 Hi, I'm Professor Stephen Nesheba and I want to elaborate a little bit for you on the reversible adiabatic expansion of an ideal gas. And namely, what I want to get at is figuring out how to express the pressure of such a gas as a function of the volume as it expands, given that we know that it's going to be following what we call an adiabat, which is a path through state space corresponding to a reversible adiabatic expansion. So here we've got our starting state, it's our reference state, pressure, temperature, and volume. It has undergone an expansion, so now it's going to be at some new pressure, temperature, and volume. In an indicator diagram, the idea is that I've got pressure here, volume on this axis. There might be our reference state, which would be that pressure and that volume and that temperature, the reference state. Since it's undergoing an expansion, we know that the gas is going to cool down. If it's a reversible process, then we say that it's going to follow this reversible adiabat and end up on this cooler isotherm. So this would be T-Rap and that would be the final temperature, which is cooler than the starting temperature. Now, the starting point on this analytically will be a result that expresses this change in temperature, which rearranged slightly from what you might have seen before. We said T over T-Rap is equal to some stuff and I've just moved the T-Rap off to the right. So here we have the final temperature, that temperature right there would be the starting temperature corresponding to that isotherm times the ratio V divided by the reference volume all raised to the minus C and this minus C is what's called the reduced constant volume peak capacity, which would be the actual constant volume peak capacity divided by N times R. So there we have that and so how do we get to really what we want here, which is expressing the pressure as a function of volume, because this is really the temperature as a function of volume to quite what we want. Well, it's easy enough to do. There's a temperature for the ideal gas in its final state, which would be PV over NR. There's the temperature for the gas in its initial state, which would be P-Rap T-Rap over NR. Now, because we're assuming that we haven't lost any gas along the way and gas constantly is a gas constant, those can go away. So that expression now turns into just this one, that PV is that PV, that P-Rap V-Rap is that and the same thing over there. There's one more step that we can do, which I'm going to just to have it, I want to have ratios of the pressure. So I'm going to divide by both sides by P-Rap. So I get P-Rap, P over P-Rap on that side and then I'm going to divide by volume on the right side. So I have V-Rap over V, that would be raised to the one power and then here's V over V-Rap, which you combine all of those terms and it turns into V-Rap over V raised to the gamma, where gamma is just defined to be one plus one over that same C that we started off with. So this is the, that's our final algebraic expression and it just says the ratio of the final pressure to the new pressure is equal to that quantity and you can kind of see that since we're assuming that this is an expansion, that volume is bigger than that volume and therefore the final pressure is going to be lower than the starting pressure, which you can see from there. Okay.