In this discussion, the Stefan-Boltzmann law will be written in terms of the energy contained in a "trapped" electromagnetic field, rather than being written in terms of the power of an "escaping" electromagnetic field (as I discussed for sunlight in a recent post). An energy-versus-temperature equation is one example of a thermodynamic equation of state. The well-known energy-versus-temp relation for a monatomic ideal gas, for instance, is U=3NkT/2. I'll look at many more examples of equations of state, and at how they are related to the "fundamental relations" of thermodynamics in a future post.
Our main man Herb Callen has a good discussion (Section 3-6 in his 2nd edition text) on the thermodynamics of radiation in an empty box:
If the walls of any 'empty' vessel are maintained at a temperature T, it is found that the vessel is, in fact, a repository of electromagnetic energy. The quantum theorist might consider the vessel as containing photons, the engineer might view the vessel as a resonant cavity supporting electromagnetic modes, whereas the classical thermodynamicist might eschew any such mechanistic models.
The empty vessel could be almost anything. A metal breadbox, a wooden breadbox, a closed cardboard box, a spherical Christmas tree ornament, or Santa Claus ornament, as long as it's hollow. Yes, it's generally going to have air in it, which is a point that doesn't get discussed much. That's not to say it will be inflated with air like a tennis ball, football or basketball. But it could be, and there's an electromagnetic field in there with that compressed air. We can just let these "vessels" be at room temperature. Otherwise we have the problem of keeping the walls uniformly at the same high or low temperature.
Even if a box has walls that are at different temperatures, there will be some sort of radiation spectrum in the box. Let's say one wall is heated so it's emitting most of the radiation in the cavity, presumably with a different emission spectrum near its surface than the spectra at the surfaces of the other walls. But is this even possible? Can the spectrum of the radiation in the box be different on different sides of the box?
Well, what we would have is the usual linear superposition (addition) of the amplitudes of the all the waves/photons in the box.,* resulting in constructive and destructive interference, which results in an equilibrium energy-versus-frequency spectrum. If we have a rectangular or square box, we can ask what the role of the radiation is in changing the temperature distributions in the five walls of our box that are not directly heated. This changing of the temps is a temporary, or transient, behavior. An equilibrium temperature distribution will come into being after the transient behavior is over, due to heat conduction in the walls and radiation from the heated wall of the box to the other walls of the box.
But the radiation in the box won't have a black-body spectrum unless the walls of the empty vessel are maintained, as Callen says, at the same temperature, which gives equality of absorption and emission rates for every part of the walls. At least this is the desired black-body cavity experimental set-up. The other thing we need for the usual, real, set-up is a little peek-a-boo hole in the box to observe the spectrum of the radiation.
But we aren't going to try to observe it here. We just want to ask, for a given temperature how much electromagnetic energy is in the vessel? The answer is given by the Stefan-Boltzmann law in yet another form, different from the three ways I wrote it in my February post. Instead of flux (power per unit area) or intensity, the total energy of the radiation is given as proportional to Kelvin-temperature-to-the-fourth-power. Callen writes this as
U = bVT4
where V is the volume of the empty vessel and b = 4σ/c = 7.56x10-16 J/m3K4, where σ is the Stefan-Boltzmann constant and c is the speed of light. (See my 12 November 2020 post, where the integral of the Planck spectral energy density function at temperature T is shown symbolically and the factor of c/4 is used. Here, we're getting rid of that factor. If you divide both sides of the above equation by V, the result is energy density.) Sooooooo, the above equation really comes from integrating the Planck spectral energy density formula over all frequencies while holding temperature constant.
What, then, is the difference between the views of the quantum theorist, the engineer, and the thermodynamicist in Callen's description above? The photons of the quantum theorist are after all normal modes of the electromagnetic field (see page 97 of these notes on the EM field), and a normal mode is a standing wave that has one of the resonant frequencies of the box. The best brief reference on this subject is Rodney Loudon's two-page discussion, "Introduction: The Photon," at the beginning of his book The Quantum Theory of Light, 3rd ed., and the five references therein.
The quantum theorist is not so much interested anymore in cavity radiation by itself as he or she is in cavity quantum electrodynamics, when there are a controlled number of atoms in a very small cavity (and no air, unless the "atoms" in the cavity are a few oxygen and nitrogen molecules) . An electrical engineer who wants to use the "vessel" as a resonant chamber for electromagnetic radiation is just like an acoustical engineer who wants to use an acoustic chamber to enhance certain frequencies and suppress others. (Here we can think about the usual "box of gas" as having acoustical modes of vibration.) Microwave ovens possess electromagnetic resonant cavities to enhance the microwave cooking frequencies, while musical instruments, for example, enhance the desired harmonics of sound waves.
But I'm getting off the subject here. The subject, which I haven't fully explained yet, is the equations of state of blackbody radiation, and how they can be used to find the entropy of blackbody radiation in terms of U and V. This is a "fundamental relation" in thermodynamics, and would be something the thermodynamicist is interested in, particularly the fact that N--whether he or she thinks of it as the number of photons in the box or number of normal modes--isn't an independent variable along with U and V. (See the quote from Callen below, and note he uses an exclamation mark.)
Besides the above equation of state for the total energy of trapped blackbody radiation in terms of temperature, there's a well-known relation between radiation pressure P, total energy U, and volume V,
P = U/3V
which you can copy and paste into an internet search box to find out more about. "It will be noted," says Callen, "that these empirical equations of state are functions of U and V, but not of N. This observation calls our attention to the fact that in the 'empty' cavity there exist no conserved particles to be counted by a parameter N. The electromagnetic radiation within the cavity is governed by a fundamental equation of the form S = S(U, V) in which there are only two rather than three independent extensive parameters!"
The relevant equation for finding the entropic fundamental relation is
ST = U + PV ,
or
S = U / T + PV / T.
From the two equations of state, we have
1/T = b1/4 (V/U)1/4
and
P/T = (1/3)
b1/4 (V/U)1/4
so that the entropy fundamental equation is
S(U, V) = (4/3) (bV)1/4 U3/4
One of the things I'm interested in is how this expression for the entropy of black-body radiation in terms of U and V compares with the logarithmic expression found by Planck, but I'll come back to that later. First, I'll solve the three problems in this section of Callen's book (Section 3.6), then I'll post some detailed typed handouts from a Statistical Physics class I took in 1995 at Texas State University that used Callen's book as the text (thank you, Mark Loewe!).
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*Professor Hazeltine said in a junior-level electrodynamics class I took at the UofTexas at Austin 30 years ago that "photons don't interact." I raised my hand and said, "What about interference and diffraction?" Hazeltine explained that "interacting" has to do with things that exert forces on each other, and photons don't exert forces on each other. I asked, "Could we still say they interact linearly then?" I don't recall his response, but he stuck with the "photons don't interact" paradigm. It's not a good paradigm, because photons, being bosons, are gregarious and will indeed interact in a certain way (a sort of attraction) that fermions don't (they sort of repulse each other). We don't call mutual boson interactions and mutual fermion interactions by the name "forces," but maybe we should, or maybe we should modify our idea of forces.
