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Question #42501

Difficulty: 8/10 | 🟢 Free

The Question

Consider an ideal gas of non-interacting heteronuclear diatomic molecules. Each molecule consists of two atoms of masses $m$ and $2m$ , separated by a bond length $r_0$ and bound by an effective spring of force constant $K$ . The gas is maintained at a temperature $T = \frac{\hbar}{k_B \ln 2} \sqrt{\frac{3K}{2m}}$ , where $k_B$ is the Boltzmann constant and $\hbar$ is the reduced Planck constant. Assuming the rotational modes are fully excited ( $k_B T \gg \frac{\hbar^2}{2\mu r_0^2}$ ) but electronic transitions are negligible, what is the molar heat capacity at constant volume, $C_V$ , of this gas?

Options

A.
$R \left[ \frac{5}{2} + 2(\ln 2)^2 \right]$
B.
$\frac{7}{2} R$
C.
$R \left[ \frac{5}{2} + \frac{1}{2}(\ln 2)^2 \right]$
D.
$R \left[ \frac{3}{2} + 2(\ln 2)^2 \right]$

Check Your Answer

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

Equipartition Theorem|Reduced Mass Effect|Atoms - Energy Level Scaling

Why This Question Is Tricky

Common Mistakes

This question tests the precise application of statistical mechanics to the Equipartition Theorem at intermediate temperatures where vibrational modes are only partially excited ('thawing').

A common trap for students is to blindly apply the high-temperature limit ( $C_V = 7/2 R$ for diatomic gases) without checking the thermal energy scale against the vibrational energy quantum $\hbar \omega$ .

Furthermore, it requires the correct identification of the reduced mass $\mu = 2m/3$ for a heteronuclear diatomic molecule to determine the true vibrational frequency and validate the rotational energy eigenvalue scale.

Failing to use the Einstein model for vibrational heat capacity or miscalculating the reduced mass leads directly to the distractor options.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Equipartition Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free