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Question #42882

Difficulty: 8/10 | 🟢 Free

The Question

A rigid cylindrical pipe of length $L$ , closed at both ends, contains a pure monoatomic gas A of molar mass $M_0$ at a pressure $P_0$ and temperature $T$ . The fundamental frequency of longitudinal acoustic resonances in the pipe is $f_0$ . A pure diatomic gas B of molar mass $4M_0$ is then slowly injected into the pipe, maintaining the temperature at $T$ , until the total pressure inside the pipe reaches $2P_0$ . Assuming ideal gas behavior, what is the frequency of the first overtone of the gas mixture in the pipe?

Options

A.
$\frac{6}{5} f_0$
B.
$\frac{9}{5} f_0$
C.
$2 \sqrt{\frac{46}{125}} f_0$
D.
$\sqrt{\frac{8}{5}} f_0$

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

Harmonic Content in Pipes|Dalton's Law of Partial Pressures|Kinetic Theory

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of acoustic resonance and the kinetic theory of gas mixtures.

A common trap is assuming the adiabatic index $\gamma$ remains constant or incorrectly averaging $\gamma$ directly instead of calculating the effective molar heat capacities ( $C_v$ and $C_p$ ).

Additionally, students often misapply Dalton's Law by failing to deduce that equal partial pressures imply equal molar quantities.

Finally, the problem tests harmonic series knowledge: students often confuse a pipe closed at both ends (where all harmonics are present, making the first overtone $2f$ ) with a standard closed-open organ pipe (where only odd harmonics are present, making the first overtone $3f$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Harmonic Content in Pipes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free