I was able to derive the formula for root-mean-squared speed of a molecule, using a basic method taught at school. However, in my textbook, there's a point that:

It can be shown that $\bar v = \sqrt{\frac{8RT}{\pi M}}$

where $R = 8.314\text{ J mol}^{-1}\text{ K}^{-1}$, T = Absolute temperature, and M = Molar mass of given gas.

I was unable to prove this. I am aware of the Maxwell-Boltzmann speed distribution curve, but haven't learnt it in detail. Can this be derived from that? Are there other methods of deriving it?


1 Answer 1


You're on the right track: you do indeed need to work out $v_{rms}$ from the Maxwell-Boltzmann distribution. There are no other methods, because, by definition, you compute a mean with respect to a probability distribution describing the underlying population, so the pdf has to come in somewhere.

The MB distribution comes from (1) the Boltzmann Distribution, justified by the analysis of the Canonical Ensemble, and, to apply this distribution, you also need to know what degeneracy function aka density of states $g(E) $ is - with each particle's energy given by $m\,v^2/2$, the number of possible states in a narrow velocity range around velocity $v$ is proportional to the volume of a spherical shell, i.e. proportional to $v^2$. So when we work out the Boltzmann distribution for the velocities, we get $p(E)\propto g(E) \exp(-E/(k\,T)) = v^2 e^{-\frac{m\,v^2}{2\,k\,T}}$, which last function is proportional to the Maxwell-Boltzman distribution $p_{MB}(v)$.

Now all you need to do is find the correct normalization constant to make $\int_0^\infty p_{MB}(v)\,\mathrm{d} v = 1$ (or look up the normalized distribution) and then work out the mean square velocity $\int_0^\infty\,v^2\, p_{MB}(v)\,\mathrm{d} v$ and you're there.

You will also need to convert the Boltzmann constant $k$ to molar gas constant $R=N_A\,k$.

  • $\begingroup$ I'm sorry I didn't make it clear earlier, I wanted to derive $v_{mean}$ not $v_{rms}$. I believe you'll edit that in, as the method you say is for $v_{mean}$. +1 Thanks! $\endgroup$
    – Abhigyan
    Commented Feb 17, 2018 at 8:20
  • 1
    $\begingroup$ @AbhigyanC The method will work for any statistic. So, with the normalized distribution, calculate $\int_0^\infty\,v\,p_{MB}(v)\,\mathrm{d} v$. $\endgroup$ Commented Feb 17, 2018 at 12:24

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