Blundell Thermal Physics · Problem 5.4

The Maxwell–Boltzmann distribution

Physics — Statistical Mechanics · maxwell-boltzmann-mean-speeds

Problem

A Maxwell--Boltzmann distribution implies that a given molecule (mass

m

) will have a speed between

v

and

v+\mathrm{d}v

with probability equal to

f(v)\,\mathrm{d}v

where

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

and the proportionality sign is used because a normalization constant has been omitted. (You can correct for this by dividing any averages you work out by

\int_0^\infty f(v)\,\mathrm{d}v

.) For this distribution, calculate the mean speed

\langle v \rangle

and the mean inverse speed

\langle 1/v \rangle

. Show that

\langle v \rangle \langle 1/v \rangle = \frac{4}{\pi}.

Related theory

Maxwell Boltzmann Distribution

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