D.28 Kirch­hoff’s law

Sup­pose you have a ma­te­r­ial in ther­mo­dy­namic equi­lib­rium at a given tem­per­a­ture that has an emis­siv­ity at a given fre­quency that ex­ceeds the cor­re­spond­ing ab­sorp­tiv­ity. Place it in a closed box. Since it emits more ra­di­a­tion at the given fre­quency than it ab­sorbs from the sur­round­ing black­body ra­di­a­tion, the amount of ra­di­a­tion at that fre­quency will go up. That vi­o­lates Plank’s black­body spec­trum, be­cause it re­mains a closed box. The case that the emis­siv­ity is less than the ab­sorp­tiv­ity goes sim­i­larly.

Note some of the im­plicit as­sump­tions made in the ar­gu­ment. First, it as­sumes lin­ear­ity, in the sense that emis­sion or ab­sorp­tion at one fre­quency does not af­fect that at an­other, that ab­sorp­tion does not af­fect emis­sion, and that the ab­sorp­tiv­ity is in­de­pen­dent of the amount ab­sorbed. It as­sumes that the sur­face is sep­a­ra­ble from the ob­ject you are in­ter­ested in. Trans­par­ent ma­te­ri­als re­quire spe­cial con­sid­er­a­tion, but the ar­gu­ment that a layer of such ma­te­r­ial must emit the same frac­tion of black­body ra­di­a­tion as it ab­sorbs re­mains valid.

The ar­gu­ment also as­sumes the va­lid­ity of Plank’s black­body spec­trum. How­ever you can make do with­out. Kirch­hoff did. He (at first) as­sumed that there are gage ma­te­ri­als that ab­sorb and emit only in a nar­row range of fre­quen­cies, and that have con­stant ab­sorp­tiv­ity $a_{\rm {g}}$ and emis­siv­ity $e_{\rm {g}}$ in that range. Place a plate of that gage ma­te­r­ial just above a plate of what­ever ma­te­r­ial is to be ex­am­ined. In­su­late the plates from the sur­round­ing. Wait for ther­mal equi­lib­rium.

Out­side the nar­row fre­quency range, the ma­te­r­ial be­ing ex­am­ined will have to ab­sorb the same ra­di­a­tion en­ergy that it emits, since the gage ma­te­r­ial does not ab­sorb nor emit out­side the range. In the nar­row fre­quency range, the ra­di­a­tion en­ergy $\dot{E}$ go­ing up to the gage plate must equal the en­ergy com­ing down from it again, oth­er­wise the gage plate would con­tinue to heat up. If $B$ is the black­body value for the ra­di­a­tion in the nar­row fre­quency range, then the en­ergy go­ing down from the gage plate con­sists of the ra­di­a­tion that the gage plate emits plus the frac­tion of the in­com­ing ra­di­a­tion that it re­flects in­stead of ab­sorbs:

\dot E = e_{\rm {g}} B + (1-a_{\rm {g}}) \dot E
\quad \Longrightarrow \quad \dot E/B = e_{\rm {g}} / a_{\rm {g}}

Sim­i­larly for the ra­di­a­tion go­ing up from the ma­te­r­ial be­ing ex­am­ined:

\dot E = e B + (1-a) \dot E
\quad \Longrightarrow \quad \dot E/B = e/a

By com­par­ing the two re­sults, $e$$\raisebox{.5pt}{$/$}$$a$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e_{\rm {g}}$$\raisebox{.5pt}{$/$}$$a_{\rm {g}}$. Since you can ex­am­ine any ma­te­r­ial in this way, all ma­te­ri­als must have the same ra­tio of emis­siv­ity to ab­sorp­tiv­ity in the nar­row range. As­sum­ing that gage ma­te­ri­als ex­ist for every fre­quency range, at any fre­quency $e$$\raisebox{.5pt}{$/$}$$a$ must be the same for all ma­te­ri­als. So it must be the black­body value 1.

No, this book does not know where to or­der these gage ma­te­ri­als, [38]. And the same ar­gu­ment can­not be used to show that the ab­sorp­tiv­ity must equal emis­siv­ity in each in­di­vid­ual di­rec­tion of ra­di­a­tion, since di­rec­tion is not pre­served in re­flec­tions.