Blackbody

The explanation by Boltzman for the energy density versus energy, frequency, or wavelength of Blackbody radiation, light emitted by all objects, was the first invocation of discrete or quantized energy levels. The explanation also involved the first use of the Boltzman population model. The basic idea is that all objects have quantized energy levels that are populated according to E=hν. E is the energy, h is Planck′s constant, ν is the frequency of light, and k_B is Boltzman′s constant, and T is the temperature in Kelvin.

\begin{equation} {P_i} = \frac{{{e^{-iE/{k_B}T}}}}{Q} \label{eq:pop} \end{equation}

In Equation \eqref{eq:pop} Q is the partition function, see Eqn. \eqref{eq:pf}

\begin{equation} Q = \sum\limits_{j = 1}^n {{e^{ - jE/{k_B}T }}} \label{eq:pf} \end{equation}

This quantized, population-weighted model leads to the energy density versus energy curve, Eqn. \eqref{eq:bbody} where c is the speed of light.

\begin{equation} {\rho _E}\left( {E,T} \right) = \frac{{8\pi E{\nu ^2}}}{{{c^3}}}\frac{{dE/h}}{{{e^{E/{k_B}T}} - 1}} \label{eq:bbody} \end{equation}

Differentiation of Equation \eqref{eq:bbody} with respect to temperature and setting the derivative to zero provides the peak maximum expression in Eqn. \eqref{eq:peak}.

\begin{equation} {E_{\max }} = 2.8214RT = 23.46T \label{eq:peak} \end{equation}

Finally, integration of Equation \eqref{eq:bbody} leads to the total intensity expression, Eqn. \eqref{eq:ints}.

\begin{equation} M = \frac{{48{\pi ^5}{{\left( {RT} \right)}^4}}}{{90{h^3}{c^3}}} \label{eq:ints} \end{equation}

Figure 1 provides the Black body energy density profile as a function of Temperature for two temperature regions, around room temperature and approaching the temperature of the sun.

Figure 1. Energy density versus energy for Blackbody radiation as given by Eqn. (3).

Temperature = 100;