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Question #42091

Difficulty: 8/10 | 🟢 Free

The Question

An ideal gas consisting of molecules of mass $m$ and number density $n$ is confined in a cylinder by a piston. The piston moves outward with a constant, small velocity $u$ ( $u \ll \bar{v}$ , where $\bar{v}$ is the mean speed of the gas molecules). Assuming the gas remains in thermal equilibrium at temperature $T$ and the collisions of the molecules with the piston are perfectly elastic, which of the following expressions gives the pressure $P$ exerted by the gas on the piston? (Let $P_0 = n k_B T$ be the static pressure of the gas).

Options

A.
$P_0 - m n u \bar{v}$
B.
$P_0 - \frac{1}{2} m n u \bar{v}$
C.
$P_0 - \frac{1}{3} m n u \bar{v}$
D.
$P_0 - 2 m n u \bar{v}$

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

Change in Velocity (Δv)|First Law Application|Molecular Degrees of Freedom|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the microscopic derivation of pressure on a moving boundary.

A common mistake is to either only adjust the momentum transfer $\Delta p = 2m(v_x - u)$ or only adjust the collision rate $dN = (v_x - u)n(v_x)dv_x$ , but not both.

Doing only one yields a correction term of $\frac{1}{2} m n u \bar{v}$ (Option B).

Another trap is using a simplified 1D model where all molecules move at $\bar{v}$ , which incorrectly gives a correction of $2 m n u \bar{v}$ (Option D).

This highlights the necessity of integrating over the Maxwell-Boltzmann distribution for both collision frequency and momentum transfer simultaneously.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Change in Velocity (Δv)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free