The speed of the sound in oxygen $\mathrm{O}_2$ at a certain temperature is $460 \mathrm{~ms}^{-1}$. The…
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)
$330 \mathrm{~ms}^{-1}$
$1420 \mathrm{~ms}^{-1}$
$500 \mathrm{~ms}^{-1}$
$650 \mathrm{~ms}^{-1}$
Solution
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 $1420 \mathrm{~m} / \mathrm{s}$