Pressure, Temperature, and RMS Speed — Question 10

Question 10

Reflection and absorption produce different beam pressures

A normal molecular beam has flux $J=2.00\times10^{22}\,\mathrm{m^{-2}s^{-1}}$, mass $m=3.00\times10^{-26}$ kg and speed 400 m/s. A fraction $r$ reflects elastically and the rest sticks, without re-emission during the measurement. Measured force per area is .360 Pa.

1. Derive pressure as a function of $r$ and infer the reflected fraction.

2. Find the allowed pressure range and the mass accumulation rate per unit area at the inferred $r$. Why is this not an equilibrium gas-pressure measurement?

Question 10 - Solution

Reflection and absorption produce different beam pressures

(a) Method and calculation

$$p=J[2rmv+(1-r)mv]=Jmv(1+r),\quad Jmv=.240\,\mathrm{Pa}.$$$$.360=.240(1+r)\ \Rightarrow\ r=.500.$$

(b) Interpretation and further calculation

$$0\le r\le1\ \Rightarrow\ .240\le p\le.480\,\mathrm{Pa}.$$$$\frac{\dot m_{\mathrm{wall}}}A=(1-r)Jm=.500(2.00\times10^{22})(3.00\times10^{-26})=3.00\times10^{-4}\,\mathrm{kg/(m^2s)}.$$Particles arrive in a directed beam and accumulate at the wall; there is no isotropic stationary molecular population.

Physical check

Absorbing a molecule transfers $mv$ normal momentum; reversing it transfers $2mv$.