Before You Begin

Question #42581

Difficulty: 8/10 | 🟢 Free

The Question

A dilute classical gas consists of particles of mass $m$ at temperature $T$ and pressure $P$ . Instead of being hard spheres, the particles interact exclusively via a repulsive central potential $V(r) = \frac{K}{r^2}$ ( $K > 0$ ). A collision is registered if the scattering angle in the center-of-mass frame exceeds a fixed threshold $\theta_0$ . If the gas is heated to a temperature $2T$ while maintaining a constant pressure $P$ , the mean free path of the gas particles changes from $\lambda_0$ to $\lambda'$ . What is the ratio $\frac{\lambda'}{\lambda_0}$ ?

Options

A.
2
B.
4
C.
8
D.
$2\sqrt{2}$

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

Mean Free Path|Impact Parameter Geometry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the transition from the standard hard-sphere kinetic theory to a dynamic cross-section model.

The primary trap is assuming the collision cross-section $\sigma$ is constant (which yields $\lambda \propto T$ at constant pressure, leading to an incorrect ratio of 2).

Students must integrate the concept of impact parameter scaling with relative velocity for an inverse-square potential ( $\sigma \propto v^{-2} \propto T^{-1}$ ) and correctly combine it with the ideal gas law ( $n_0 \propto 1/T$ at constant $P$ ) to find the true scaling $\lambda \propto T^2$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Mean Free Path
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free