Prove that a good absorber is a good emitter
Answers
Explanation:
A Good Absorber is a Good Emitter
According to the Stefan-Boltzmann law, the energy radiated by a blackbody radiator per second per unit area is proportional to the fourth power of the absolute temperature and is given by

For hot objects other than ideal radiators, the law is expressed in the form:

where e is the emissivity of the object (e = 1 for ideal radiator). If the hot object is radiating energy to its cooler surroundings at temperature Tc, the net radiation loss rate takes the form

In this relationship the term with Tc represents the energy absorbed from the environment. This expression explicitly assumes that the same coefficient e applies to both the emission into the environment and the absorption from the environment. That is, a good emitter is a good absorber and vice versa; the same coefficient can be used to characterize both processes. Why is that true?
Perhaps the most fundamental conceptual way to approach this question is to observe that a hot object placed in a room must ultimately come to thermal equilibrium with the room. The hot object will initially emit more energy into the room than it absorbs from the room, but that will cause the temperature of the room to rise and the temperature of the object to drop. But when they reach the same temperature, we can conclude that the amount of energy absorbed on average is exactly the same as the energy emitted. That is, the expression above for net energy radiated to the environment must give us zero when T=Tc.
The above argument is based upon the Second Law of Thermodynamics in the form that states that heat will not spontaneously flow from a cold object to a hot object. If the absorption coefficient were higher than the emission coefficient for the object, then it could absorb net energy from the room even when its temperature were higher than the room.
But suppose you wanted to argue that a good absorber must be a good emitter based on the microscopic processes involving the atoms in the surface of an object. Then it becomes quantum question and involves the following ideas:
1. All electromagnetic radiation can be considered to be quantized, existing as photons with energy given by the Planck hypothesis, E=hf.
2. In order for a solid (or any matter, but I am assuming we are talking about solids) to absorb a photon of given energy hf , it must have a pair of energy levels separated by that amount of energy hf, so that the photon elevates the system from the lower member of the pair to the upper.
3. For visible light or near visible, then energy level pairs involved in most absorption are electron energy levels, so that when you absorb a green photon of photon energy 2.2 eV, you are causing an electron very near the surface of the solid to jump upward 2.2 eV. It can't do it unless there is a level at 2.2 eV up to receive it.
..QUESTION..
Prove that a good absorber is a good emitter.
..ANSWER..
Thermal radiation, process by which energy, in the form of electromagnetic radiation, is emitted by a heated surface in all directions and travels directly to its point of absorption at the speed of light; thermal radiation does not require an intervening medium to carry it.
Thermal radiation ranges in wavelength from the longest infrared rays through the visible-light spectrum to the shortest ultraviolet rays. The intensity and distribution of radiant energy within this range is governed by the temperature of the emitting surface. The total radiant heat energy emitted by a surface is proportional to the fourth power of its absolute temperature (the Stefan–Boltzmann law).
ㅤㅤThe rate at which a body radiates (or absorbs) thermal radiation depends upon the nature of the surface as well. Objects that are good emitters are also good absorbers (Kirchhoff’s radiation law). A blackened surface is an excellent emitter as well as an excellent absorber. If the same surface is silvered, it becomes a poor emitter and a poor absorber. A blackbody is one that absorbs all the radiant energy that falls on it. Such a perfect absorber would also be a perfect emitter.
ㅤㅤThe heating of the Earth by the Sun is an example of transfer of energy by radiation. The heating of a room by an open-hearth fireplace is another example. The flames, coals, and hot bricks radiate heat directly to the objects in the room with little of this heat being absorbed by the intervening air. Most of the air that is drawn from the room and heated in the fireplace does not reenter the room in a current of convection but is carried up the chimney together with the products of combustion.
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