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Q.
The ratio of the velocity of sound in hydrogen gas to helium gas at room temperature if their $\gamma $ are $\frac{7}{5}$ and $\frac{5}{3}$ respectively .
NTA AbhyasNTA Abhyas 2020
Solution:
From Newton-Laplace equation, we know that speed of sound in a gas is given as
$v=\sqrt{\frac{\gamma R T}{M_{\omega }}}$
Here we observe that $R$ and $T$ are same and equal for both the gases. As a result they will cancel out when we take the ratio of speeds of sound in these two gases as follows:-
$\frac{v_{H_{2}}}{v_{H_{e}}}=\sqrt{\frac{\left(\gamma \right)_{H_{2}} \left(M_{\omega }\right)_{H_{e}}}{\left(\gamma \right)_{H_{e}} \left(M_{\omega }\right)_{H_{2}}}}$
Putting $\gamma _{H_{2}}=\frac{7}{5}$ and $\gamma _{H e}=\frac{5}{3}$ and molecular weights of both the gases we get:-
$=\sqrt{\frac{7}{5} \times \frac{3}{5} \times \frac{4}{2}}=\sqrt{\frac{42}{25}}$