 So, the final thing that we're going to look at in this lecture is another Carnot cycle, a reversible cycle that exists, well it does not exist, but it's one that we study as kind of a theoretical abstract, and that is the Carnot refrigerator heat pump cycle. So, similar to the Carnot heat engine, it consists of four reversible processes, and we'll start by taking a look at it on a PV diagram. So, the first, again, what we have is the cycle is operating between two isotherms, and the first process within the cycle is one of an expansion, and this is an adiabatic reversible expansion. We then undergo, so that takes us from state one to state two. It's basically the reverse cycle of what we looked at for the Carnot heat engine. The next part of the cycle, going from two to three, is one where we do isothermal expansion, and remember whenever you do expansion the gas gets cold, so in order to keep it isothermal, we have to add heat, and that will come from the low temperature source, and then we go into a compression process that is adiabatic, and that takes us from three up to four, and given that it's adiabatic and it's compression, you have heat generated, it cannot escape because it's adiabatic and consequently we go from isotherm T low to isotherm T high, the gas gets hotter, and the final step is an isothermal compression process, taking us from state four to state one, and again in order for us to remain isothermal through a compression process, we need to reject heat, and so that heat is going out as QH. So you can see it's very similar to what we saw before when we looked at the Carnot heat engine, except the cycle is operating in the other direction. So what we're going to do now, we're going to take a look at the coefficient of performance for the Carnot refrigerator or heat pump, and we'll write out the expression, so the coefficient of performance refrigeration, and I put REV there to denote the fact that this is for a reversible cycle, so that's an expression for the coefficient of performance, and the coefficient of performance for a heat pump, again for a reversible cycle such as the Carnot cycle. So those are the coefficients of performance that we would be examining as idealizations for any real process, and in reality, if we're looking at a refrigeration cycle with irreversibilities in it, which all refrigeration cycles would have, the coefficient of performance for irreversible cycles are going to be less than the coefficient of performance for the Carnot cycle, and similarly for a heat pump. So what the Carnot refrigerator heat pump cycle does is it gives us an idealization with which we can compare to, and the reversible coefficient performances are basically the best performance we would be able to achieve. So that concludes today's lecture. Thank you very much.