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Question #42582

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated container holds a mixture of two ideal gases, Gas $A$ and Gas $B$ , in thermal equilibrium at temperature $T$ . The molecules of Gas $A$ have mass $m$ and diameter $d$ , while the molecules of Gas $B$ have mass $4m$ and diameter $2d$ . The number density of Gas $A$ is $n$ and that of Gas $B$ is $2n$ . Let $Z_{AA}$ be the average collision frequency of a single Gas $A$ molecule with other Gas $A$ molecules. Let $Z_{AB}$ be the average collision frequency of a single Gas $A$ molecule with Gas $B$ molecules. Assuming the molecules behave as rigid spheres, what is the ratio $\frac{Z_{AB}}{Z_{AA}}$ ?

Options

A.
$\frac{9\sqrt{10}}{8}$
B.
$\frac{9\sqrt{5}}{4}$
C.
$\frac{27\sqrt{2}}{8}$
D.
$\frac{9}{2}$

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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|Frame of Reference|Vector Vector Direction

Why This Question Is Tricky

Common Mistakes

This question targets a quintessential trap in the kinetic theory of gases: the origin of the $\sqrt{2}$ factor in the collision frequency formula.

Students often memorize $Z = \sqrt{2} n \sigma v_{avg}$ without realizing that the $\sqrt{2}$ emerges specifically because the reduced mass of identical particles is $\mu = m/2$ , which modifies the average relative velocity $\langle v_{rel} \rangle = \sqrt{8kT/\pi\mu}$ .

When asked to find the collision frequency between different gases, students commonly fall into one of two traps: they either blindly apply the $\sqrt{2}$ factor to the mixture (leading to Option D), or they incorrectly use the scalar sum of the individual average speeds instead of the proper vector-derived relative velocity distribution (leading to Option C).

The correct approach requires returning to the fundamental Maxwell-Boltzmann relative velocity integral via reduced mass.

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