 So as we continue to explore relationships between thermodynamic variables, we're at a point now where we can combine a few results that we've seen previously to get an equation that is going to be quite handy and useful as we move forward. So let's start with the first law of thermodynamics. So if you recall the first law in the differential form, I could say energy is heat plus work or in this case I'll say change in energy, differential change in energy is dQ plus dW, differential change in heat and differential change in work. For cases where we're only interested in PV work, we know how to write down what the work is equal to as PV work, that's minus P external times the change in volume. This is often not very convenient to know the external pressure of a system, we'd rather deal with the internal pressure of a system. So let's assume that we're doing some process that's reversible. Under reversible conditions, we can say two things. First of all, the internal pressure and the external pressure are the same. If I'm doing the process slowly and reversibly enough, the internal pressure and external pressure are in equilibrium with each other the whole time, equilibrium and reversible are different words for the same thing. Then P external can be written as internal pressure P. Also, since we're doing it reversibly, I'll write dQ just to remind me that we've done this under reversible condition, I'll write that as dQ reversible. That might be enough to remind you of another equation that we've seen previously. The Clausius theorem, remember, was a relationship between the heat for a reversible process and the entropy change for that process. Specifically for a reversible process, dQ over T is dS. We phrase that at the time as saying 1 over T is an integrating factor that turned this inexact differential dQ into an exact differential dS. So of course, I can rearrange that and I can say dQ reversible is dS times T. That's what I'll use to transform this equation. So instead of dQ reversible, I'll use the Clausius theorem to say T dS and I've still got a minus P dV. So at this point, my equation looks like this. dU is equal to T dS minus P dV, so I'll put that equation in a box. That's the equation that we'll actually call the fundamental equation. We'll run across several equations that we'll call fundamental equations. This is in particular the fundamental equation for you, the fundamental equation for the internal energy. This is actually a truly significant result that we'll come back to over and over. Significant for a couple of reasons. Number one, it's a useful way of telling us how the energy of a system changes. If we know how it's temperature, pressure, volume, and entropy, we know the values of those and perhaps how they're changing as well. So that's a useful thing to do. We often want to know how much the energy changes as we change some other properties. Of course, we already had an equation that told us how much the energy changed as a system underwent some change. But this equation, the first law, even though that's important enough to be called the first law of thermodynamics, that equation dealt with inexact differentials. So it was convenient. It was path dependent on the right side of this equation. So it's often not convenient to use that expression. This expression is much more convenient in that it's written in terms of only state variables. So this equation is no longer path dependent at all. So if we know, it doesn't matter what path we take to get from the initial conditions to the final conditions. If we know how much the entropy changed and the volume changed and what the temperature and pressure are, we can use this expression to calculate the energy. I'll point out as a corollary to that statement, because this expression has only state variables and is not path dependent at all. It doesn't actually matter what path we use to derive at this result. We derive this result as being true for a reversible path. But this expression is true. Once we've got this expression, it's true even if we had taken some irreversible path to get there. This expression might have looked different in the irreversible case, but the differences between the path dependent of the Q portion and the W portion would end up canceling out. This expression is always true, regardless of whether it's a reversible process or an irreversible process. So this expression will turn out to be very useful as we go forward. In fact, the fact that this expression is so simple looking, so relatively simple looking in terms of just combining these four different thermodynamic variables is a significant fact on its own. So that's something we'll also pay a little bit of attention to, and that's what we'll look at next.