 At this point, we've developed quite a catalog of different thermodynamic properties. We've seen flavors of the energy like energy, enthalpy, free energies, like the Helmholtz and Gibbs free energy. Of course, we've also got more basic fundamental thermodynamic properties like the temperature and entropy pressure volume. We've got some derived quantities like heat capacities, constant pressure, constant volume, and we're going to be seeing more thermodynamic variables as we continue. So here's a list of thermodynamic quantities that we can discuss. It's very common when thinking about thermodynamics to run across problems that require you to think about how one thermodynamic property changes in response to another one changing. So for example, how quickly does the energy change as the temperature changes? Often, we need to specify what's being held constant. So how does the internal energy change as we change the temperature while holding the volume constant or while holding something else constant? So many times when we run across variables like this, they're ones that we have already seen. In fact, du dt at constant v, that is the constant volume heat capacity. We've seen ones like if I ask how quickly the Gibbs free energy changes as I change the temperature. If I ask that at constant pressure, then the answer is negative s, negative the entropy. So sometimes we run across a question of how quickly something is changing with respect to something else at constant something else and we know the answer. There's other cases where we don't know the answer, at least not yet. So if I were to ask you, what about how the internal energy changes as I change the temperature but at constant pressure rather than constant volume? That one, we don't yet know the answer to, although we'll figure out the answer to that. Other cases, we don't always have to ask questions about energy. We can have any one of these variables changing with respect to any of the others. So I could ask, for example, how does the pressure change as I change the temperature of a system maybe while holding the volume of the system constant? And again, that's a reasonable thing to ask, but we don't yet know the answer. So we've got a couple of approaches. We could take them one by one, determine what each one of these is and put them in a list and start keeping track of the ones that we already know the answer to until we have written down all the ones we're likely to care about. But there's clearly going to be quite a few of them. If we ask ourselves how quickly does something change as I modify something else while holding something else constant, even with just the 10 variables that we understand so far, that's 10 choices for what x could be or for what y could be or for what z could be. So there's many hundreds of possibilities for these thermodynamic derivatives that might be interesting. So that's going to be a lot of work to write down many hundreds of these thermo derivatives. And it's far too long a list to memorize the answers to. So what we need to be able to do instead is to come up with some techniques for when we run across a thermodynamic relationship that we would like to know the answer to, how do we go about finding out what the answer is if it's one of these that we don't know anything about just yet. So that's where we're headed next is to develop a set of tools and techniques and methods for taking a thermodynamic relationship of this sort and determining mathematically what that should be equal to.