Kirchhoff's law of thermal radiation
Encyclopedia
- See also Kirchhoff's lawsKirchhoff's lawsThere are several Kirchhoff's laws, all named after Gustav Robert Kirchhoff:* Kirchhoff's circuit laws* Kirchhoff's law of thermal radiation* Kirchhoff's equations* Kirchhoff's three laws of spectroscopy* Kirchhoff's law of thermochemistry-See also:...
for other laws named after Kirchhoff.
In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, Kirchhoff's law of thermal radiation, or Kirchhoff's law for short, is a general statement equating emission and absorption in heated objects, proposed by Gustav Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...
in 1859, following from general considerations of thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
and detailed balance
Detailed balance
The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes : At equilibrium, each elementary process should be equilibrated by its reverse process....
.
An object at some non-zero temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
radiates electromagnetic energy. If it is a perfect black body
Black body
A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...
, absorbing all light that strikes it, it radiates energy according to the black-body radiation formula. More generally, it is a "grey body" that radiates with some emissivity
Emissivity
The emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...
multiplied by the black-body formula. Kirchhoff's law states that:
- At thermal equilibrium, the emissivityEmissivityThe emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...
of a body (or surface) equals its absorptivityAbsorptivityThe term absorptivity has two meanings:*In chemistry, it usually refers to Molar absorptivity : the constant \epsilon used in the Beer-Lambert law, A=\epsilon c l, where A is the absorbance, c is the concentration of the solution, and l is the path length....
.
Here, the absorptivity (or absorbance) is the fraction of incident light (power) that is absorbed by the body/surface. In the most general form of the theorem, this power must be integrated over all wavelengths and angles. In some cases, however, emissivity and absorption may be defined to depend on wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
and angle, as described below.
Kirchhoff's Law has a corollary
Corollary
A corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
: the emissivity cannot exceed one (because the absorptivity cannot, by conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
), so it is not possible to thermally radiate more energy than a black body, at equilibrium. In negative luminescence
Negative luminescence
Negative luminescence is a physical phenomenon by which an electronic device emits less thermal radiation when an electric current is passed through it than it does in thermal equilibrium . When viewed by a thermal camera, an operating negative luminescent device looks colder than its environment.-...
the angle and wavelength integrated absorption exceeds the material's emission, however, such systems are powered by an external source and are therefore not in thermal equilibrium.
This theorem is sometimes informally stated as a poor reflector is a good emitter, and a good reflector is a poor emitter. It is why, for example, lightweight emergency thermal blankets are based on reflective metallic coatings
Optical coating
An optical coating is one or more thin layers of material deposited on an optical component such as a lens or mirror, which alters the way in which the optic reflects and transmits light. One type of optical coating is an antireflection coating, which reduces unwanted reflections from surfaces, and...
: they lose little heat by radiation.
Theory
If we consider an ideal situation in which an enclosure with perfectly reflecting walls contains radiation with a certain amount of energy, then at equilibrium, this "photon gas" will have a Planck distribution of energies. This will be true even though the walls are perfectly reflecting due to the very small amount of interaction between the photons themselves. The equilibration process will take a considerable amount of time, but the distribution of energies and radiation density will ultimately approach a Planck distribution.The next step is to realize that, as a result of the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
, any enclosure at thermal equilibrium must also have a Planck distribution of radiation. If this were not true, then we could bring that system in contact with the above ideal system, both at the same temperature, and by connecting them through an optical filter, we can have a net amount of radiation pass from one body to the other. For example, suppose in the second system, the density of photons at narrow frequency band around wavelength
were higher than that of a black body at that temperature. If a filter that passed only that frequency band was inserted in an opening that connected the two bodies, then there would be a net transfer of photons, and their energy, from the second system to the first. This is in violation of the second law of thermodynamics, which states that there can be no net transfer of energy between two bodies at the same temperature.In the second system, therefore, at each frequency, the walls must absorb and emit energy in such a way as to maintain the black body distribution. The absorptivity
Absorptivity
The term absorptivity has two meanings:*In chemistry, it usually refers to Molar absorptivity : the constant \epsilon used in the Beer-Lambert law, A=\epsilon c l, where A is the absorbance, c is the concentration of the solution, and l is the path length....
is the ratio of the energy absorbed by the wall to the energy incident on the wall, for a particular wavelength. This will be proportional to 
where 
is the intensity of black body radiation at wavelength 
and temperature 
. The emissivityEmissivity
The emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...
of the wall is defined as the ratio of emitted energy to the amount that would be radiated if the wall were a perfect black body. That will be
where 
is the emissivity at wavelength 
. These two quantities must be equal, or else the distribution of photon energies in the cavity will deviate from that of a black body. This yields Kirchoff's law:
By a similar, but more complicated argument, it can be shown that, since black body radiation is equal in every direction (isotropic), the emissivity and the absorptivity, if they happen to be dependent on direction, must again be equal for any given direction.
Average and overall absorptivity and emissivity data are often given for materials with values which differ from each other. For example, white paint is quoted as having an absorptivity of 0.16, while having an emissivity of 0.93. This is because the absorptivity is averaged with weighting for the solar spectrum, while the emissivity is weighted for the emission of the paint itself at normal ambient temperatures. The absorptivity quoted in such cases is being calculated by:
while the average emissivity is given by:
Where
is the emission spectrum of the sun, and 
is the emission spectrum of the paint. Although, by Kirchoff's law, 
in the above equations, the above averages 
and 
are not generally equal to each other. The white paint will serve as a very good insulator against solar radiation, because it is very reflective of the solar radiation, and although it therefore emits poorly in the solar band, its temperature will be around room temperature, and it will emit whatever radiation it has absorbed in the infrared, where its emission coefficient is high.