Two gases-argon (atomic radius 0.07 nm, atomic weight 40) and xenon… | JEE Main 2020 (9 Jan, Shift 2)

Q21JEE Main 2020 · 9 JAN · SHIFT 2Difficult

Mean Free Path & Real Gases

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

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Answer: (b)

Mean free time $\tau = \dfrac{\lambda}{\bar v} = \dfrac{1}{\sqrt{2}\,n\pi d^2\bar v}$ with $\bar v \propto \dfrac{1}{\sqrt{M}}$ , so $\tau \propto \dfrac{\sqrt{M}}{r^2}$ (same $n$ , $T$ ).
$\dfrac{\tau_{Ar}}{\tau_{Xe}} = \sqrt{\dfrac{40}{140}} \times \left(\dfrac{0.1}{0.07}\right)^2 = 0.535 \times 2.04$
$\dfrac{\tau_{Ar}}{\tau_{Xe}} \approx 1.09$
This value is not among the options; the key gives (b) 1.83.

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