Blundell Thermal Physics · Problem 6.5

Pressure

Physics — Statistical Mechanics · molecular-flux-average-energy

Problem

(a) Given that the number of molecules hitting unit area of a surface per second with speeds between

v

and

v + \mathrm{d}v

and angles between

\theta

and

\theta + \mathrm{d}\theta

to the normal is

\frac{1}{2}\,v\,n\,f(v)\,\mathrm{d}v\,\sin\theta\cos\theta\,\mathrm{d}\theta,

show that the average value of

\cos\theta

for these molecules is

\tfrac{2}{3}

. (b) Using the results above, show that for a gas obeying the Maxwellian distribution (i.e.,

f(v) \propto v^2 e^{-mv^2/2k_\mathrm{B}T}

) the average energy of all the molecules is

\tfrac{3}{2}k_\mathrm{B}T

, but the average energy of those hitting the surface is

2k_\mathrm{B}T