The Maxwell-Boltzmann Distribution

Definition: Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution of an ideal gas is the function

$$f(v)=4\pi {\left(\frac{m}{2\pi k_{B}T}\right)}^{3/2}{v}^2{\mathrm{e}}^{-m{v}^2/2kT},$$

where $T$ is the Temperature of the gas, $m$ is the mass of a single gas particle and $k_B$ is the Boltzmann Constant.

Theorem: Speed of Particles in an Ideal Gas

The probability that the speed of a given particle in an ideal gas is between two values $v_1$ and $v_2$ is given by integrating its Maxwell-Boltzmann distribution from $v_1$ to $v_2$:

$${\int }_{v_1}^{v_2}f(v)\mathrm{dv}$$

PROOF

TODO

Theorem: Average Speed of the Particles in an Ideal Gas

The average speed of the particles in an ideal gas is

$$v_{\text{avg}}=\sqrt{\frac{8k_{B}T}{\text{uppi}m}},$$

where $T$ is the gas Temperature, $m$ is the mass of a single gas particle and $k_B$ is the Boltzmann Constant.

PROOF

TODO

Theorem: Root-Mean-Square Speed of the Particles in an Ideal Gas

The root-mean-square speed of the particles of an ideal gas is

$$v_{\text{rms}}=\sqrt{\frac{3k_{B}T}{m}},$$

where $k_B$ is the Boltzmann Constant, $T$ is the Temperature and $m$ is the mass of a single gas particle.

NOTE

The root-mean-square speed can alternatively be expressed via the Molar Gas Constant $R$ and the molar mass $M$ of a single particle:

$$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$$

PROOF

TODO