Question

Two gases-argon $($atomic radius $0.07\text{nm}$, atomic weight $40)$ and xenon $($atomic radius $0.1\text{nm}$, atomic weight $140)$ have the same number density and are at the same temperature. The ratio of their respective mean free paths, is closest to:

• A. $3.67$
• B. $1.83$
• C. $2.3$
• D. $4.67$
• E. None of the above

Answer 1

The correct option is E None of the above

$\text{(Bonus)}$
Mean free path of a gas molecule is given by

$\lambda =\frac{1}{\sqrt{2}\pi d^{2}n}$

Here, $n$, = number of collisions per unit volume
$d=$ diameter of the molecule

If average speed of molecule is $v$ then
Mean free time, $\tau =\frac{\lambda }{v}$

$\Rightarrow \tau =\frac{1}{\sqrt{2}\pi n{d}^{2}v}=\frac{1}{\sqrt{2}\pi n{d}^2}\sqrt{\frac{M}{3RT}}(\because v=\sqrt{\frac{3RT}{M}})$

$\therefore \tau \propto \frac{\sqrt{M}}{d^2}$

$\therefore \frac{{\tau }_1}{{\tau }_2}=\frac{\sqrt{M_1}}{d_1^2}\times \frac{d_2^2}{\sqrt{M_2}}$

$=\sqrt{\frac{40}{140}}\times {\left(\frac{0.1}{0.07}\right)}^2=1.09$

So, the nearest answer is $1$.