First slide

Mean free path

Question

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

Moderate

A

3.67
B

4.67
C

1.09
D

2.3

Solution

$\begin{array}{l} M e a n F r e e P a t h = λ = \frac{1}{\sqrt{2} π n_{v} d^{2}} \\ M e a n F r e e T i m e = τ = \frac{λ}{v} = \frac{1}{\sqrt{2} π n_{v} d^{2} v} \\ \Rightarrow τ = \frac{1}{\sqrt{2} π n_{v} d^{2}} \times \sqrt{\frac{M}{3 R T}} \\ ∴ \frac{τ_{1}}{τ_{2}} = \sqrt{\frac{M_{1}}{M_{2}}} \times \frac{d_{2}^{2}}{d_{1}^{2}} \\ \Rightarrow \frac{τ_{A r}}{τ_{X e}} = \sqrt{\frac{40}{140}} \times {(\frac{0.1}{0.07})}^{2} = \sqrt{\frac{2}{7}} \times \frac{1}{0.49} = 1.09 \end{array}$

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