આદર્શ વાયુઓ \(\mathrm{He}, \mathrm{CH}_4\), અને \(\mathrm{CO}_2\) માં પ્રસરતા ધ્વનિ તરંગને ધ્યાનમાં લો. બધા વાયુઓ માટે ગુણોત્તર \(\frac{P}{\rho}\) સમાન છે, જ્યાં \(P\) દબાણ અને \(\rho\) ઘનતા છે. વાયુઓમાં ધ્વનિની ઝડપનો ગુણોત્તર \(\mathrm{v}_{\mathrm{He}}: \mathrm{v}_{\mathrm{CH}_4}: \mathrm{v}_{\mathrm{CO}_2}\) કેટલો થશે?

A \(\sqrt{\frac{7}{5}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}\)
B \(\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{7}{5}}\)
C \(\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}\)
D \(\sqrt{\frac{4}{3}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{7}{5}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{v}_{\text {sound }}=\sqrt{\frac{\gamma \mathrm{p}}{\rho}} \\ \gamma=1+\frac{2}{\mathrm{f}} \\ \gamma_{\mathrm{He}}=\frac{5}{3} \\ \gamma_{\mathrm{CH}_4}=\gamma_{\mathrm{CO}_2} \approx 1.33=\frac{4}{3}\) (Experimental data)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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