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The Stefan Boltzmann law is a physical law, which indicates the achievement radiated thermally from a black body as a function of its temperature.

Stefan Boltzmann law

Each body, whose temperature lies over the absolute zero, sends radiant heat. A black body is an idealized body, which can absorb all radiation meeting it completely (absorption factor = 1). After the kirchhoffschen radiation law reached therefore also its emissivity the value 1 and it sends concerned the thermal achievement maximally possible at the temperature. The Stefan Boltzmann law indicates, which radiating power emits P a black body of the surface A and the absolute temperature T. It is

P = \ sigma \ cdot A \ cdot T^4

with the Stefan Boltzmann constants \ sigma. The radiating power of a black body is thus proportional to the fourth power of its absolute temperature: a doubling of the temperature causes that the eradiated power rises around the factor 16.

The Stefan Boltzmann constant is a natural constant and its numerical value amounts to in accordance with CODATA 2000

\ sigma = \ frac {2 \ pi^5k^4} {15h^3c^2} = (5 {,} 670,400 \ pm 0 {,} 000,040) \, \ cdot \, 10^ {- 8} \, \ mathrm {\ frac {W} {m^2 K^4}}.

K are (not with \ sigma the one which can be confounded) the Boltzmann constant, h the Planck quantum of action and C the speed of light.

Derivation

For derivation one proceeds from the spectral radiance of a black body and integrates her both over the entire semi-infinite space, into which the regarded two dimensional element radiates, and over all frequencies, in order to receive the specific radiant emittance M^o (T):

M^o (T) = \ int_ {\ nu=0} ^ {\ infty} \ int_ {semi-infinite space} \ frac {2 h \ nu^ {3}} {c^2} \ frac {1} {e^ {\ left (\ frac {h \ nu} {kT} \ right)}- 1} \, \ cos (\ beta) \, \ sin (\ beta) \ mathrm {D} \ beta \ mathrm {D} \ varphi \, \ mathrm {D} \ nu.

The cosine factor considers with the fact the circumstance that with radiation into any through \ beta and \ varphi given direction only the projection standing perpendicularly on this direction appears \ cos (\ beta) \ mathrm {to D} A of the surface \ mathrm {D} A as effective jet surface. The term \ sin (\ beta) \ mathrm {D} \ beta \ mathrm {D} \ varphi is a solid angle element.

Since the black body is in principle a vague emitter and its spectral radiance therefore direction-independent, the integral has the value over the semi-infinite space \ pi. For the integration over the frequencies it is to be noted that

\ int_ {0} ^ {\ infty} \ frac {x^3} {e^ {x} - 1} \, \ mathrm {D} x = \ frac {\ pi^4} {15}.

Still if one integrates in such a way received specific radiant emittance M^o (T) over the radiating surface, one receives the Stefan Boltzmann law in the form indicated above.

Non--black bodies

The Stefan Boltzmann law applies only to black bodies. If a non-black body is given, which radiates direction-independent (Lambert emitter so mentioned) and its emissivity \ varepsilon (T) for all frequencies the same value has (grey body so mentioned), then is radiating power delivered of this

P = \ varepsilon (T) \ cdot \ sigma \ cdot A \ cdot T^4Grauer Lambert emitter.

If the emissivity is temperature-dependent, then the entire radiating power is no longer strictly proportional to the fourth power of the absolute temperature because of this additional temperature dependence.

For an emitter, with which the direction-independentness and/or the frequency-independentness are not given to the emission, the integral must be computed individually using the regularities concerned for the determination of the hemispherical entire eating ion degree \ epsilon (T). Many bodies deviate only little from the ideal Lambert emitter; if the emissivity in the frequency range, in which the body delivers a noticeable portion of its radiating power, leaves itself to only little varied, the Stefan Boltzmann law at least approach to use.

Historical

The Stefan Boltzmann law was experimentally discovered in the year 1879 by Josef Stefan and deduced 1884 by Ludwig Boltzmann theoretically by thermodynamic considerations from the classical electromagnetic theory of the radiation. In the year 1900, thus 21 years after the Stefan Boltzmann law, discovered Max Planck the Planck radiation law designated after it, from which the Stefan Boltzmann law follows simply by integration over all directions and wavelengths. The Planck radiation law could attribute the Stefan Boltzmann constant also for the first time with the introduction of the quantum of action h to fundamental natural constants.

Example

Outside of the terrestrial atmosphere a surface aligned to the sun receives an irradiancy from S = 1,367 W/m2 (solar constant). One determines the temperature T of the sun surface on the assumption that the sun is in sufficient approximation a black body. The sun radius amounts to R = 6.963 108m, the middle distance between earth and sun is D = 1.496 1011 M.

The radiating power P delivered by the sun surface penetrates one concentrically around the sun put socket pad of the radius D with the irradiancy S, thus altogether P=4pD2 amounts to S = 3.845 1026 W (luminosity of the sun). After the Stefan Boltzmann law the temperature of the radiating surface amounts to

T = \ sqrt [4] {\ frac {P} {\ sigma A}} = \ sqrt [4] {\ frac {S \ cdot 4 \ pi D^2} {\ sigma \ cdot 4 \ pi R^2}} = \ sqrt [4] {\ frac {S \ cdot D^2} {\ sigma \ cdot R^2}} = \ sqrt [4] {\ frac {1367 \ cdot 2 {,} 238 \ cdot 10^ {22}} {5 {,} 670 \ cdot 10^ {- 8} \, \ cdot \, 4 {,} 844 \ 10^ cdot {17}}} \; \ mathrm {K} = 5777 \, \ mathrm {K}

In such a way determined temperature of the sun surface is called effective temperature. The temperature, which a equivalent large black body must have, is to be delivered around the same radiating power as the sun.

See also

The Stefan Boltzmann law makes a statement about altogether the radiating power delivered by a black body on all frequencies. The allocation on individual frequencies and/or wavelengths is described by the Planck radiation law.

Literature

  • Stefan, J.: Over the relationship between the radiant heat and the temperature, in: Minutes of the meeting of the mathematical-scientific Classe of the imperial academy of the sciences, Bd. 79 (Vienna 1879), P. 391-428.
  • Boltzmann, L.: Derivative of the Stefan' law, concerning the dependence of the radiant heat on the temperature from the electromagnetischen light theory, in: Annals of physics and chemistry, Bd. 22 (1884), P. 291-294

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