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Stefan–Boltzmann constant

Not to be confused with 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, and it was later derived in 1884 by 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 temperature (K) to be converted to units for intensity (W⋅m−2), which is power per unit area.

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

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

σ =5.670374419...×10−8 W⋅m−2⋅K−4.

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 calcm−2day−1K−4:

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

In US customary units the Stefan–Boltzmann constant is

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

σ = 2 π 5 k B 4 15 h 3 c 2 = π 2 k B 4 60 3 c 2 , {\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}}}\,,}
where

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:

σ = 2 π 5 R 4 15 h 3 c 2 N A 4 = 32 π 5 h R 4 R 4 15 A r ( e ) 4 M u 4 c 6 α 8 , {\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−3Θ−4

A related constant is the radiation constant (or radiation density constant) a {\displaystyle a} , which is given by

a = 4 σ c = 7.5657 × 10 15 e r g c m 3 K 4 = 7.5657 × 10 16 J m 3 K 4 . {\displaystyle 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}} .}
  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(PDF) (2nd ed.). 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. Retrieved2019-05-20.
  6. Çengel, Yunus A. (2007). Heat and Mass Transfer: a Practical Approach (3rd ed.). McGraw Hill.
  7. Radiation constant from ScienceWorld

Stefan–Boltzmann constant
Stefan Boltzmann constant Language Watch Edit Not to be confused with Boltzmann constant The Stefan Boltzmann constant also Stefan s constant a physical constant denoted by the Greek letter s 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 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 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 exitanceValue EditSince the 2019 redefinition of the SI base units the Stefan Boltzmann constant is given exactly rather than measured in experiment The value is given in SI units by s 5 670374 419 10 8 W m 2 K 4 5 In cgs units the Stefan Boltzmann constant is s 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 s 1 170937 10 7 cal cm 2 day 1 K 4 In US customary units the Stefan Boltzmann constant is 6 s 1 713441 10 9 BTU hr 1 ft 2 R 4 The Stefan Boltzmann constant is defined in terms of other fundamental constants ass 2 p 5 k B 4 15 h 3 c 2 p 2 k B 4 60 ℏ 3 c 2 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 where kB 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 s 5 670374 419 184 429 453 970 996 731 889 230 875 840 122 970 291 30 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 s 2 p 5 R 4 15 h 3 c 2 N A 4 32 p 5 h R 4 R 4 15 A r e 4 M u 4 c 6 a 8 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 R is the universal gas constant NA is the Avogadro constant R is the Rydberg constant Ar e is the relative atomic mass of the electron Mu is the molar mass constant 1 g mol by definition a is the fine structure constant Dimensional formula M1T 38 4 A related constant is the radiation constant or radiation density constant a displaystyle a which is given by 7 a 4 s c 7 5657 10 15 e r g c m 3 K 4 7 5657 10 16 J m 3 K 4 displaystyle 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 Edit Krane Kenneth 2012 Modern Physics John Wiley amp Sons p 81 Stefan Boltzmann Law Encyclopaedia Britannica Halliday amp 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 amp 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 Cengel Yunus A 2007 Heat and Mass Transfer a Practical Approach 3rd ed McGraw Hill Radiation constant from ScienceWorldExternal links Edit Media related to Stefan Boltzmann constant at Wikimedia Commons Retrieved from https en wikipedia org w index php title Stefan Boltzmann constant amp oldid 1053285940, wikipedia, wiki, book,

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