Kirchhoff's Law of Thermal Radiation - Theory

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 a 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. The absorptivity 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 emissivity 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.