Thermal spectrum of a warmer body in a colder room

Here are a few questions about heat that I've been wondering about.

Suppose I had a piece of glass (just as an example) at room temperature, let's say $$T_0 = 293$$ K, and I moved it into a dark room that was $$T_R = 1$$ K. I assume the glass will radiate heat until it and the room are the same temperature.

What does its thermal spectrum look like during that process?

• I think it's a nice courtesy to name your variables, e.g., $T_0=293\,$K and then something like $T_R = 1\,$K. Otherwise, we have to spend time doing it for you in an answer. Maybe also called out $\epsilon$. (It also a good gauge of the questioner, as confused variables and strange symbol are tells....).
– JEB
Commented Jun 26 at 16:55
• Do you mean the variation of the spectrum with time? Or with temperature? Commented Jun 30 at 12:02
• Well, I assume both, because the temperature should drop over time. IE, what's the spectrum at 100 K, 50 K, 10 K, etc Commented Jun 30 at 12:04
• " I assume the glass will radiate heat until it and the room are the same temperature." The glass will to radiate afterwards too, but the energy emitted is the same as the energy gained. Commented Jul 1 at 15:39
• A full theory of time evolution of thermal radiation is hard because it is non-equilibrium dynamics, however if everything is slow enough (as in most cases) probably it can be easily modeled Commented Jul 1 at 15:40

Suppose I had a piece of glass (just as an example) at room temperature, let's say T0=293 K, and I moved it into a dark room that was TR=1 K. I assume the glass will radiate heat until it and the room are the same temperature.

What does its thermal spectrum look like during that process?

You have correctly noted that we are dealing here with a process, that is development in time - and the radiation spectrum would change over time.

A body in thermal equilibrium with its environment has emission spectrum close to black body radiation. It is not exactly black body radiation, since no object absorbs all the radiation incident on it, but black body is often a good approximation.

A large (macroscopic), object not in equilibrium with its environment also emits a spectrum close to black body spectrum - the reason for that is that the radiation emitted is only a small part of radiation trapped inside the object, which is in equilibrium with it, i.e., it is black body radiation as well. (This allows us to determine temperature of stars... while the deviations from the black body spectrum allow us top determine their chemical composition.)

Thus, the spectrum will remain nearly black body throughout the cooling process - provided that the cooling is slow, which is usually the case, when it is cooling via contact with air, a poor heat conductor. The shape of the spectrum will gradually evolve with the temperature, which is the parameter determining the shape of the black body curve:

Remark
I don't know, if there are any specific caveats related to the type of the material chosen in this question (a piece of glass), so I apologize, if thsi answer is too generic.

• Thanks! If the room were a vacuum would that change anything vs if it was filled with air? Commented Jul 1 at 21:51
• @MikeHelland In vacuum the scenario outlined in the answer is even more precise, as the energy losses are only through the radiation. In air one in principle has to account for heat conduction (through the air), and convection. There may be also heat conduction through the support/suspension that holds the object. Commented Jul 3 at 11:37