Before You Begin

Question #42562

Difficulty: 8/10 | 🟢 Free

The Question

Consider a classical system of non-interacting particles of mass $m$ (in $\text{kg}$ ) constrained to move in a one-dimensional conservative force field. The potential energy of a particle is given by $U(x) = \alpha x^4$ , where $\alpha$ is a positive constant with units of $\text{J} \cdot \text{m}^{-4}$ and $x$ is the position in meters. The system is in thermal equilibrium at an absolute temperature $T$ (in $\text{K}$ ). If $k_B$ is the Boltzmann constant, what is the average total mechanical energy per particle?

Options

A.
k_B T
B.
$\frac{3}{4} k_B T$
C.
$\frac{1}{2} k_B T$
D.
$\frac{5}{4} k_B T$

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|Conservative Force Field|Calculus (Integration)

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental limits of the Equipartition Theorem.

Students often blindly memorize that each 'degree of freedom' contributes $\frac{1}{2}k_B T$ to the total energy, incorrectly assuming this applies to any potential well.

This trap exposes the misconception by using an anharmonic potential $U(x) = \alpha x^4$ .

The equipartition result of $\frac{1}{2}k_B T$ strictly requires the energy term to be quadratic in the coordinate or momentum.

By forcing the student to evaluate the partition function integral for a quartic potential, they must realize that $\langle U \rangle = \frac{1}{4}k_B T$ , yielding a total average energy of $\frac{3}{4}k_B T$ instead of the harmonic oscillator's $k_B T$ .

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