KCET · Physics · Waves and Sound
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 is
- A \(\sqrt{\frac{5}{42}}\)
- B \(\sqrt{\frac{5}{21}}\)
- C \(\frac{\sqrt{42}}{5}\)
- D \(\frac{\sqrt{21}}{5}\)
Answer & Solution
Correct Answer
(C) \(\frac{\sqrt{42}}{5}\)
Step-by-step Solution
Detailed explanation
Velocity of sound in a gas
\(v=\sqrt{\frac{\gamma p}{d}}\)
\(\frac{v_{\mathrm{H}_{2}}}{V_{\mathrm{He}}}=\sqrt{\frac{\gamma_{\mathrm{H}_{2}} \times d_{\mathrm{He}}}{d_{\mathrm{H}_{2}} \times \gamma_{\mathrm{He}}}}\)
\(\frac{\mathrm{H}_{2}}{\mathrm{He}}=\sqrt{\frac{7 \times 3 \times 2}{5 \times 5}}\)
\(\text {As } \frac{d_{\mathrm{He}}}{d_{\mathrm{H}_{2}}}=2 \)
\( \therefore \frac{v_{\mathrm{H}_{2}}}{v_{\mathrm{He}}}=\frac{\sqrt{42}}{5}\)
\(v=\sqrt{\frac{\gamma p}{d}}\)
\(\frac{v_{\mathrm{H}_{2}}}{V_{\mathrm{He}}}=\sqrt{\frac{\gamma_{\mathrm{H}_{2}} \times d_{\mathrm{He}}}{d_{\mathrm{H}_{2}} \times \gamma_{\mathrm{He}}}}\)
\(\frac{\mathrm{H}_{2}}{\mathrm{He}}=\sqrt{\frac{7 \times 3 \times 2}{5 \times 5}}\)
\(\text {As } \frac{d_{\mathrm{He}}}{d_{\mathrm{H}_{2}}}=2 \)
\( \therefore \frac{v_{\mathrm{H}_{2}}}{v_{\mathrm{He}}}=\frac{\sqrt{42}}{5}\)
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