Question

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 .

• A. $\sqrt{\frac{42}{25}}$
• B. $\frac{\sqrt{21}}{5}$
• C. $\sqrt{\frac{7}{3}}$
• D. $\sqrt{\frac{3}{7}}$

Answer 1

Answer: (A)

From Newton-Laplace equation, we know that speed of sound in a gas is given as
$v=\sqrt{\frac{\gamma RT}{M_{\omega }}}$
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 ${\gamma }_{He}=\frac{5}{3}$ and molecular weights of both the gases we get:-
$=\sqrt{\frac{7}{5}\times \frac{3}{5}\times \frac{4}{2}}=\sqrt{\frac{42}{25}}$