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)

$\because v_{\text{rms }}=\sqrt{\frac{3RT}{M}}$ and $v_{\text{sound }}=\sqrt{\frac{\gamma RT}{M}}$,
$v_{\text{rms }}=2v_{\text{sound }}$
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 $C_p=\frac{n_1{C}_{p_1}+n_2{C}_{p_2}}{n_1+n_2}$
$\therefore \gamma =\frac{C_P}{C_V}=\frac{n_1{C}_{p_1}+n_2{C}_{p_2}}{n_1{C}_{V_1}+n_2{C}_{V_2}}$
$\therefore \frac{3}{2}=\frac{2\left(\frac{5}{2}R\right)+n\left(\frac{7}{2}R\right)}{2\left(\frac{3}{2}R\right)+n\left(\frac{5}{2}R\right)}$
$\Rightarrow \frac{3}{2}=\frac{10+7n}{6+5n}$
$\Rightarrow n=2$