MHT CET · Physics · Waves and Sound
What is the ratio of the velocity of sound in hydrogen \(\left(\gamma=\frac{7}{5}\right)\) to that in helium \(\left(\gamma=\frac{5}{3}\right)\) at the same temperature? (Molecular weight of hydrogen and helium is 2 and 4 respectively)
- A \(\frac{\sqrt{42}}{5}\)
- B \(\frac{5}{\sqrt{42}}\)
- C \(\frac{\sqrt{21}}{5}\)
- D \(\frac{5}{\sqrt{21}}\)
Answer & Solution
Correct Answer
(A) \(\frac{\sqrt{42}}{5}\)
Step-by-step Solution
Detailed explanation
Velocity of sound is given by
\(\mathrm{V}=\sqrt{\frac{\gamma \mathrm{RT}}{\mathrm{M}}} \therefore \frac{\mathrm{V}_{\mathrm{H}}}{\mathrm{V}_{\mathrm{He}}}=\sqrt{\frac{\gamma_{\mathrm{H}}}{\gamma_{\mathrm{He}}} \cdot \frac{\mathrm{M}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{H}}}}(\mathrm{T}\) is the same for both)
\(
=\sqrt{\frac{7}{5} \times \frac{3}{5} \times \frac{4}{2}}=\frac{\sqrt{42}}{5}
\)
\(\mathrm{V}=\sqrt{\frac{\gamma \mathrm{RT}}{\mathrm{M}}} \therefore \frac{\mathrm{V}_{\mathrm{H}}}{\mathrm{V}_{\mathrm{He}}}=\sqrt{\frac{\gamma_{\mathrm{H}}}{\gamma_{\mathrm{He}}} \cdot \frac{\mathrm{M}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{H}}}}(\mathrm{T}\) is the same for both)
\(
=\sqrt{\frac{7}{5} \times \frac{3}{5} \times \frac{4}{2}}=\frac{\sqrt{42}}{5}
\)
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