% The Maxwell-Boltzmann distribution plotted at different temperatures reveals that the most probable velocity v_p of ideal gas particles scales with the square root of temperature.

\documentclass{standalone}

\usepackage{pgfplots,siunitx}
\pgfplotsset{compat=newest}

\def\kB{1.38e-23} % Boltzmann constant
\def\mu{1.66e-27} % unified atomic mass unit/Dalton (symbols: m_u or Da)

\def\maxwellboltzmann#1{4 * pi * (\mu / (2 * pi * \kB * #1))^(3/2) * x^2 * exp(-\mu * x^2 / (2 * \kB * #1))}

\begin{document}

\begin{tikzpicture}

  \begin{axis}[
      domain = 0:8000,
      xlabel = {$v$ [\si{\metre\per\second}]},
      ylabel = $P(v)$,
      smooth, thick,
      axis lines = left,
      every tick/.style = {thick},
    ]

    \addplot[color=red,samples=100]{\maxwellboltzmann{100}};

    \addplot[color=yellow]{\maxwellboltzmann{300}};

    \addplot[color=blue]{\maxwellboltzmann{1000}};

    \legend{\SI{100}{\kelvin}, \SI{300}{\kelvin}, \SI{1000}{\kelvin}}

  \end{axis}
\end{tikzpicture}

\end{document}
