Stefan-Boltzmann Constant
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Stefan%E2%80%93Boltzmann Constant

Log-log graphs of peak emission wavelength and radiant exitance vs. black-body temperature - red arrows show that 5780 K black bodies have 501 nm peak and 63.3 MW/m2 radiant exitance

The Stefan-Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter ? (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.[1] 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, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann.[2] 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.[3] "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures".[4] The Stefan-Boltzmann constant can be used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan-Boltzmann constant allows for temperature (K) to be converted to units for intensity (W?m-2), which is power per unit area.

The value of the Stefan-Boltzmann constant is given in SI units by

? = .[5]

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:[6]

? ? .

The value of the Stefan-Boltzmann constant is derivable as well as experimentally determinable; see Stefan-Boltzmann law for details. It can be defined in terms of the Boltzmann constant as

${\displaystyle \sigma ={\frac {2\pi ^{5}k_{\rm {B}}^{4}}{15h^{3}c^{2}}}={\frac {\pi ^{2}k_{\rm {B}}^{4}}{60\hbar ^{3}c^{2}}}=5.670\,374\,419...(exact)\,\times 10^{-8}\ {\textrm {J}}{\cdot }{\textrm {m}}^{-2}{\cdot }{\textrm {s}}^{-1}{\cdot }{\textrm {K}}^{-4},}$

where:

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

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

where:

Dimensional formula: M1T-3?-4

A related constant is the radiation constant (or radiation density constant) a which is given by:[7]

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

## References

1. ^ Krane, Kenneth (2012). Modern Physics. John Wiley & Sons. p. 81.
2. ^ "Stefan-Boltzmann Law". Encyclopædia Britannica.
3. ^ Halliday & Resnick (2014). Fundamentals of Physics (10th Ed). John Wiley and Sons. p. 1166.
4. ^ Eisberg, Resnick, Robert, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Ed) (PDF). John Wiley & Sons. Archived from the original (PDF) on 2014-02-26.
5. ^ "2018 CODATA Value: Stefan-Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved .
6. ^ Heat and Mass Transfer: a Practical Approach, 3rd Ed. Yunus A. Çengel, McGraw Hill, 2007
7. ^ Radiation constant from ScienceWorld