Stefan–Boltzmann 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 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.^[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; it was formally derived in 1884 by his former student and collaborator, the 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 quickly 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 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 temperature (K) to be converted to units for 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

σ = 5.670374419...×10⁻⁸ W⋅m⁻²⋅K⁻⁴.^[5]

In cgs units the Stefan–Boltzmann constant is

σ = 5.670374...×10⁻⁵ erg⋅cm⁻²⋅s⁻¹⋅K⁻⁴.

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

σ = 1.170937...×10⁻⁷ cal cm⁻²⋅day⁻¹⋅K⁻⁴.

In US customary units the Stefan��Boltzmann constant is^[6]

σ = 1.713441...×10⁻⁹ BTU⋅hr⁻¹⋅ft⁻²⋅°R⁻⁴.

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

σ = 5.67037441918442945397099673188923087584012297029130...×10⁻⁸ J⋅m⁻²⋅s⁻¹⋅K⁻⁴.

The CODATA recommended value [ref?] 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: M¹T⁻³Θ⁻⁴

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

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}} .

References[]

^ Krane, Kenneth (2012). Modern Physics. John Wiley & Sons. p. 81.
^ "Stefan-Boltzmann Law". Encyclopædia Britannica.
^ Halliday & Resnick (2014). Fundamentals of Physics (10th ed.). John Wiley and Sons. p. 1166.
^ Eisberg, Resnick, Robert, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (PDF) (2nd ed.). John Wiley & Sons. Archived from the original (PDF) on 2014-02-26.
^ "2018 CODATA Value: Stefan–Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
^ Çengel, Yunus A. (2007). Heat and Mass Transfer: a Practical Approach (3rd ed.). McGraw Hill.
^ Radiation constant from ScienceWorld

External links[]

Media related to Stefan–Boltzmann constant at Wikimedia Commons

[ModernKrane-1] Krane, Kenneth (2012). Modern Physics. John Wiley & Sons. p. 81.

[Britannica-2] "Stefan-Boltzmann Law". Encyclopædia Britannica.

[Fundamentals-3] Halliday & Resnick (2014). Fundamentals of Physics (10th ed.). John Wiley and Sons. p. 1166.

[Eisberg,_Resnick-4] Eisberg, Resnick, Robert, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (PDF) (2nd ed.). John Wiley & Sons. Archived from the original (PDF) on 2014-02-26.

[physconst-sigma-5] "2018 CODATA Value: Stefan–Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

[6] Çengel, Yunus A. (2007). Heat and Mass Transfer: a Practical Approach (3rd ed.). McGraw Hill.

[7] Radiation constant from ScienceWorld

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