 So our understanding of heat capacities of solids at the moment is we have this simple prediction by DuLong and Petit or the Ecopartition theorem that the capacities 3R. And that's true in the classical limit when the temperature is high enough that the system can be treated classically. And then the Einstein model of the solids essentially just treating each of the vibrational degrees of freedom like a harmonic oscillator tells us this more complicated expression that gives us a temperature-dependent heat capacity. So this explains why many of the values are close to the Ecopartition theorem prediction and some of the values are a little too low. It can explain some of these other values that are too large. And I'll highlight some of the values that do appear too large on this table. So we see lead, for example, 26.4, tin at 26.9, uranium at 27.7. Those are all larger than the Ecopartition theorem prediction of 24.9. So if we have a heat capacity that the Einstein model tells us should asymptotically approach 3R as the temperature increases, how do we explain values that are larger than 3R? If we remind ourselves of where that value of 3R came from in the Ecopartition theorem, where that comes from is we have, if we count degrees of freedom in a solid, translational, rotational, vibrational degrees of freedom, there's only a few translational and rotational degrees of freedom. In our solid that contains a large number of atoms, most of those 3N degrees of freedom are vibrational degrees of freedom, each of which contributes a factor of r to the heat capacity, 1 half r from the potential energy, 1 half r from the kinetic energy. Altogether, those 3N factors of r give me a total heat capacity of 3N times r for atoms or 3 times r if I've divided by the number of atoms of the substance. So if the heat capacity is 3R, that implies not just that all these modes are completely classical and are completely contributing to the heat capacity, but there must be some additional degrees of freedom that are contributing to the heat capacity as well. In fact, if we plot the temperature-dependent heat capacity of these various metals, once we reach a temperature where the equilibrium theorem is reached and we continue a little further, we see that the heat capacity eventually will begin to rise a little bit more. And in fact, it is true that we have some additional degrees of freedom that we haven't accounted for here that are now beginning to contribute to the heat capacity. So these degrees of freedom that we've discussed so far, those are the atomic degrees of freedom, the motion of the nuclei. I'll call those the nuclear degrees of freedom because if I pick an atom up and I move it around, if I translate the atom or if I rotate the crystal or if I cause a vibrational motion by moving atoms relative to each other, those are all moving the nuclei of the atoms. There's also electronic degrees of freedom when the electrons are involved in the energetic excitations. So those electronic degrees of freedom, we've already seen those a little bit when we talked about spectroscopy. If I excite a molecule from one electronic state to another electronic state, those are excitations of the electrons. And remember that the equilibrium theorem requires these degrees of freedom to be classical in order to use the equilibrium theorem estimate. Translations are always classical. Rotations, those are classical as long as the temperatures for diatomic molecules. We saw that those are classical as long as the temperature exceeds the rotational temperature. So that was, roughly speaking, somewhere in the vicinity of a few kelvin or a few tens of kelvin. The vibrational temperature in diatomic molecules was thousands of kelvin, but in these solids, as we've seen, the vibrational temperature tends to be on the vicinity of a few hundred kelvin. So as long as we're bigger than a few hundred kelvin, the vibrations get treated classically. The electronic degrees of freedom, the excitations to lift molecules from one electronic state up to the other, because electrons are so light in mass, those excitations are so high in energy, usually taking visible or ultraviolet photons, for example. Those electronic excitations typically don't begin to be classical until we're well above thousands of kelvin. So in fact, that's what we're seeing here is this rise from 0 up to the equipartition theorem limit of 3R happens as we cause the vibrational, as we increase the temperatures large enough to cause the vibrational motions to be treated classically. But the electronic motions don't begin to be treated classically until much higher temperatures. So how many additional degrees of freedom do we have? That totally depends on the substance. It depends on the electronic structure of the substances. Maybe each of these atoms has only one valence electron that has an accessible electronic state, or maybe it has more than one electron. And that'll depend on the details of the metal or the solid that we're talking about. But what we can tell from the data is for the materials like lead and tin and uranium that have started this rise up above 3R down at temperatures like room temperature, what that means is they have anomalously low lying electronic states that don't require us to heat up to 1,000 Kelvin or more to begin to excite them. But lead and tin and uranium have some electronic states that are accessible even at temperatures of 300 Kelvin. So as a summary of what we understand about these heat capacities now, many solids have equipartition theorem values for their heat capacities. They might be too low if the vibrations in the solid are not yet classical at a particular temperature, or they might be higher than predicted by the equipartition theorem if they're not just fully classical at that temperature, but also have some low lying electronic excited states.