Mean Free Path — Question 6

Question 6

Different stationary targets contribute additive collision hazards

A fast probe travels through two types of effectively stationary scatterers. Their number densities are $\nu_A=2.00\times10^{20}$ and $\nu_B=1.00\times10^{20}\,\mathrm{m^{-3}}$; cross sections are $\sigma_A=1.00\times10^{-19}$ and $\sigma_B=3.00\times10^{-19}\,\mathrm{m^2}$. Independent hazards add: $1/\lambda=\nu_A\sigma_A+\nu_B\sigma_B$. No identical-moving-gas $\sqrt2$ factor applies.

1. Find the mean free path and the probability that a collision, when it occurs, is with B.

2. Find survival through 3.00 cm. If B is removed, calculate the new mean free path and survival.

Question 6 - Solution

Different stationary targets contribute additive collision hazards

(a) Method and calculation

$$h_A=20.0\,\mathrm{m^{-1}},\quad h_B=30.0\,\mathrm{m^{-1}},\quad\lambda=1/(20+30)=.0200\,\mathrm m.$$$$P(B\mid\text{collision})=h_B/(h_A+h_B)=.600.$$

(b) Interpretation and further calculation

$$S(.0300)=e^{-50(.0300)}=e^{-1.5}=.223130.$$$$\lambda_{A\mathrm{\ only}}=1/20=.0500\,\mathrm m,\quad S_A=e^{-.600}=.548812.$$A less numerous species can dominate collisions when its cross section is larger.

Physical check

Hazards, rather than mean free paths, add for independent competing collision mechanisms.