Kirchhoff's law of thermal radiation

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. This law turned out to be inconsistent yielding the ultraviolet catastrophe. The correct form of the law was found by Max Planck in 1900, assuming quantized emission of radiation, and is termed Planck's law.[7] This marks the advent of quantum mechanics.

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.[8]

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 ${\displaystyle \lambda }$ 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.[9] Hence absorptivity and emissivity must be equal. The absorptivity ${\displaystyle \alpha _{\lambda }}$ 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 ${\displaystyle \alpha _{\lambda }E_{b\lambda }(\lambda ,T)}$ where ${\displaystyle E_{b\lambda }(\lambda ,T)}$ is the intensity of black body radiation at wavelength ${\displaystyle \lambda }$ and temperature ${\displaystyle T}$. Independent of the condition of thermal equilibrium, 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. The emitted energy is thus ${\displaystyle \varepsilon _{\lambda }E_{b\lambda }(\lambda ,T)}$ where ${\displaystyle \varepsilon _{\lambda }}$ is the emissivity at wavelength ${\displaystyle \lambda }$. 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:

${\displaystyle \alpha _{\lambda }=\varepsilon _{\lambda }}$

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.[9]

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.[10] 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:

${\displaystyle \alpha _{\mathrm {sun} }=\displaystyle {\frac {\int _{0}^{\infty }\alpha _{\lambda }(\lambda )I_{\lambda \mathrm {sun} }(\lambda )\,d\lambda }{\int _{0}^{\infty }I_{\lambda \mathrm {sun} }(\lambda )\,d\lambda }}}$

while the average emissivity is given by:

${\displaystyle \varepsilon _{\mathrm {paint} }={\frac {\int _{0}^{\infty }\varepsilon _{\lambda }(\lambda ,T)E_{b\lambda }(\lambda ,T)\,d\lambda }{\int _{0}^{\infty }E_{b\lambda }(\lambda ,T)\,d\lambda }}}$

Where ${\displaystyle I_{\lambda \mathrm {sun} }}$ is the emission spectrum of the sun, and ${\displaystyle \varepsilon _{\lambda }E_{b\lambda }(\lambda ,T)}$ is the emission spectrum of the paint. Although, by Kirchhoff's law, ${\displaystyle \varepsilon _{\lambda }=\alpha _{\lambda }}$ in the above equations, the above averages ${\displaystyle \alpha _{\mathrm {sun} }}$ and ${\displaystyle \varepsilon _{\mathrm {paint} }}$ 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.

Gustav Kirchhoff stated his law in several papers in 1859 and 1860, and then in 1862 in an appendix to his collected reprints of those and some related papers.[13]

Prior to Kirchhoff's studies, it was known that for total heat radiation, the ratio of emissive power to absorptive ratio was the same for all bodies emitting and absorbing thermal radiation in thermodynamic equilibrium. This means that a good absorber is a good emitter. Naturally, a good reflector is a poor absorber. For wavelength specificity, prior to Kirchhoff, the ratio was shown experimentally by Balfour Stewart to be the same for all bodies, but the universal value of the ratio had not been explicitly considered in its own right as a function of wavelength and temperature.

Kirchhoff's original contribution to the physics of thermal radiation was his postulate of a perfect black body radiating and absorbing thermal radiation in an enclosure opaque to thermal radiation and with walls that absorb at all wavelengths. Kirchhoff's perfect black body absorbs all the radiation that falls upon it.

Every such black body emits from its surface with a spectral radiance that Kirchhoff labeled I (for specific intensity, the traditional name for spectral radiance).

Kirchhoff's postulated spectral radiance I was a universal function, one and the same for all black bodies, only depending on wavelength and temperature.

The precise mathematical expression for that universal function I was very much unknown to Kirchhoff, and it was just postulated to exist, until its precise mathematical expression was found in 1900 by Max Planck. It is nowadays referred to as Planck's law.

Then, at each wavelength, for thermodynamic equilibrium in an enclosure, opaque to heat rays, with walls that absorb some radiation at every wavelength:

For an arbitrary body radiating and emitting thermal radiation, the ratio E / A between the emissive spectral radiance, E, and the dimensionless absorptive ratio, A, is one and the same for all bodies at a given temperature. That ratio E / A is equal to the emissive spectral radiance I of a perfect black body, a universal function only of wavelength and temperature.

• Sakuma–Hattori equation
• Wien's displacement law

• Evgeny Lifshitz and L. P. Pitaevskii, Statistical Physics: Part 2, 3rd edition (Elsevier, 1980).
• F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill: Boston, 1965).