Physics · Ch 10 — Thermal Properties of Matter

Kirchhoff's Law of Radiation

10.12

Section 10.11 introduced two separate properties of any surface -- its absorptive power $a$ and its

emissive power $e$ -- and noted, without proof, that good absorbers also tend to be good emitters.

Kirchhoff's law of radiation is the precise statement of exactly why this pairing is not a mere

coincidence but a physical necessity.

Kirchhoff's law states that, for any surface in thermal equilibrium, at any given wavelength and

temperature, the ratio of that surface's emissive power to its absorptive power is one and the same

fixed value for every surface -- and that this common value is precisely equal to the emissive power that

a perfectly black body would have at that same wavelength and temperature:

\frac{e_{\text{surface}}}{a_{\text{surface}}} = e_{\text{black body}} = \text{a fixed value, the same for every surface}

The immediate and important consequence of this is that a surface's emissive power and absorptive power

are not independent of one another -- a surface that is a good absorber ( $a$ close to $1$ ) at a given

wavelength must also be a good emitter ( $e$ close to $1$ ) at that same wavelength, and a surface that is

a poor absorber must also be a poor emitter. Neither property can be large while the other stays small,

for a body in thermal equilibrium.

This linkage follows from a simple thermal-equilibrium argument. Imagine two surfaces facing one another

inside a closed, insulated enclosure, left long enough to reach a common, steady temperature. If one

surface absorbed strongly but emitted only weakly (or vice versa), it would continuously gain more radiant

energy than it gave back, and so would keep warming relative to the other surface -- but this would

violate the basic requirement that, once thermal equilibrium is reached, no further net change of

temperature can occur anywhere in an isolated system. The only way this contradiction is avoided, for

every surface and at every wavelength, is if a surface's ability to absorb radiation and its ability to

emit radiation are locked together in the fixed ratio Kirchhoff's law describes.

A practical, everyday illustration: place a black, matte cooking pot and a bright, polished pot of

otherwise identical size, material, and contents on identical stove burners. The black pot heats up

noticeably faster (its high absorptive power lets it take in the stove's radiant heat efficiently) -- and,

by Kirchhoff's law, that same high absorptive power is inseparably linked to an equally high emissive

power, so once removed from the stove, the black pot also cools down faster than the polished one,

losing its heat by radiation more readily. The polished pot, conversely, both heats up and cools down more

slowly, for the very same underlying reason.

Kirchhoff's law also accounts for the Fraunhofer lines, the numerous narrow dark lines that cross the …