 So we have this equation that we've called the fundamental equation for the internal energy. du is equal to Tds minus pdv. The fact that the change in the internal energy is represented as something times the change in the entropy and something else added to something else times the change in the volume, that suggests that it might be natural to think of the energy varying as a function of the entropy and the volume. If these two change, then that tells us how the energy will change. And given that that's true, once I have a function u written as a function of s and v, then just like any function, I could write the differential change in u. How does that relate to the change in s and change in v, even if I didn't know the fundamental equation? That's going to be change in u is going to come from some change in s and then the energy will change at some rate as I change the entropy. du ds at constant v times ds. And it will also have some contribution from when I change the volume. The energy will change as I change the volume at constant s. So change s and leave v constant or change v and leave s constant. Either one of those will affect the entropy, I'm sorry, the energy add those two together and you get the total change, the total differential for the internal energy. Comparing this expression that I just wrote down and the fundamental equation, we've got du is equal to something times ds and something times dv, same thing here, du is equal to something times ds and something times dv. That allows us to identify these two derivatives in front of the ds and in front of the dv. And it tells us that du ds at constant v must be equal to t. This term is equal to this term and du dv at constant s is equal to negative pressure. So those are two new results that we've obtained using the fundamental equation. The energy, as I change the volume of the system, if I do that adiabatically, isentropically, if I change the volume of the system without changing the entropy, its energy will change at a rate that is negative the pressure. Likewise, if I change the entropy at constant volume, the energy will change proportional to the temperature. This expression is actually not completely new. We've seen that the change in entropy as I change the energy at constant volume, that's one over t. So we've previously seen that result, that's just the reciprocal of this result. If I take du ds and turn it upside down, I get ds du. And so instead of t, I have one over t. So that result's not new, but this result is new. That's one thing the fundamental equation is useful for, is telling us how this property of the energy changes as I change the s, or as I change the volume. However, just because I have energy written as a function of s and v doesn't mean I couldn't also think of energy as a function of other variables, it might be more convenient to think of how the energy changes as I change the temperature as I change the pressure. Temperature and pressure are much more accessible, more convenient variables to use experimentally. I can change the temperature of the system relatively easily. I can change the pressure exerted on a system relatively easily. Pressure is a little more convenient than changing the volume directly. But we can certainly affect the volume of the system directly. Changing the entropy of the system, thinking of the entropy as an independent variable is a little bit difficult. If I were to say go into lab and make sure the entropy of your beaker is 500 joules per Kelvin, I don't think any of us would have any idea how to do that directly. There's no entropy meter we can use to see whether the entropy of the system is what we want it to be or not. So temperature and pressure are maybe much more convenient variables to think about experimentally. So let's think about how the energy changes as a function of T and P rather than these less convenient variables S and V. So I can use the same trick. I can say the total differential of U is equal to some amount due to the change in temperature. That's DUDT while holding pressure constant. And some amount due to the change in pressure, that would be DUDP while holding temperature constant. So that expression is fine. I can write U as a function of any variables I want. I can write the differential in terms of those variables. The question then becomes what's the equivalent of the fundamental theorem? And that I'll just give you a preview. We're not able to derive this expression just yet. We'll be able to do that quite shortly actually. But if I were to write down an expression like this one for how much the energy changes in response to a change in temperature and pressure, that's a much less convenient equation. So let me look it up and make sure I get the coefficients correct. It ends up looking quite a bit more complicated. So there is an expression. Energy will change as some amount times the change in temperature plus some different amount times the change in pressure. Notice that to write this down, I've had to use some variables that we're not familiar with just yet. As a preview, this quantity alpha is something called the thermal expansion coefficient. That's the same as this alpha here. The kappa here is an isothermal compressibility coefficient. So those we haven't considered yet. We haven't defined those yet, so you shouldn't know what those are. But those are just properties of some material that we can either calculate or we can look up, as we'll see shortly. The main thing to notice for now is this collection of things in front of the DT is fairly complicated. Heat capacity, pressure, volume, thermal expansion coefficient. The coefficient in front of DP is also complicated in volume. Volumes and pressures and temperatures and these two coefficients that we haven't talked about. So we can define how the energy changes as a function of the change in the temperature and the change in pressure. But it's an uglier expression. It's not nearly as simple as just temperature times dS, minus pressure times dV. So what that tells us is for some reason, the energy is more naturally more cleanly expressed as a function of S and V than it is as a function of T and P. So in that sense, this is the more natural way to think about the internal energy as a function of S and V, because the fundamental equation is much simpler than this alternate version of the equation. So what we say is that S and V are the natural variables of the internal energy, because when we write the energy or the differential change in the energy as a function of S and V, then the equations come out nice and simple and straightforward. If we try to write them in other terms and other variables like T and P or T and V or S and P, any other combination of these variables, then the equations will come out less convenient, less clean looking. So we'll see as we go on why it is that S and V happen to be the natural variables of the internal energy. But for now, we'll just point out that when we're thinking about internal energy, S and V are the natural variables. If we want to deal with temperature and pressure, what that means is it's easier not to deal with the internal energy but with some other properties as well. So that's where we're heading next is to be able to define properties whose natural variables might be T and P or T and V or some other quantities.