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The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan , who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.
Boltzmann constant: The Boltzmann constant, k, is one of seven fixed constants defining the International System of Units, the SI, with k = 1.380 649 x 10 −23 J K −1. The Boltzmann constant is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule).
The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. [ 8 ] : 163 For a fluid consisting of only one kind of particle, the number density n is given by n = ∫ f d 3 p . {\displaystyle n=\int f\,d^{3}\mathbf {p} .}
The solution of the above integral yields a remarkably elegant equation for the total emissive power of a blackbody, the Stefan-Boltzmann law, which is given as, = where is the Steffan-Boltzmann constant.
Z is the partition function, corresponding to the denominator in equation 1; m is the molecular mass of the gas; T is the thermodynamic temperature; k B is the Boltzmann constant. This distribution of N i : N is proportional to the probability density function f p for finding a molecule with these values of momentum components, so:
However, because black-body radiation increases rapidly with temperature (as the fourth power of temperature, given by the Stefan–Boltzmann law), radiation pressure due to the temperature of a very hot object (or due to incoming black-body radiation from similarly hot surroundings) can become significant. This is important in stellar interiors.
The universal form of the bound was originally found by Jacob Bekenstein in 1981 as the inequality [1] [2] [3], where S is the entropy, k is the Boltzmann constant, R is the radius of a sphere that can enclose the given system, E is the total mass–energy including any rest masses, ħ is the reduced Planck constant, and c is the speed of light.
Boltzmann's equation = is the realization that the entropy is proportional to with the constant of proportionality being the Boltzmann constant. Using the ideal gas equation of state ( PV = NkT ), It follows immediately that β = 1 / k T {\displaystyle \beta =1/kT} and α = − μ / k T {\displaystyle \alpha =-\mu /kT} so that the ...