
In
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, Kirchhoff's law of thermal radiation refers to wavelength-specific radiative
emission and
absorption by a material body in
thermodynamic equilibrium
Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...
, including radiative exchange equilibrium. It is a special case of
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.
"Reciprocal relations" occur betw ...
as a consequence of the
time reversibility
In mathematics and physics, time-reversibility is the property (mathematics), property of a process whose governing rules remain unchanged when the direction of its sequence of actions is reversed.
A deterministic process is time-reversible if th ...
of microscopic dynamics, also known as
microscopic reversibility.
A body at
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
radiates
electromagnetic energy
In physics, and in particular as measured by radiometry, radiant energy is the energy of electromagnetic and gravitational radiation. As energy, its SI unit is the joule (J). The quantity of radiant energy may be calculated by integrating radia ...
. A perfect
black body in thermodynamic equilibrium absorbs all light that strikes it, and radiates energy according to a unique law of radiative emissive power for temperature (
Stefan–Boltzmann law
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 Lu ...
), universal for all perfect black bodies. Kirchhoff's law states that:
Here, the dimensionless coefficient of absorption (or the absorptivity) is the fraction of incident light (power) at each spectral frequency that is absorbed by the body when it is radiating and absorbing in thermodynamic equilibrium.
In slightly different terms, the emissive power of an arbitrary opaque body of fixed size and shape at a definite temperature can be described by a dimensionless ratio, sometimes called the
emissivity
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
: the ratio of the emissive power of the body to the emissive power of a black body of the same size and shape at the same fixed temperature. With this definition, Kirchhoff's law states, in simpler language:
In some cases, emissive power and absorptivity may be defined to depend on angle, as described below. The condition of thermodynamic equilibrium is necessary in the statement, because the equality of emissivity and absorptivity often does not hold when the material of the body is not in thermodynamic equilibrium.
Kirchhoff's law has another corollary: the emissivity cannot exceed one (because the absorptivity cannot, by
conservation of energy
The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
), so it is not possible to thermally radiate more energy than a black body, at equilibrium. In
negative luminescence 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 thermodynamic equilibrium.
Principle of detailed balance
Kirchhoff's law of thermal radiation has a refinement in that not only is thermal emissivity equal to absorptivity, it is equal ''in detail''. Consider a leaf. It is a poor absorber of green light (around 470 nm), which is why it looks green. By the principle of detailed balance, it is also a poor emitter of green light.
In other words, if a material, illuminated by black-body radiation of temperature
, is dark (well absorbing) at a certain frequency
, then its own thermal radiation will be strong (well emitting) at the same frequency
and the same temperature
.
More generally, all
intensive properties are balanced in detail. So for example, the absorptivity at a certain incidence direction, for a certain frequency, of a certain polarization, is the same as the emissivity at the same direction, for the same frequency, of the same polarization. This is the principle of detailed balance.
History
Before Kirchhoff's law was recognized, it had been experimentally established that a good absorber is a good emitter, and a poor absorber is a poor emitter. Naturally, a good reflector must be a poor absorber. This is why, for example, lightweight
emergency thermal blankets are based on reflective
metallic coatings: they lose little heat by radiation.
Kirchhoff's great insight was to recognize the universality and uniqueness of the function that describes the black body emissive power. But he did not know the precise form or character of that universal function. Attempts were made by
Lord Rayleigh and
Sir James Jeans 1900–1905 to describe it in classical terms, resulting in
Rayleigh–Jeans law
In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength ''λ'', it is
B_\l ...
. This law turned out to be inconsistent yielding the
ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century and early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of en ...
. The correct form of the law was found by
Max Planck
Max Karl Ernst Ludwig Planck (; ; 23 April 1858 – 4 October 1947) was a German Theoretical physics, theoretical physicist whose discovery of energy quantum, quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial con ...
in 1900, assuming quantized emission of radiation, and is termed
Planck's law. This marks the advent of
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.
Theory
In a blackbody enclosure that contains electromagnetic radiation with a certain amount of energy at thermodynamic equilibrium, this "
photon gas" will have a
Planck distribution of energies.
One may suppose a second system, a cavity with walls that are opaque, rigid, and not perfectly reflective to any wavelength, to be brought into connection, through an optical filter, with the blackbody enclosure, both at the same temperature. Radiation can pass from one system 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 the first system. If the optical filter passed only that frequency band, 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 requires that there can be no net transfer of heat 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. Hence absorptivity and emissivity must be equal. The
absorptivity of the wall is the ratio of the energy absorbed by the wall to the energy incident on the wall, for a particular wavelength. Thus the absorbed energy is
where
is the intensity of black-body radiation at wavelength
and temperature
. Independent of the condition of thermal equilibrium, the
emissivity
The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
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. The emitted energy is thus
where
is the emissivity at wavelength
. For the maintenance of thermal equilibrium, 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 Kirchhoff'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 Kirchhoff'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.
Planck's derivation

Historically, Planck derived the black body radiation law and detailed balance according to a classical thermodynamic argument, with a single heuristic step, which was later interpreted as a quantization hypothesis.
In Planck's set up, he started with a large
Hohlraum at a fixed temperature
. At thermal equilibrium, the Hohlraum is filled with a distribution of EM waves at thermal equilibrium with the walls of the Hohlraum. Next, he considered connecting the Hohlraum to a single small
resonator
A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a reso ...
, such as Hertzian resonators. The resonator reaches a certain form of thermal equilibrium with the Hohlraum, when the spectral input into the resonator equals the spectral output at the resonance frequency.
Next, suppose there are two Hohlraums at the same fixed temperature
, then Planck argued that the thermal equilibrium of the small resonator is the same when connected to either Hohlraum. For, we can disconnect the resonator from one Hohlraum and connect it to another. If the thermal equilibrium were different, then we have just transported energy from one to another, violating the second law. Therefore, the spectrum of all black bodies are identical at the same temperature.
Using a heuristic of quantization, which he gleaned from Boltzmann, Planck argued that a resonator tuned to frequency
, with average energy
, would contain entropy