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Maxwell-Boltzmann Distribution

The kinetic energy distribution of electrons, ions and neutrals is very important for characterizing a gas. From statistical arguments and also from experiments the so called Maxwell Boltzmann distribution can be derived (see fig. [*]). This indicates the portion of the gas constituents having a specific velocity $v$ in thermal equilibrium ($m$ denotes the mass of the particles). At www.gs68.de you can plot a Maxwell Boltzmann Distribution online.


\begin{displaymath}
f(v) = \frac{4}{\sqrt{\pi}}  \left(\frac{m}{2 {\ensuremath...
...ight)^{3/2}
 v^2  e^{\frac{-m v^2}{2 {\ensuremath{k_B}}T}}.
\end{displaymath} (1.7)

Figure: Maxwell Boltzmann distribution
\resizebox{12cm}{!}{\includegraphics{maxwell_boltzmann}}

The shape of the distribution gets wider for higher temperatures and also the peak decreases. It is possible to calculate a mean speed $\overline{v}$ which is equal to:


\begin{displaymath}
\overline{v} = \int^\infty_0 v  f(v) \mbox{d}v
= \sqrt{\frac{8 {\ensuremath{k_B}}T}{\pi m}}.
\end{displaymath} (1.8)

The mean speed is always higher than the most probable speed due to the skewness of the distribution. For Argon atoms at $20^\circ$ C (=295K) $\overline{v}$ = 394m/s.


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Next: Collision Parameters Up: Gases and Ideal Gases Previous: Kinetic Gas Theory   Contents   Index

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