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Question #42584

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly absorbing wall of work function $\phi$ ( $\phi \gg kT$ ) separates a vacuum from an ideal monoatomic gas (mass $m$ , number density $n$ , temperature $T$ ). Gas molecules striking the wall with normal kinetic energy $E_x = \frac{1}{2}mv_x^2 \ge \phi$ are absorbed, transferring their momentum to the wall.

Simultaneously, the vacuum side of the wall is illuminated by a photon beam of intensity $I$ , which is completely absorbed. The radiation pressure from the photons exactly balances the partial pressure exerted only by the absorbed gas molecules.

Assuming the gas follows the Maxwell-Boltzmann velocity distribution $f(v_x) = n \sqrt{\frac{m}{2\pi kT}} e^{-mv_x^2/2kT}$ , which of the following represents the intensity $I$ ?

(Hint: For large $v_0$ , use the approximation $\int_{v_0}^{\infty} v^2 e^{-\alpha v^2} dv \approx \frac{v_0}{2\alpha} e^{-\alpha v_0^2}$ )

Options

A.
$c n k T \sqrt{\frac{\phi}{\pi k T}} e^{-\phi / k T}$
B.
$c n k T \sqrt{\frac{k T}{4 \pi \phi}} e^{-\phi / k T}$
C.
$2 c n k T \sqrt{\frac{\phi}{\pi k T}} e^{-\phi / k T}$
D.
c n \phi e^{-\phi / k T}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Pressure-Kinetic Energy Relation|Einstein's Photoelectric Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of kinetic pressure as a momentum flux integral over a velocity distribution, rather than a simple macroscopic average.

The primary trap (Option B) is the assumption that the partial pressure of high-energy molecules can be found by multiplying the average gas pressure ( $P = nkT$ ) by the fraction of molecules exceeding the threshold energy $\phi$ .

This incorrect approach ignores that higher-velocity molecules transfer proportionally more momentum per collision and collide more frequently, leading to a missing factor of $2\phi/kT$ .

Additionally, it couples the stochastic nature of gas collisions with macroscopic radiation pressure, forcing students to rigorously apply the Maxwell-Boltzmann distribution to a boundary condition.

Quick Facts About This Question

Question Information

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