Search results
Results from the WOW.Com Content Network
Exact solutions to the Boltzmann equations have been proven to exist in some cases; [15] this analytical approach provides insight, but is not generally usable in practical problems. Instead, numerical methods (including finite elements and lattice Boltzmann methods ) are generally used to find approximate solutions to the various forms of the ...
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.
Combining the formulas for the Schwarzschild radius of the black hole, the Stefan–Boltzmann law of blackbody radiation, the above formula for the temperature of the radiation, and the formula for the surface area of a sphere (the black hole's event horizon), several equations can be derived. The Hawking radiation temperature is: [2] [20] [21]
However, for an ideal gas the standard Chapman–Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under shock wave conditions. Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation – his famous H-theorem.
Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ.This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the ...
It is a linearized Poisson–Boltzmann model, which assumes an extremely simplified model of electrolyte solution but nevertheless gave accurate predictions of mean activity coefficients for ions in dilute solution. The Debye–Hückel equation provides a starting point for modern treatments of non-ideality of electrolyte solutions.
From Boltzmann's equation one may also analytically derive (analytical) mathematical models for properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas).
The surface emits a radiative flux density F according to the Stefan–Boltzmann law: = where σ is the Stefan–Boltzmann constant. A key to understanding the greenhouse effect is Kirchhoff's law of thermal radiation. At any given wavelength the absorptivity of the atmosphere will be equal to the emissivity. Radiation from the surface could be ...