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Question #42594

Difficulty: 8/10 | 🟢 Free

The Question

A closed organ pipe of length $L$ and radius $r$ , made of a material with a coefficient of linear expansion $\alpha$ , is placed in a gas chamber. Initially, the chamber contains pure Helium gas (molar mass $4 \text{ g/mol}$ ) at temperature $T_0$ , and the pipe resonates at its fundamental frequency $f_0$ . Neon gas (molar mass $20 \text{ g/mol}$ ) is then injected into the chamber until its partial pressure equals the partial pressure of Helium. Simultaneously, the chamber's temperature is raised to $3T_0$ . Assuming both gases are ideal and monoatomic, and $\alpha T_0 \ll 1$ , what is the new fundamental frequency of the pipe?

Options

A.
$f_0 (1 - 2\alpha T_0)$
B.
$f_0$
C.
$f_0 (1 - 3\alpha T_0)$
D.
$f_0 (1 + 2\alpha T_0)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Dalton's Law of Partial Pressures|Harmonic Content in Pipes|End Correction

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Dalton's Law of partial pressures, the thermodynamics of the speed of sound, and the geometric scaling of end corrections.

A common trap is assuming that the end correction $e = 0.6r$ is a static constant that does not thermally expand, or completely forgetting the thermal expansion of the pipe when the speed of sound remains unchanged.

By carefully balancing the increase in temperature with the increase in the effective molar mass of the gas mixture, the speed of sound is kept constant.

This isolates the thermal expansion of the pipe's effective length $L_{eff} = L + 0.6r$ , forcing students to recognize that both the physical length and the radius (and thus the end correction) scale by the same thermal expansion factor $(1 + \alpha \Delta T)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Dalton's Law of Partial Pressures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free