Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is \(\sqrt{2}\) times the speed of sound, then the value of \(n\) will be

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Answer & Solution

Correct Answer

(B) \(2\)

Detailed explanation

\(v _{ s }=\sqrt{\frac{\gamma RT }{ M }}\)

\(v _{ mms }=\sqrt{\frac{3 RT }{ M }}\)

\(\frac{ v _{ s }}{ v _{ rms }}=\sqrt{\frac{\gamma}{3}}=\frac{1}{\sqrt{2}} \Rightarrow \frac{\gamma}{3}=\frac{1}{2} \Rightarrow \gamma=\frac{3}{2}\)

\(\gamma=1+\frac{2}{ f _{\operatorname{mix}}}\)

\(f _{\operatorname{mix}}=\frac{2 \times 3+ n \times 5}{ n +2}=\frac{6+ n \times 5}{( n +2)}\)

\(\gamma=1+\frac{2( n +2)}{6+ n \times 5}=\frac{6+5 n +2 n +4}{6+5 n }\)

\(\gamma=\frac{7 n +10}{6+5 n }=\frac{3}{2}\)

\(14 n +20=18+15 n\)

\(n =2\)

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