 So we have this expression that I've told you is very important that we'll end up using in many different ways called the fundamental equation for the internal energy. And this equation, remember, is what illustrated to us that the energy, the internal energy, can be thought of as a natural function of the variables s and v because this equation has a particularly simple form. But there's more than one type of energy that we are concerned with. For example, there's a form of energy called the enthalpy that we've seen as more convenient to work with than the internal energy when we're doing things at constant pressure. So it turns out there's also a form of the fundamental equation that applies not just to the internal energy, but also to the enthalpy and to every different type of energy that we'll consider. And the good news is those are fairly easy to obtain for different forms of the energy. So for example, if I want to know what the equivalent of this statement is for the enthalpy, I just have to remember the definition of the enthalpy. Enthalpy is internal energy plus this product p times v. A differential expression, if I take the differential of the left side to get dh, take the differential of the right, differential of u is just du. Then differential of p and v, the product pv, either one of those could be changing. So if I use the product rule, the differential of pv is pdv plus vdp. But I know something about du. The fundamental equation tells me what du is equal to. du is equal to tds minus pdv. So that has come from the fundamental equation to that I need to add pdv and vdp for these two terms. But this minus pdv and this plus pdv are equal with opposite signs, so they cancel each other out. So what I'm left with is just the statement that the differential change in the enthalpy, dh, is equal to tds plus vdp. So that's an equation of the same form, a differential change in h instead of u is equal to some combination of the differential changes in two other properties. So this is the fundamental equation for the enthalpy. It differs from the differential, from the fundamental equation for the internal energy only in this last term. They both have a tds as the first term and then we've got a plus vdp rather than a minus pdv. And just like this expression told us that the internal energy was a natural function of s and v, s and v are the natural variables of the internal energy. This expression tells us that the natural variables of the enthalpy are the entropy and the pressure, h is a natural function of s and p. So notice those natural variables are different. This fundamental equation also tells us two other things immediately. Since we're thinking of h as a function of s and p and asking how it changes as s changes and as p changes, the differential of h is in a more general sense some rate of change as I change s and some rate of change as I change p, each multiplied by the relative change in s and in p while holding the other one constant. So that would be true of any two-dimensional function. If I compare the terms in this expression and the fundamental equation that we've just derived, we can see there's something multiplying a ds and something multiplying a dp. So if I just equate those two coefficients, we've discovered that this derivative dhds at constant p, that must be equal to the temperature. Likewise, dhdp at constant s, this derivative, that must be equal to the volume. So here's two derivatives that we haven't seen before. How quickly does the enthalpy change when I change the entropy while holding the pressure constant? How quickly does the enthalpy change when I change the pressure while holding the entropy constant? Those have these particularly simple answers. So those may or may not be something we're interested in. There's thermodynamic derivatives. You can imagine now how many of this type of thermodynamic derivative there are. The derivative of some thermodynamic property with respect to some other thermodynamic property while holding a third one constant. There's literally hundreds of these derivatives we could write down. We've got these two new ones. We've seen some more previously that came from the fundamental equation for the energy. We'll see many more as we go along. But this gives you a taste of how we determine what those derivatives are when we need them. The other thing I'll point out that's important about this fundamental equation for the enthalpy, now that we have it, is we've succeeded in using the enthalpy. We've succeeded in turning the energy, which is a natural function of S and V, into the enthalpy, which is a natural function of S and P. So it's perhaps not surprising since enthalpy is the property we use when we're doing something at constant pressure, it's convenient and perhaps not surprising that its natural variables include the pressure rather than the volume. And in fact, taking something whose natural variables are S and V, adding this PV product has managed to switch the natural variables from V to the other one in this product to P. So we're still left with the fact that having entropy as a natural variable is not terribly convenient. It's going to be much easier when we're able to talk about functions whose natural variables are not the entropy but the temperature. So as a sort of food for thought and a preview of what's to come in the next few video lectures, you can ask yourself what would we do to these energies in order to transform the natural variable of S into a natural variable of T? What would we do to these functions to make that transformation? So we'll see that in a lecture coming up.