My name is Alexander FufaeV and here I write about:
Maxwell Velocity Distribution of Gas Particles
Important Formula
$$f(\class{blue}{v}) ~=~ \sqrt{\frac{2}{\pi}} \, \left( \frac{\class{brown}{m}}{k_{\text B} \, T} \right)^{3/2} \, \class{blue}{v}^2 \, \mathrm{e}^{ - \frac{\class{brown}{m}\,\class{blue}{v}^2}{2k_{\text B} \, T} }$$
$$f(\class{blue}{v}) ~=~ \sqrt{\frac{2}{\pi}} \, \left( \frac{\class{brown}{m}}{k_{\text B} \, T} \right)^{3/2} \, \class{blue}{v}^2 \, \mathrm{e}^{ - \frac{\class{brown}{m}\,\class{blue}{v}^2}{2k_{\text B} \, T} }$$
What do the formula symbols mean?
Distribution function
$$ f(\class{blue}{v}) $$
Relative frequency density \(f(\class{blue}{v})\). The area under this curve, between two velocities, indicates the percentage \(p(v_1, v_2)\) of the particles in the velocity interval under consideration:
\[ p(v_1, v_2) ~=~ \int_{v_1}^{v_2} f(\class{blue}{v}) \, \text{d}v \]
The unit of \(f(\class{blue}{v})\) is \( \mathrm{s}/\mathrm{m} \).
Velocity
$$ \class{blue}{\boldsymbol v} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Velocity of a particle of the gas.
Mass
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$
Mass of a particle of the gas.
Temperature
$$ T $$ Unit $$ \mathrm{K} $$
Temperature of the gas in Kelvin.
Boltzmann Constant
$$ k_{\text B} $$ Unit $$ \frac{\mathrm J}{\mathrm K} = \frac{\mathrm{kg} \,\mathrm{m}^2}{\mathrm{s}^2 \, \mathrm{K}} $$Boltzmann constant is a physical constant from many-particle physics and has the following exact value:
$$ k_{\text B} ~=~ 1.380 \, 649 ~\cdot~ 10^{-23} \, \frac{\mathrm{J}}{\mathrm{K}} $$
The Maxwell velocity distribution is a statistical distribution of the velocities \( \class{blue}{v} \) of particles with mass \( \class{brown}{m} \) in an ideal gas at a specific temperature \( T \). This distribution was developed by James Clerk Maxwell and is a crucial component of kinetic gas theory.
$$ \begin{align} f(\class{blue}{v}) ~=~ \sqrt{\frac{2}{\pi}} \, \left( \frac{\class{brown}{m}}{k_{\text B} \, T} \right)^{3/2} \, \class{blue}{v}^2 \, \mathrm{e}^{ - \frac{\class{brown}{m}\,\class{blue}{v}^2}{2k_{\text B} \, T} } \end{align} $$