Pressure, Temperature, and RMS Speed — Question 7

Question 7

An oblique molecular beam transfers only normal momentum to a smooth wall

A dilute collimated beam has number density $\nu=10^{20}\,\mathrm{m^{-3}}$, molecular mass $m=4.00\times10^{-26}$ kg and speed 500 m/s. It strikes a flat wall at angle $60^\circ$ to the normal and reflects specularly. Use incident number flux $J=\nu v\cos\theta$.

1. Derive the normal force per wall area and evaluate it.

2. Find the ratio to normal incidence and explain why two cosine factors appear. Is there a tangential force for ideal specular reflection?

Question 7 - Solution

An oblique molecular beam transfers only normal momentum to a smooth wall

(a) Method and calculation

$$\Delta p_{n,\mathrm{wall}}=2mv\cos\theta,\quad\frac FA=J\Delta p_n=2\nu mv^2\cos^2\theta.$$$$F/A=2(10^{20})(4\times10^{-26})(500)^2(1/2)^2=.500\,\mathrm{Pa}.$$

(b) Interpretation and further calculation

$$\frac{p(60^\circ)}{p(0)}=\cos^2 60^\circ=\frac14,\quad p(0)=2.00\,\mathrm{Pa}.$$$$\Delta p_t=0\ \Rightarrow\ F_t/A=0.$$One cosine reduces the incidence rate per wall area; the other reduces normal momentum transfer per collision.

Physical check

This directed-beam force is not the isotropic-equilibrium formula with a $1/3$ factor.