Statistical Thermodynamics and Rate Theories/Blackbody Radiation

Blackbody radiation is the phenomenon where a heated object will emit electromagnetic radiation

As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans.

$B_{\nu }(\nu ,T)=\frac{8\pi {\nu }^2}{c^3}\times ⟨E(\nu ,T)⟩$

Rayleigh–Jeans Law

The average thermal energy of a classical harmonic oscillator is predicted by equipartition theorem to be,

$⟨E(\nu ,T)⟩=k_{B}T$

Combined with the Rayleigh model, the blackbody intensity is predicted to be,

$B_{\nu }(\nu ,T)=\frac{8\pi {\nu }^2}{c^3}\times k_{B}T$

This purely classical theory is known as the Rayleigh–Jeans Law. Although it provides a reasonable description of the intensity of blackbody radiation at low frequencies, it incorrectly predicts that the intensity will continue to increase at higher frequencies, although experiments show that the intensity should reach a maximum and then decline at high frequencies. This is known as the ultraviolet catastrophe.

Planck's Law

$B_{\nu }(\nu ,T)=\frac{8\pi {\nu }^2}{c^3}\times ⟨E(\nu ,T)⟩$

If we derive the average energy for the oscillator based on the quantum harmonic oscillator, this equations becomes

$B_{\nu }(\nu ,T)=\frac{8\pi {\nu }^2}{c^3}\times \frac{h\nu }{\exp \left(\frac{h\nu }{k_{B}T}\right)-1}$

Planck's law states that

$B_{\nu }(\nu ,T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{(h\nu /k_{B}T)}-1},$

where

B_ν(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.

h is the Planck constant;

c is the speed of light in a vacuum;

k_B is the Boltzmann constant;

$\nu$ is the frequency of the electromagnetic radiation;

T is the absolute temperature of the body.