Auguste Piccard image courtesy of
Deutsches
Bundesarchiv under the
Creative
Commons Attribution-ShareAlike 3.0 Germany License
Today is the birthday of
- Giovanni Alfonso Borelli (1608-1679).
- Auguste Piccard (1884-1962).
Today’s Problem
The probability distribution for molecular speeds in an ideal gas is given by the Maxwell-Boltzmann speed distribution function
\[f(v)=4\pi v^2\left(\frac{m}{2\pi kT}\right)^{3/2} e^{-mv^2/2kT}\]
Use this to calculate the average speed in an ideal gas.
Answer
The average speed will be given by the integral
\[\bar{v} = \int_{0}^{\infty}vf(v)dv = 4\pi\left(\frac{m}{2\pi kT}\right)^{3/2}\int_{0}^{\infty}v^{3}e^{-mv^2/2kT}dv\]
With the change of variables
\[x=v\sqrt{\frac{m}{2kT}}\]
the integral becomes
\[\bar{v} = 4\sqrt{\frac{2kT}{\pi m}}\int_{0}^{\infty}x^{3}e^{-x^2}dx\]
With the change in variables \(y=x^2\) the integral becomes
\[\frac{1}{2}\int_{0}^{\infty}ye^{-y}dy\]
Using integration by parts, this evaluates to \(1/2\). So the average speed is
\[\bar{v} = \sqrt{\frac{8kT}{\pi m}}\]
© 2026 Stefan Hollos and Richard Hollos