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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 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.
At the same time, gravity will attempt to contract the system even further, and will do so on a free-fall time = / /, where is the universal gravitational constant, is the gas density within the region, and = / is the gas number density for mean mass per particle (μ = 3.9 × 10 −24 g is appropriate for molecular hydrogen with 20% helium by ...
The general equation can then be written as [6] = + + (),. where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions.
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.
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 ...
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.