TS EAMCET · Physics · Kinetic Theory of Gases

Consider a gas with molar mass $M .$ If the sound at frequency $f$ is introduced to a tube of this gas at temperature $T$ then an internal acoustic standing wave is set up with nodes separated by $L .$ The adiabatic constant $(γ = \frac{C_{p}}{C_{v}})$ is

A $\frac{M f^{2} L^{2}}{R T}$
B $\frac{M f^{2} L^{2}}{4 R T}$
C $\frac{4 M f^{2} L^{2}}{R T}$
D $\frac{3 M f^{2} L^{2}}{2 R T}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{4 M f^{2} L^{2}}{R T}$

Step-by-step Solution

Detailed explanation

Recall the formula of the fundamental frequency of the standing wave, f=v2L, and the formula of sound waves in terms of absolute temperature and molecular mass, v=3RTM, ⇒f=γRTM2L⇒f2×4L2=γRTM⇒γ=f2×4L2MRT

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Consider a gas with molar mass M. If the sound at frequency f is introduced to a tube of this gas at temperature T then an internal acoustic standing wave is set up with nodes separated by L. The adiabatic constant γ=CpCv is

Step-by-step Solution

Consider a gas with molar mass $M .$ If the sound at frequency $f$ is introduced to a tube of this gas at temperature $T$ then an internal acoustic standing wave is set up with nodes separated by $L .$ The adiabatic constant $(γ = \frac{C_{p}}{C_{v}})$ is