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Question #42560

Difficulty: 8/10 | 🟢 Free

The Question

For a certain diatomic gas, the molar heat capacity at constant volume is given by $C_V = \frac{f(T)}{2} R$ , where $f(T)$ represents the effective number of degrees of freedom. Due to the gradual unfreezing of vibrational modes at high temperatures, $f(T)$ varies non-linearly with temperature $T$ as $f(T) = 5 + \frac{(T - T_0)^2}{2T_0^2}$ for $T \ge T_0$ . Calculate the change in internal energy of $n = 2 \text{ moles}$ of this gas when it is heated from $T = T_0$ to $T = 3T_0$ at constant volume.

Options

A.
$\frac{34}{3} R T_0$
B.
16 R T_0
C.
12 R T_0
D.
14 R T_0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Equipartition Theorem|Energy of Vibrational Modes|Calculus (Integration)

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental understanding of the Equipartition Theorem during the transition phase of vibrational mode activation.

A common trap is for students to blindly apply the state formula $U = \frac{f}{2}nRT$ to calculate $\Delta U = \frac{f_2}{2}nRT_2 - \frac{f_1}{2}nRT_1$ , which yields the incorrect trap answer of $16 R T_0$ .

This algebraic shortcut is physically invalid because $U = \frac{f}{2}nRT$ is derived under the assumption that $f$ is constant from absolute zero.

When degrees of freedom are temperature-dependent (unfreezing), the internal energy must be rigorously integrated using $dU = n C_V dT$ .

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