Maxwell speed distribution

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The probability that speed of a molecule of mass m lies in the range v to v+dv is given by

P(v)dv = 4 \pi v^2 dv \left( \frac{m}{2 \pi k_B T} \right)^{3/2} \exp (-mv^2/2k_B T)

where T is the temperature and k_B is the Boltzmann constant. The maximum of this distribution is located at

v_{\rm max} = \sqrt{\frac{2k_BT}{m}}

The mean speed is given by

\overline{v} = \frac{2}{\sqrt \pi} v_{\rm max}

and the root-mean-square speed by

v_{\rm rms} = \sqrt \frac{3}{2} v_{\rm max}


  1. J. C. Maxwell "", British Association for the Advancement of Science 29 Notices and Abstracts 9 (1859)
  2. J. C. Maxwell "", Philosophical Magazine 19 pp. 19 (1860)
  3. J. C. Maxwell "", Philosophical Magazine 20 pp. 21 (1860)
  4. J. Clerk Maxwell "On the Dynamical Theory of Gases", Philosophical Transactions of the Royal Society of London 157 pp. 49-88 (1867)
  5. J. S. Rowlinson "The Maxwell-Boltzmann distribution", Molecular Physics 103 pp. 2821 - 2828 (2005)