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.[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.


Not to be confused with 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 value of the Stefan–Boltzmann constant 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 ≈ 5.6704×10−5 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 ≈ 11.7×10−8 cal cm−2⋅day−1⋅K−4.

In US customary units the Stefan–Boltzmann constant is:[6]

σ ≈ 1.714×10−9 BTU⋅hr−1⋅ft−2⋅°R−4.

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

where:

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

where:

Dimensional formula: M1T−3Θ−4

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

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 2019-05-20.
  6. Heat and Mass Transfer: a Practical Approach, 3rd Ed. Yunus A. Çengel, McGraw Hill, 2007
  7. Radiation constant from ScienceWorld

LET ME BEGIN BY REQUESTING AN ENTIRELY SEPARATE, IF MARGINALLY CONNECTED ENTRY. My original query was for "Stefan's Constant" or "Stefan's Law," not the Stefan Boltzmamn Constant. Stefan's Constant/Law was cursed by the succession of the law Stefan and his student Boltzmann later derived. Stefan's Law and Constant are separate, as follows:

With Planck’s constant, h or ɦ, relating every spectral line of energy (heat/light) to a radiant frequency, a succession of formulas ensued for describing energy in the new universe. Firsm Slovenian physicist Ludwig Boltzmann derived his Boltzmann Constant, k, relating the average kinetic energy of gas particles to the gas’s temperature. It is today one of seven “defining constants,” continuously more precisely measured and variously combined with other descriptors, that determine basic scientific units. The Boltzmann Constant’s current value, which relates energy in Joules and degrees Kelvin, is agreed today to be exactly k = 1.380649×10−23 J/K. Jozef Stefan, Boltzmann’s teacher, also a Slovenian physicist, derived his constant, a, based on those of Planck and Boltzmann. Stefan’s Law relates the energy, e, that a blackbody radiates per unit area to its thermodynamic temperature T, demonstrating that the energy density e of a blackbody is proportional to aT4, i.e., Stefan’s Constant a times blackbody temperature to the fourth power. Solving then for a, Stefan’s Constant, a volume’s energy relative to temperature, is:

a = π2 k4 / 15 ɦ3 c3 = 7.57 x 10--16 J / m3 K4, where
ɦ is Planck’s constant h/2π,

k is the Boltzmann constant, c is the speed of light, and K is temperature in Kelvins. This very fundamental measure, a, employs an impressive body of essential variables, including: the speed of light, Planck’s energy-frequency constant, Boltzmann’s gas constant and Kelvin temperature. It is important to know at least that Stefan’s Constant can express energy and energy density, and that it is not the Stefan-Boltzmann Constant. Its existence has oddly dropped out of the literature and was not very settled when it was recognized. There seems to be no confusion about what the nature of a blackbody’s energy density is but physicists have been fast and loose about what they call the law that describes it. Not infrequently they have used Stefan for Stefan-Boltzmann and vice versa. Phillip M. Morse (Professor of Physics, MIT, derives Stefan's Law and correctly identifies it as such on page 81 of his Thermal Physics, Revised Edition, (W. A. Benjamin, Inc., NYC & Amsterdam, 1965). On page 376 of Fundamentals of statistical and thermal physics (McGraw-Hill Book Co., New York, 1965) R. Reif, Professor of Physics, University of California, Berkeley, correctly derives Stefan's Law for the energy density, then mistakenly calls it the Stefan-Boltzmann Law, which he later derives and correctly names on page 388. R. W. Christy and Agnar Pytte (Dartmouth College), in their The structure of matter (W. A. Benjamin, Inc., NYC & Amsterdam, 1965), derive the Stefan-Boltzmann Law on page 299 and, by a minority mistake, call it Stefan's Law. In his Quantum Mechanics (Princeton University Press, 1992) by P. J. E. Peebles, Albert Einstein Professor of Physics, Princeton University, derives energy density on page 21 and calls it the Stefan-Boltzmann Law, though the constant derived is Stefan's a. Thus, while Stefan’s Law was once not only better known but at least occasionally correctly named, it has stirred a good deal of nomenclatural confusion, which seems unabated today. Its relevance is that total energy in a blackbody is and, must be aT4. Unless energy on a blackbody cavity’s surfaces are equal, the Second Law of Thermodynamics predicts that energy will flow over an electromagnetic gradient (manifested by photons) from higher to lower energy, with the lower energy surface absorbing energy until that surface reaches the blackbody’s equilibrium temperature. By definition this equilibrium temperature is that of the higher energy surface (aT4), and cannot exceed it. The last qualification is critical. As the lower energy surface absorbs higher energy photons from the more energetic surface, its own energy increases, causing a diminished gradient flow from higher to lower energy until the lower energy surface equals the higher (blackbody) temperature and energy stops flowing. While energy has stopped flowing it exists throughout the cavity in the stable wave state. Here, for those who are interested, is the derivation of Stefan’s Constant. It equals twice the momentum, p, component perpendicular to the wall of photons reflected off the wall times half the perpendicular velocity component, where the perpendicular velocity component tells us how many photons strike the wall per second only half having a perpendicular velocity component toward the wall. The pressure P for a photon gas exerted in the x-direction on area A of the wall will be summed over all i = 1 to N photons: P = ½ ∑ 2 pix vix / V = ⅓ U/V = ⅓ e. U is just ∑ pix vix + piy viy + piz viz for photons. So for random motion, the summation over only one component is one-third of this sum. Note that radiation pressure on the wall does not require one photon incident on the wall to be absorbed and another emitted. The result is the same for a photon reflected off the wall, just like calculating the pressure on a wall for a perfect gas with the assumption of elastic collisions with the wall. A gas of point particles like this has a pressure of ⅔ e, however, since the average kinetic energy of the perfect gas particles is ½ the product of the momentum and the velocity and not the product of the momentum and the velocity as with photons. Thus, T dS = dU + P dV = d(Ve) + P dV = V de + e dV + ⅓ e dV = V (de/dT) dT + (4/3) e dV. Also, T dS = T (∂S/∂T)V dT + T (∂S/∂V)T dV, so matching up the terms for dT and for dV, (∂S/∂T)V = (V/T) de/dT (∂S/∂V)T = 4e/3T Since the order of differentiation does not matter, it is also the case that {∂[(∂S/∂T)V]/ ∂V}T = {∂[(∂S/∂V)T]/ ∂T}V {∂[(V/T) de/dT]/ ∂V}T = {∂[4e/3T]/ ∂T}V (1/T) de/dT = (4/3T) de/dT – 4e/3T2 de/dT = 4e/T, yielding e = aT4


This is from a very busy industrial physicist from whom I have taken notes. His name is Dr. Charles R. Anderson, who may be available to direct further inquiry on the matter if he has trouble finding time to do so himself.

I am an old journalist who does what young journalists today never do, question and DIG!

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