 So among the many thermodynamic relationships we can consider, the easiest or the simplest ones to write down are the ones that follow directly from the fundamental equations for the four different types of energy. So remember we have fundamental equations for energy and enthalpy, home holds free energy and Gibbs free energy. We'll start with Gibbs free energy as the one we've seen most recently. Fundamental equation for the Gibbs free energy tells us immediately two thermodynamic derivatives. In fact, I'll write those down and then I'll remind you of where they come from. So if we write down, if we use the fundamental equation and using the analogy of if we think of a function of two variables, the differential of that function has one piece that tells how it's changing in response to a change in x. Another piece that tells us how much that function is changing in response to a change in the variable y. And crucially the coefficient that's in front of the dx tells us this derivative df dx. Coefficient in front of dy tells us df dy while holding the other variable constant. So if we just look at these fundamental equations, we can read directly off of them the dg dt at constant other variable at constant p. That's equal to this coefficient of negative s. Likewise, the coefficient here in front of the dp must be the rate at which g is changing with respect to p, the pressure. So dg dp at constant t is equal to volume. So for each of these fundamental equations that we have seen. So writing those down, du is equal to tds minus pdv. dh is equal to tds plus vdp. Da is minus sdt minus pdv. Each of those fundamental equations allows us to write down some derivatives just like we've done for the Gibbs free energy. So for the internal energy, du ds at constant v is equal to this coefficient t. du dv at constant s is equal to negative pressure. The enthalpy equation proceeding similarly, dh ds at constant p is equal to temperature, dh dp at constant entropy is equal to volume. And then for the third one, Helmholtz free energy, dA dt at constant v is equal to negative s. And dA dv at constant t is equal to negative pressure. So all together, if we know what the fundamental equations are, they allow us to write down these eight thermodynamic relationships. So that's eight thermodynamic relationships, a couple of which we've seen before, some of which are new. The thing to remember here is not that here's eight more relationships to add to a list and to memorize. The most easy thing to do is if you know what the fundamental equations are, which we've had occasion to use multiple times. If you remember the fundamental equations, you can determine these derivatives immediately. So if we ever need dh ds at constant p, we recognize from the fundamental equation exactly what that derivative is equal to. Even easier than memorizing four different fundamental equations. If you remember how to get from one of these to the other, for example, enthalpy is energy plus pv, so that swaps the roles of the p and the v in the fundamental equation. Just by memorizing one of these fundamental equations, knowing how to get the others very easily, you can also very easily proceed to get these eight thermodynamic relationships. So these relationships, I'm calling those the fundamental thermodynamic relationships because they flow from the fundamental equations, are perhaps the easiest ones to derive and make use of. And next, we'll explore some techniques for determining some additional thermodynamic relationships.