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Question #80259

Difficulty: 5/10 | 🟢 Free

The Question

Consider one mole of a gas of non-interacting diatomic molecules in thermal equilibrium at a high temperature $T$ . The molecules have standard 3D translational degrees of freedom. The rotational energy levels are given by $E_J = B J(J+1)$ with degeneracy $g_J = 2J+1$ . Due to a nuclear spin selection rule, only states with EVEN quantum numbers ( $J = 0, 2, 4, \dots$ ) are accessible. The vibrational motion is 1D, with kinetic energy $K = \frac{p^2}{2\mu}$ and a highly anharmonic potential energy $V(x) = \lambda x^4$ , where $x$ is the bond displacement and $\lambda > 0$ . Assuming $k_B T \gg B$ and $k_B T \gg$ vibrational energy spacings, what is the molar heat capacity at constant volume, $C_V$ , of this gas? ( $R$ is the universal gas constant)

Options

A.
$\frac{13}{4} R$
B.
$\frac{11}{4} R$
C.
$\frac{7}{2} R$
D.
3 R

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

Molar Heat Capacity (Cv vs Cp)|Energy Level Scaling|Quantization of Angular Momentum

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's fundamental understanding of statistical mechanics versus rote memorization of the equipartition theorem.

The primary trap is the nuclear spin selection rule restricting rotational states to even $J$ .

Students intuitively assume that halving the number of accessible microstates will halve the rotational energy and heat capacity (leading to $C_{v,rot} = R/2$ ).

However, a constant scaling factor in the partition function ( $Z \to Z/2$ ) results in an additive constant in $\ln Z$ , which vanishes upon differentiation with respect to temperature, leaving the heat capacity unchanged.

Additionally, the anharmonic $x^4$ potential forces students to derive the generalized equipartition contribution ( $C_{v,vib} = 3/4 R$ ) rather than blindly applying the harmonic oscillator result ( $R$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Molar Heat Capacity (Cv vs Cp)
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free