Question

A plot of distribution function $f\left(v\right)$ vs $\text{speed}\left(v\right)$ for $N_2\left(300\text{K}\right),O_2\left(400\text{K}\right),H_2\left(300\text{K}\right)$ is given below. Points $1$, $2$ and $3$ in the following plot respectively correspond to:
($v_{mp}$ is most probable velocity)

• A. $v_{mp}$ of $N_2\left(300\text{K}\right);v_{mp}$ of $H_2\left(300\text{K}\right);v_{mp}$ of $O_2\left(400\text{K}\right)$
• B. $v_{mp}$ of $H_2\left(300\text{K}\right);v_{mp}$ of $N_2\left(300\text{K}\right);v_{mp}$ of $O_2\left(400\text{K}\right)$
• C. $v_{mp}$ of $O_2\left(300\text{K}\right);v_{mp}$ of $N_2\left(300\text{K}\right);v_{mp}$ of $H_2\left(400\text{K}\right)$
• D. $v_{mp}$ of $N_2\left(300\text{K}\right);v_{mp}$ of $O_2\left(300\text{K}\right);v_{mp}$ of $H_2\left(400\text{K}\right)$

Answer 1

The correct option is D $v_{mp}$ of $N_2\left(300\text{K}\right);v_{mp}$ of $O_2\left(300\text{K}\right);v_{mp}$ of $H_2\left(400\text{K}\right)$

We know that
$v_{mp}=\sqrt{\frac{2RT}{M}}$
Where, $R$ is the universal gas constant, $T$ is temperature and $M$ is molar mass.
$\Rightarrow v_{mp}\propto \sqrt{\frac{T}{M}}$
So, $v_{mp}$ for $N_2,O_2$ and $H_2$
$v_{mp}\left(N_2\right)\propto \sqrt{\frac{300}{28}}$
$v_{mp}\left(O_2\right)\propto \sqrt{\frac{400}{32}}$ and $v_{mp}\left(H_2\right)\propto \sqrt{\frac{300}{2}}$
Thus,
$v_{mp}$ of $N_2\left(300\text{K}\right)<v_{mp}$ of $O_2\left(400\text{K}\right)<v_{mp}$ of $H_2\left(300\text{K}\right)$
So, option $\left(d\right)$ is correct answer.

Why this question? Tips: Remember the variation of $v_{mp}$ with different parameter. $v_{mp}=\sqrt{\frac{2P}{\rho }}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2KT}{M}}$