# 6.6: What is the origin of the Planck Function?

Recall that molecules have a wide range of speeds and thus a wide range of energies. The Maxwell–Boltzmann Distribution, which gives the distribution of molecules as a function of energy, is given approximately by the equation:

$f(E)=\frac{2 \sqrt{E}}{\sqrt{\pi}(k T)^{3 / 2}} \exp (-E / k T)$

where f is the probability that a molecule has an energy within a small window around E, T is the absolute temperature, and k is the Boltzmann constant. The above equation, when integrated over all energies, gives the value of 1.

The functional form of this distribution is shown below:

All objects—gas, liquid, or solid—emit radiation. If we think of radiation as photons, we would say that these photons have a distribution of energies, just like molecules do. However, photons cannot have continuous values of photon energy; instead, the photon energy is quantized, which means that it can have only discrete energy values that are different by a very very small amount of energy. When this quantized distribution is assumed, then the distribution of spectral irradiance leaving a unit area of the object’s surface per unit time per unit wavelength interval into a hemisphere is called the Planck Distribution Function of Spectral Irradiance:

$P_{e}(\lambda)=\frac{2 \pi h c^{2}}{\lambda^{5}} \frac{1}{\exp (h c / \lambda k T)-1}$

where h is Planck’s constant, c is the speed of light, k is the Boltzmann constant (1.381 x 10–23 J K–1), T is the absolute temperature, and λ is the wavelength. The integral of this function over all wavelengths leads to the Stefan–Boltzmann Law Irradiance, which gives the total radiant energy per unit time per unit area of the object’s surface emitted into a hemisphere.