Stefan–Boltzmann constant
# Stefan–Boltzmann constant

The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature. The theory of thermal radiation lays down the theory of quantum mechanics, by using physics to relate to molecular, atomic and sub-atomic levels. Slovenian physicist Josef Stefan formulated the constant in 1879; it was formally derived in 1884 by his former student and collaborator, the Austrian physicist Ludwig Boltzmann. The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box. "The amount of thermal radiation emitted increases quickly and the principal frequency of the radiation becomes higher with increasing temperatures". The Stefan–Boltzmann constant can be used to measure the amount of heat that is emitted by a black body, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan–Boltzmann constant allows for units of temperature (K) to be converted to units of intensity $(W⋅m^{−2}),$ which is power per unit area.

## Value

Since the 2019 redefinition of the SI base units, the Stefan–Boltzmann constant is given exactly rather than in experimental values. The value is given in SI units by

In cgs units the Stefan–Boltzmann constant is

In thermochemistry the Stefan–Boltzmann constant is often expressed in $cal⋅cm^{−2}⋅day^{−1}⋅K^{−4}:$

In US customary units the Stefan–Boltzmann constant is

The Stefan–Boltzmann constant is defined in terms of other fundamental constants as

$\sigma ={\frac {2\pi ^{5}k_{\rm {B}}^{4}}{15h^{3}c^{2}}}={\frac {\pi ^{2}k_{\rm {B}}^{4}}{60\hbar ^{3}c^{2}}}\,,$

where

* $k_B$ is the Boltzmann constant,
* h is the Planck constant,
* ħ is the reduced Planck constant, and
* c is the speed of light in vacuum,

giving

    $\sigma ={\frac {5\,454\,781\,984\,210\,512\,994\,952\,000\,000\pi ^{5}}{29\,438\,455\,734\,650\,141\,042\,413\,712\,126\,365\,436\,049}}\,\mathrm {J} \cdot \mathrm {m} ^{-2}\cdot \mathrm {s} ^{-1}\cdot \mathrm {K} ^{-4}$

The CODATA recommended value prior to 20 May 2019 (2018 CODATA) was calculated from the measured value of the gas constant:

$\sigma ={\frac {2\pi ^{5}R^{4}}{15h^{3}c^{2}N_{\rm {A}}^{4}}}={\frac {32\pi ^{5}hR^{4}R_{\infty }^{4}}{15A_{\rm {r}}({\rm {e}})^{4}M_{\rm {u}}^{4}c^{6}\alpha ^{8}}},$

where

* R is the universal gas constant
* $N_A$ is the Avogadro constant
* $R_∞$ is the Rydberg constant
* $A_r(e)$ is the "relative atomic mass" of the electron
* $M_u$ is the molar mass constant (1 g/mol by definition)
* α is the fine-structure constant.

Dimensional formula: $M^1T^{−3}Θ^{−4}$

A related constant is the radiation constant (or radiation density constant) $a$, which is given by

$a={\frac {4\sigma }{c}}=7.5657\times 10^{-15}\mathrm {erg\cdot cm^{-3}\cdot K^{-4}} =7.5657\times 10^{-16}\mathrm {J\cdot m^{-3}\cdot K^{-4}} .$

It provides the electromagnetic energy density $u$ according to $u=aT^{4}.$

