My name is Alexander FufaeV and here I write about:

# Maxwell Velocity Distribution of Gas Particles

## Important Formula

## 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.

$$ \begin{align} f(\class{blue}{v}) ~=~ \sqrt{\frac{2}{\pi}} \, \left( \frac{\class{brown}{m}}{k_{\text B} \, T} \right)^{3/2} \, \class{blue}{v}^2 \, \mathrm{e}^{ - \frac{\class{brown}{m}\,\class{blue}{v}^2}{2k_{\text B} \, T} } \end{align} $$