 So there are a lot of different relationships between different thermodynamic variables, some of which we've seen before and a lot more of which we'll see in the future. And to understand these and to find the connections between these and derive new ones, we need to understand some basic identities about partial derivatives. So that's where we'll start trying to understand some of these partial derivative identities. The first of which is one that you've heard of before, the chain rule. And you likely know what that is for ordinary derivatives. So if I have a function derivative of that function df dx, I can write that as the derivative of the function with respect to a different variable multiplied by the derivative of that other variable with respect to the one I was first interested in. So that's what the chain rule is for ordinary derivatives. We're more interested in partial derivatives because most of our thermodynamic quantities depend on more than one variable and it turns out the chain rule is very similar in partial derivatives. I can still write df dx is equal to df dy times dy dx just with partial derivatives. And since these are partial derivatives, I have to take the partial derivatives while holding something else constant. And as long as all these partial derivatives are taken while holding the same thing constant, for thermodynamic variables, these x's and y's and z's, they might be temperatures or volumes or pressures or something like that. But as long as each of these derivatives is taken while holding the same thing constant, it's the same chain rule that you're more familiar with for ordinary derivatives. So what good this is, is it allows us to rewrite a derivative with respect to some variable x that we might not be comfortable with or we might not know anything about. We can rewrite that as a derivative with respect to a different variable that we might know something more about. So for example, let's say we're interested in how quickly the free energy, the Gibbs free energy of something changes as I change its volume isothermally. So that's a fairly reasonable thing to consider. I have some object at constant temperature, somehow the volume of that object changes, whether it's a box of gas or it's a solid object that I'm compressing or something like that. As the volume changes, we might want to know how much is the free energy of that object changing. This is not a derivative that we know anything about yet. The natural variables of G are pressure and temperature, not volume. So this is not something we know anything about immediately, but we can rewrite using the chain rule, we can rewrite derivative of G with respect to V as the derivative of G with respect to something else, and it's up to us to choose what to put here. We do know something about the derivative of the Gibbs free energy with respect to pressure. So if I put pressure in these two places, DGDP, DPDV, both at constant temperature, chain rule tells me that's the same thing as the derivative I was originally interested in. And now, as a reminder, since we know the fundamental equation for the Gibbs free energy, that fundamental equation reminds us that if we want the derivative of G with respect to P at constant T, DGDP at constant T is equal to this coefficient V. So I know this derivative, DGDP at constant T is equal to the volume. The second term is DPDV at constant T. So here is a thermodynamic relationship. We've just concluded that the derivative of the Gibbs free energy with respect to volume at constant temperature is equal to the volume multiplied by the derivative of pressure with respect to volume at constant temperature. Volume, of course, we understand. DPDV at constant T, that's not yet a thermodynamic relationship that we can give a name to, but that's at least something that we can picture measuring in the laboratory. It turns out that's actually very closely related to one of the named thermodynamic quantities that we've discussed previously, but in order to understand that connection, we're going to need an additional identity for partial derivative. So that's the thing we'll cover next.