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)

$v_S=\sqrt{\frac{\gamma RT}{M}}$
$v_{rms}=\sqrt{\frac{3RT}{M}}$
$\frac{v_s}{v_{rms}}=\sqrt{\frac{\gamma }{3}}=\frac{1}{\sqrt{2}}\Rightarrow \frac{\gamma }{3}=\frac{1}{2}\Rightarrow \gamma =\frac{3}{2}$
$\gamma =1+\frac{2}{f_{\text{mix. }}}$
$f_{\text{mix. }}=\frac{2\times 3+n\times 5}{n+2}=\frac{6+n\times 5}{(n+2)}$
$\gamma =1+\frac{2(n+2)}{6+n\times 5}=\frac{6+5n+2n+4}{6+5n}$
$\gamma =\frac{7n+10}{6+5n}=\frac{3}{2}$
$14n+20=18+15n$
$n=2$