# Kirchhoff’s Law of Thermal Radiation

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

As a result of this law, heat cannot spontaneously flow from cold system to hot system and the second law of thermodynamics is still satisfied.

In general, both the emissivity, ε,  and the absorptivity, α, of a surface depend on the temperature and the wavelength of the radiation. Kirchhoff’s law of thermal radiation, postulated by a German physicist Gustav Robert Kirchhoff, states that the emissivity and the absorptivity of a surface at a given temperature and wavelength are equal.

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

This law must be also valid in order to satisfy the Second Law of Thermodynamics. As was written, all bodies above absolute zero temperature radiate some heat. Two objects radiate heat toward each other. But what if a colder object with high emissivity radiates toward a hotter object with very low emissivity? This seems to violate the Second Law of Thermodynamics, which states that heat cannot spontaneously flow from cold system to hot system without external work being performed on the system. The paradox is resolved by the fact that each body must be in direct line of sight of the other to receive radiation from it. Therefore, whenever the cool body is radiating heat to the hot body, the hot body must also be radiating heat to the cool body. Moreover, the hot body will radiate more energy than cold body. The case of different emissivities is solved by the Kirchhoff’s Law of thermal radiation, which states that object with low emissivity have also low absorptivity. As a result, heat cannot spontaneously flow from cold system to hot system and the second law is still satisfied.

Emissivity
The emissivity, ε, of the surface of a material is its effectiveness in emitting energy as thermal radiation and varies between 0.0 and 1.0.

By definition, a blackbody in thermal equilibrium has an emissivity of ε = 1.0. Real objects do not radiate as much heat as a perfect black body. They radiate less heat than a black body and therefore are called gray bodies. To take into account the fact that real objects are gray bodies, the Stefan-Boltzmann law must include emissivity. Quantitatively, emissivity is the ratio of the thermal radiation from a surface to the radiation from an ideal black surface at the same temperature as given by the Stefan–Boltzmann law. Emissivity is simply a factor by which we multiply the black body heat transfer to take into account that the black body is the ideal case.

The surface of a blackbody emits thermal radiation at the rate of approximately 448 watts per square metre at room temperature (25 °C, 298.15 K). Real objects with emissivities less than 1.0 (e.g. copper wire) emit radiation at correspondingly lower rates (e.g. 448 x 0.03 = 13.4 W/m2). Emissivity plays important role in heat transfer problems. For example, solar heat collectors incorporate selective surfaces that have very low emissivities. These collectors waste very little of the solar energy through emission of thermal radiation.

Absorptivity
Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, value of absorptivity is in the range 0 < α < 1.

From its definition, a blackbody, which is an idealized physical body, absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. That is, a blackbody is a perfect absorber. Since for real objects the absorptivity is less than unity, a real object can not absorb all incident light. The incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In general, the absorptivity and the emissivity are interconnected by the Kirchhoff’s Law of thermal radiation, which states:

For an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity.

emissivity ε = absorptivity α

Note that visible radiation occupies a very narrow band of the spectrum from 0.4 to 0.76 nm, we cannot make any judgments about the blackness of a surface on the basis of visual observations. For example, consider white paper that reflects visible light and thus appear white. On the other hand it is essentially black for infrared radiation (absorptivity α = 0.94) since they strongly absorb long-wavelength radiation.

References:
Heat Transfer:
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3. U.S. Department of Energy, Thermodynamics, Heat Transfer and Fluid Flow. DOE Fundamentals Handbook, Volume 2 of 3. May 2016.

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