 So we've already made use of the fundamental equations for the four different types of energy to derive some thermodynamic relationships, the ones I've called the fundamental thermodynamic relationships. But they are going to continue to provide us more gifts. They're going to allow us to derive a few additional thermodynamic relationships as well. So let me explain that with going back to this basic example of a two-dimensional function. If we think of these energies as functions of their natural variables, S and V, or T and P, or whichever, any function of two variables, we've used multiple times the idea that we can write down the differential as df dx times dx plus df dy times dy. That's what let's us say, for example. And I'll go ahead and write down this statement dg dt. Is negative S. And from here, dg dt is equal to V. Two derivatives we've seen before. We can read those derivatives off the coefficients. Coefficient in front of dx is the derivative with respect to x. The coefficient in front of dy is the derivative with respect to y. But now let's think about second derivatives. If I ask you what is the second derivative of f with respect to x and y. So the mixed second partial derivative, df dx dy. d squared f dx dy. We can think of that in two different ways. We can think of that as the x derivative of the y derivative. So first take the y derivative, then take the x derivative. Or we could think of it the other way around, the y derivative of the x derivative. First take the x derivative and then take the y derivative. And it doesn't matter what order I take those derivatives in. I'll get the same answer in either way. The second mixed partial derivative is the same regardless of which order I take it in. What that means is, looking at this diagram, if I take the first derivative with respect to x and then take dy, I'll get the second mixed derivative. Or if I take this coefficient, df dy, and take dx of it, the x derivative of it, I'm going to get the same result. So I can start with either one of these coefficients, take a derivative, and get to the same answer. What that means for these fundamental equations is, if I'm thinking about the second derivative of g with respect to t and p, I can either take the first derivative with respect to t and then take the derivative with respect to p. So take the pressure derivative of this thing. So take the pressure derivative of s with a negative sign. Gives me minus ds dp. Or if I prefer, I can do it the other way around. I can use this first pressure derivative and then take the temperature derivative, temperature derivative of v. Either one of these should be this mixed second partial derivative. Notice that they don't look anything like one another. Negative ds dp at constant t must be equal to dv dt at constant p because it doesn't matter what order I take these mixed partial second derivatives in. So that's a thermodynamic relationship that's very useful. We might not have known anything about how the entropy changes as I change the pressure isothermally. But I know it's related to, it's the negative of how quickly the volume changes as I change the temperature at constant pressure. So what that means is we've, this fundamental equation has given us a new thermodynamic relationship minus ds dp at constant t is equal to dv dt at constant pressure. So and I can read that directly off the fundamental equation now that I know where it comes from. Derivative of this with respect to this at constant this is equal to derivative of this coefficient with respect to the other variable while holding this one constant. So I can do the same with a little less work now for the rest of these fundamental equations. Just taking the derivative of each coefficient with respect to the opposite variable while holding its own variable constant, I learned that dt dv at constant s is equal to negative dp ds at constant v from the fundamental equation for the internal energy. Enthalpy tells me that dt dp at constant s is equal to dv ds at constant p. And the Helmholtz free energy tells me negative derivative of s with respect to v at constant t is equal to negative derivative of p with respect to t at constant v. So that's a collection of thermodynamic relationships, relationships between one thermodynamic derivative and a completely different looking thermodynamic derivative that are true and followed directly from these fundamental equations. So again, it's very useful to have in mind either the fundamental equations or memorize one and be able to write down the rest of them easily. And then when you need one of these relationships, you can obtain it very easily from the fundamental equations that you know how to write down. Each of these four relationships is useful. We'll have occasion to use them again. The name of these relationships is called Maxwell relations. So each one of these four is called a Maxwell relation. And again, the idea behind these Maxwell relations is each one of them is equal to the second derivative of an energy with respect to two different variables. And either way I have of writing down that second derivative is bound to give me the same answer. So that's what gives me this relationship between the two halves of the Maxwell relation.