The ratio of the speed of sound in helium gas to that in nitrogen gas at $295$ K is ( Ratio of specific heats $γ$ for helium is $\frac{5}{3}$ and that for nitrogen is $\frac{7}{5}$. Molecular weight for helium and nitrogen are $4$ and $28$ respectively.)

Question

A pipe open at both ends has a fundamental frequency $f$ in air. The pipe is now dipped vertically in water drum to half of its length. The fundamental frequency of the air column is now equal to:

Answer 1

Step 1: Numeric values:
$\frac{\gamma}{M}$ for helium is $\frac{5/3}{4} = 0.4167$. For nitrogen it is $\frac{7/5}{28} = 0.05$.

Step 2: Ratio:
$\frac{0.4167}{0.05} = 8.33 = \frac{25}{3}$.

Step 3: Root:
Speed ratio $= \sqrt{8.33} = 2.89 = \frac{5}{\sqrt3}$. The temperature cancels, so the value 295 K is not needed.

Final Answer:
The ratio is 5 over root 3, option (D). \[ \boxed{\frac{5}{\sqrt{3}}} \]

The ratio of the speed of sound in helium gas to that in nitrogen gas at \(295\) K is ( Ratio of specific heats \(γ\) for helium is \(\frac{5}{3}\) and that for nitrogen is \(\frac{7}{5}\). Molecular weight for helium and nitrogen are \(4\) and \(28\) respectively.)

Show Hint

The Correct Option is D

Solution and Explanation

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