Pressure, Temperature, and RMS Speed — Question 6

Question 6

Bulk drift must be removed before inferring thermal pressure

In the laboratory, molecules have $\mathbf v=u\hat x+\mathbf c$, with $u=300$ m/s, $\langle\mathbf c\rangle=0$ and isotropic random rms speed $c_{\mathrm{rms}}=400$ m/s. Density is $1.50\,\mathrm{kg/m^3}$. Use thermal pressure $p=\rho\langle c^2\rangle/3$.

1. Find laboratory rms speed and thermal pressure.

2. Find $\rho\langle v_x^2\rangle$ and separate it into convective and thermal contributions. Why is it not simply the equilibrium pressure?

Question 6 - Solution

Bulk drift must be removed before inferring thermal pressure

(a) Method and calculation

$$\langle v^2\rangle=u^2+\langle c^2\rangle=300^2+400^2=250000.$$$$v_{\mathrm{rms,lab}}=500\,\mathrm{m/s},\quad p=1.50(400^2)/3=80000\,\mathrm{Pa}.$$

(b) Interpretation and further calculation

$$\rho\langle v_x^2\rangle=\rho u^2+\rho\langle c_x^2\rangle=135000+80000=215000\,\mathrm{Pa}.$$$$\rho\langle v_x^2\rangle=\rho u^2+p.$$This laboratory momentum flux includes directed transport. Thermal pressure depends on random motion relative to the local mean flow.

Physical check

A Galilean boost changes laboratory kinetic statistics but not the comoving temperature.