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Question #42599

Difficulty: 8/10 | 🟢 Free

The Question

A vertical, perfectly insulated cylinder of cross-sectional area $A$ contains a gas mixture consisting of $n$ moles of a monoatomic gas and $n$ moles of a rigid diatomic gas at an initial pressure $P_0$ and volume $V_0$ . A frictionless, insulated piston of mass $M$ rests in equilibrium on the gas. A sudden downward impulse $J$ is imparted to the piston. Assuming the initial velocity of the piston is much less than the speed of sound in the gas mixture, what is the maximum additional compression $\Delta x$ of the gas?

Options

A.
$\frac{J}{A} \sqrt{\frac{2 V_0}{3 M P_0}}$
B.
$\frac{J}{A} \sqrt{\frac{V_0}{M P_0}}$
C.
$\frac{J}{A} \sqrt{\frac{5 V_0}{7 M P_0}}$
D.
$\frac{J}{A} \sqrt{\frac{3 V_0}{5 M P_0}}$

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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

Equation of State for Mixtures|Work Done in Non-PV Diagrams|Collision Impulses

Why This Question Is Tricky

Common Mistakes

This question tests the ability to model a rapid, impulsive thermodynamic process using mechanical equivalents (the acoustic approximation of a gas as a spring).

The primary trap lies in the calculation of the restoring force.

Students often incorrectly use the isothermal bulk modulus ( $B = P_0$ ) instead of the adiabatic one, leading to Option B.

Additionally, students may incorrectly calculate the mixture's adiabatic index $\gamma$ by simply averaging the individual $\gamma$ values, or they might mistakenly assume the gas is purely monoatomic or diatomic, leading to Options C or D.

The problem also elegantly requires the realization that the linear work terms (gravity, atmospheric pressure) perfectly cancel the linear term of the gas pressure expansion, leaving only the quadratic 'spring' energy term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Equation of State for Mixtures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free