) = \frac{R}{2} \left[ 5 + 2 \left(\frac{T_0}{T}\right)^2 e^{-T_0/T} \right]$ , where $T_0$ is a characteristic vibrational activation temperature and $R$ is the universal gas constant. The gas is heated at constant volume from an initial temperature $T_1 = \frac{T_0}{\ln 4}$ to a final temperature $T_2 = \frac{T_0}{\ln 2}$ . What is the change in the internal energy of the gas?

Options

A.
$\frac{R T_0}{4} \left( \frac{5}{\ln 2} + 1 \right)$
B.
$\frac{5 R T_0}{4 \ln 2}$
C.
$\frac{13 R T_0}{8 \ln 2}$
D.
$\frac{R T_0}{4} \left( \frac{5}{\ln 2} + \ln 2 \right)$

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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

Energy of Vibrational Modes|Molar Heat Capacity (Cv vs Cp)|Energy Band Gap Temperature Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental thermodynamic definition of internal energy change $\Delta U = \int C_v(T) dT$ in regimes where the classical Equipartition Theorem breaks down.

Students commonly fall into the trap of assuming $U = \frac{f}{2}RT$ and either treating $f$ as a constant (ignoring vibrational modes entirely) or incorrectly substituting a temperature-dependent $f(T)$ directly into the algebraic formula without integrating.

By introducing a temperature-dependent vibrational activation term modeled after the Einstein solid, the problem forces students to perform a functional integration of $C_v(T)$ , penalizing those who rely on classical ideal gas shortcuts.

Quick Facts About This Question

Question Information

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