Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

$ v _{ S }=\sqrt{\frac{\gamma RT }{ M }} $
$ v _{ rms }=\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 _{\text {mix. }}}$
$ f _{\text {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$