Question

Two moles of helium are mixed with $n$ moles of hydrogen. The root mean square $\left(rms\right)$ 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

The correct option is C $2$

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