The Distribution of Molecular Speeds — Question 10

Question 10

A signed velocity component has a different distribution from speed

An isotropic equilibrium gas has Cartesian component density $h(v_x)=e^{-v_x^2/(2\sigma^2)}/(\sqrt{2\pi}\sigma)$ with $\sigma=250$ m/s. Use $\operatorname{erf}(1/\sqrt2)=.682689$ and independent equal-variance components.

1. Find $P(|v_x|<250\,\mathrm{m/s})$, $P(v_x>250\,\mathrm{m/s})$ and the three-dimensional rms speed.

2. Find $\langle|v_x|\rangle$ by integration and compare it with the mean speed $\overline v=\sqrt{8/\pi}\sigma$. Why can $h$ peak at zero while the speed density does not?

Question 10 - Solution

A signed velocity component has a different distribution from speed

(a) Method and calculation

$$P(|v_x|<\sigma)=.682689,\quad P(v_x>\sigma)=\tfrac12(1-.682689)=.1586555.$$$$v_{\mathrm{rms}}=\sqrt{\langle v_x^2+v_y^2+v_z^2\rangle}=\sqrt3\sigma\approx433.013\,\mathrm{m/s}.$$

(b) Interpretation and further calculation

$$\langle|v_x|\rangle=\frac2{\sqrt{2\pi}\sigma}\int_0^\infty v_xe^{-v_x^2/(2\sigma^2)}dv_x=\sqrt{2/\pi}\sigma\approx199.471\,\mathrm{m/s}.$$$$\overline v=2\langle|v_x|\rangle\approx398.942\,\mathrm{m/s}.$$A speed shell in three-dimensional velocity space contributes a $4\pi v^2$ factor; a one-component interval does not.

Physical check

Isotropy gives zero mean signed velocity, not zero mean speed or zero mean absolute component.