Question

Two moles of helium are mixed with $n$ moles of hydrogen. The root mean square (rms) speed of gas molecules in the mixture is $\sqrt{2}$ times the speed of sound in the mixture. Then, the value of $n$ is

• A. 1
• B. 3
• C. 2
• D. 3/2

Answer 1

Answer: (C)

$\because v_{\text {rms }}=\sqrt{\frac{3 R T}{M}}$ and $v_{\text {sound }}=\sqrt{\frac{\gamma R T}{M}}$,
$v_{\text {rms }}=2 v_{\text {sound }}$
i.e. $ \gamma=\frac{3}{2}=$ ratio of $\frac{C_{p}}{C_{V}}$ for the mixture
$C_{V}=\frac{n_{1} C_{V_{1}}+n_{2} C_{V_{2}}}{n_{1}+n_{2}}$
and $ C_{p}=\frac{n_{1} C_{p_{1}}+n_{2} C_{p_{2}}}{n_{1}+n_{2}}$
$\therefore \gamma=\frac{C_{P}}{C_{V}}=\frac{n_{1} C_{p_{1}}+n_{2} C_{p_{2}}}{n_{1} C_{V_{1}}+n_{2} C_{V_{2}}}$
$\therefore \frac{3}{2}=\frac{2\left(\frac{5}{2} R\right)+n\left(\frac{7}{2} R\right)}{2\left(\frac{3}{2} R\right)+n\left(\frac{5}{2} R\right)}$
$\Rightarrow \frac{3}{2}=\frac{10+7 n}{6+5 n}$
$\Rightarrow n=2$