Hello, [COUGH] excuse me, and good morning. Today, we're going to talk about black body radiation or equilibrium radiation. So, you recall from our discussion of quantum mechanics that we introduced the Planck black body function or Planck distribution as it's called. And if you've had a heat transfer course, you've been exposed to this. It turns out, that photons, equilibrium photons in any event, photons in an equilibrium situation can be treated as an ideal gas. There are two exceptions to this, one is that photons are bosons and Bose- Einstein statistics must be used. And, but one difference is that photons don't interact with each other, so that no approximation is made by neglecting interparticle forces. The second thing is that, photons can be absorbed and emitted by the walls of a container, so that there's no constraint on the number of photons, even in the Grand Canonical representation. And what that means is that gamma is equal to 0, gamma being you'll recall mu over kT. So we'll derive the energy distribution, the Planck function. So recall that for a non interacting gas, for a boson, that the number of particles in a given energy state K, is equal to the degeneracy of that state divided by e to the gamma plus epsilon over KT divided by -1. However, since there's no constraint on N, gamma 0, this becomes the degeneracy divided by e to the ke of k over kT-1. Now, like the particle in the box, it turns out that the individual quantum states of the photons are very closely spaced. And so you can make the continuous approximation, you can assume energy is continuously distributed. In which case you can write, rather than a Nk equals a certain value, you can write dN is equal to dg over e Epsilon over kT-1, dg here is the degeneracy for the energy increment epsilon to epsilon, d epsilon. Now, the degeneracy is almost the same for a particle in the box. In fact, it is the same except for a factor of 2, and this factor of 2 arises because an electromagnetic field can be polarized in two independent directions. Now, for a molecular gas, we related dN to d epsilon using Newtonian physics. That is, we said, that epsilon is equal to one half mc squared or if we wrote it in terms of a momentum, p squared was equal to 2m epsilon. However, photons are relativistic. And you have to use de Broglie's relationship, p is equal to epsilon over c. So as a result, we get that epsilon squared is equal to h squared c squared, over 4 times the volume to the 2/3 power all times n squared. So as a result dg is 8 pi v / h cubed c cubed, all times epsilon squared d epsilon. So if we substitute that expression into dN and use the relationship that the energy of a photon is equal to h Nu the frequency of the photon. Then we get for the number distribution that an incremental number of photons, divided by the volume is, 8 pi over c cubed times the frequency squared over e to the h mu over kT-1, all times d Nu. And that's the number of photons per unit volume in the frequency range Nu to Nu+ d Nu, but we're looking for the spectral energy density. And since we can write that the spectral energy, the energy per differential frequency at frequency Nu d Nu is equal to h Nu dN over V. So then, putting that together we get that the spectral energy density that's the energy per unit frequency and per unit volume is 8 pi h Nu cubed over c cubed, all times 1 over e to the minus h Nu over kT-1. And this is Planck's Law. So, you can look at the limits for high and low frequency easily, for large frequency then h Nu / kT is large, get the so-called Wien Formula. And four small frequency h Nu / kT is much less than 1, we get the Rayleigh- Jeans Formula. So, we can get the total energy, the total radiant energy per unit volume by integrating over the energy distribution function Plancks function. And we get, if we perform that integration we get a, pi to the fifth over 15kT to the 4th hc cubed. And that t to the 4th should be familiar to you from heat transfer. Now we could have attained this relationship by first evaluating the partition function and getting the fundamental relation. Energy is then one of the equations of state in the Grand Canonical Representation. So for a boson with gamma equals 0, the log of the partition function is equal to this expression at the bottom of the slide. So, doing the integral we get 8 pi to the 5thv over 45 times hc over kT cubed. So the fundamental relation becomes this expression at the top of the slide, so, S for the Canonical Representation, we've replaced u with one over T. And if you work the numbers, it turns out that you get PV / T, interesting, the energy we already derived. The pressure is the partial vests with V constant 1 over T. And that turns out to be this 8 pi to the fifth over 45 hc cubed all times kT to the 4th. And there is, radiation does exert a pressure and if you're satellite in outer space, you've got to take it into account. To obtain the total number of photons, you have to differentiate with respect to mu before setting mu equal to 0. And if you do that you get this expression 1.202 16 piV / (hc/kT) cubed. So, you can also get the entropy, which is given by this expression. Finally, and this is perhaps one of the most important outcomes of this analysis, you can derive the Stefan-Boltzman relationship that is used in heat transfer. So for an ideal gas the flux of particles across any given plane is nc bar over 4 and we'll derive this expression later on. It's obtained by integration of the Maxwellian velocity distribution function. So by analogy, you can show that the energy flux of photons, energy per unit time and area, is equal to this first expression, q energy per unit time and area in SI units. That would be watts per square meter is 2 pi to 5th k to the 4th divided by 15 h cube c squared, all times T to the 4th. And the co-efficient of T to the 4th, we call sigma and that's the so-called Stefan-Boltzman constant, which can be evaluated once and for all. And then SI units is 5.669 times 10 to the -8 watts per meter squared per degree K to the 4th. So, that's it for this video. Thanks for listening and have a great day.
