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Question: Answered & Verified by Expert
The speed of the sound in oxygen $\mathrm{O}_2$ at a certain temperature is $460 \mathrm{~ms}^{-1}$. The speed of the sound in helium (He) at the same temperature will be (assume both the gases to be ideal)
PhysicsWaves and SoundAP EAMCETAP EAMCET 2021 (24 Aug Shift 1)
Options:
  • A $330 \mathrm{~ms}^{-1}$
  • B $1420 \mathrm{~ms}^{-1}$
  • C $500 \mathrm{~ms}^{-1}$
  • D $650 \mathrm{~ms}^{-1}$
Solution:
2265 Upvotes Verified Answer
The correct answer is: $1420 \mathrm{~ms}^{-1}$
For Oxygen,
Molar mass, $M_1=32$
Heat capacity ratio at constant pressure to that at constant volume,
$$
\gamma_1=C_p / C_V=7 / 5
$$
[for diatomic gas]
Speed of sound, $v_1=460 \mathrm{~m} / \mathrm{s}$
For Helium,
Molar mass, $M_2=4$
Heat capacity ratio $\left(C_p / C_V\right)$,
$$
\gamma_2=\frac{5}{3} \quad \text { [for monoatomic gas] }
$$
Let speed of sound $=v_2$
Using the expression of speed of sound in an ideal gas at certain temperature $(T)$ is
$$
\begin{array}{rlrl}
& v=\sqrt{\frac{\gamma R T}{M}} \\
\therefore & v_1 & =\sqrt{\frac{\gamma_1 R T}{M_1}}...(i) \\
\text { and } & v_2 & =\sqrt{\frac{\gamma_2 R T}{M_2}}...(ii)
\end{array}
$$
Dividing Eq. (i) by Eq. (ii), we get
$$
\frac{v_1}{v_2}=\sqrt{\frac{\gamma_1 R T}{M_1}} \times \sqrt{\frac{M_2}{\gamma_2 R T}}=\sqrt{\frac{\gamma_1}{\gamma_2} \times \frac{M_2}{M_1}}
$$
Substituting the values, we get
$$
\frac{460}{v_2}=\sqrt{\frac{7}{5} \times \frac{4}{32} \times \frac{3}{5}} \Rightarrow v_2=1420 \mathrm{~m} / \mathrm{s}
$$
Hence, speed of sound in helium gas at the same temperature is $1420 \mathrm{~m} / \mathrm{s}$.

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