Carnot efficiency 2: Reversing the cycle

In the last video, I showed you that the definition of efficiency, eta, is the work that we do given the amount of heat we are given to work with. And we showed that for an engine, that could also be rewritten as 1 minus Q2 over Q1, or essentially, 1 minus the heat we output from our engine divided by the amount of heat we input from our engine. Now we applied this formula to a Carnot cycle, and we said, hey, for a Carnot engine, we could get an efficiency of this. So let me write this here. So the efficiency for Carnot, eta for Carnot, is 1 minus T2 over T1. To get this result, we had to use the fact that we were dealing with the Carnot cycle to use, you know, we were moving along these isotherms, and so I was able to take the natural log of them and do all of that, and I was able to get this for the efficiency of a Carnot engine. And let me be very clear. This is the efficiency that can only be attained by a Carnot engine. The other definitions of efficiency-- so when I just defined efficiency as equal to the work performed divided by the heat, let me call it the heat input-- or when I defined it as the net heat in. So Q1 minus Q2 over Q1. This applies to all heat engines. This is true for all heat engines, including the Carnot engine. A heat engine is an engine that operates on heat. I probably should have said that a while ago. And this engine that I made, this Carnot engine, is definitely a engine that is operating on heat, because it's taking heat here, and later it releases the heat down here. The cycle just shows what's happening to that engine. And I just want to make that distinction, too. The engine is the actual physical thing. The cycle just describes what's happening to it. So with that said, I said that this is only true for a Carnot heat engine. Now, what I'm about to embark on-- and I don't know if I'm going to finish it in this video. It might take into the next video to do it properly-- is to show you that if we're operating a heat engine between two temperature sources-- so I have my hot temperature source, I'll call that TH for T hot, and it's transferring some heat, Q1, and some other heat is coming out at Q2, and I'm performing some work, and then my other cold temperature reservoir, I'll call that T cold, is down here. And that's where I'm releasing the heat, too. I'm going to show over the next few videos that the most efficient engine is this theoretical Carnot engine. That no engine can get more efficient than this. So if this is a Carnot engine, this is the most efficient engine. Or this is the ideal, where nothing is lost. Well, I'll go into that in more detail. No engine can get more efficient than this Carnot heat engine. So to get there, to prove it to you, I'm just going to play with the Carnot engine a little bit, just to show you some of the tools that it has at its disposal. So one of the things-- let me just draw a PV diagram. In the Carnot cycle we've done so far, we've kind of always moved in one direction. We had our isothermal expansion. It went something like that. That was isothermal. Then we had our adiabatic expansion-- and the whole time we were going in that direction-- and it went like that. Then we had our isothermal contraction. It went something like this. And then we had our adiabatic contraction, to get to where we were to begin with. So then we went back like that. And the whole time, we went in this kind of clockwise direction. We went in the clockwise direction, and we took in heat up here-- because we were doing work-- we took in heat to keep our temperature constant, and then we released heat here to keep our temperature from going up from Q2. And so if I were to draw this another way-- well, I just did one like that, but let me draw it like this. I could also depict it like this, where that's my engine, this is my hot reservoir-- let me put this as T1-- T1 is up here. It transferred Q1 to my Carnot engine. My Carnot engine did some work, and then left over, it transferred into my cold reservoir, T2, it transferred Q2. This is another way of depicting what went on in this Carnot cycle. And here I've actually drawn the engine. Now, one of the tools I want to show you is that this is a reversible reaction. Or that we can take this and go the other way around. And it's dependent upon an assumption that I threw out a long time ago. So when I first drew these, I kind of introduced you to the idea of a quasistatic process. And quasistatic just means, look, you do it really slowly, so that you can always say that you're close enough to equilibrium that your macro state variables are always defined. And that was the whole justification for dealing with pebbles like this. Instead of just doing it wholesale, instead of just moving all the pebbles, and just getting to this state, from A to B kind of jumping, I wanted to do it gradually, so that I would be defined at every point in between. That's what quasistatic did for us. And when I actually made the video quasistatic processes, I said, you know, quasistatic processes, for the most part, are reversible. And sometimes I used the words interchangeably. Now, by definition, our theoretical Carnot cycle is said to be, not only is it quasistatic, but it is also reversible. Which means at any point in time-- let's say we've moved a couple of pebbles, and we've gotten right here. If we want to, if we're in the mood, we can add some pebbles back, and just follow this right back to where we were. That's what reversible means. It means you can reverse something. Now, what has to be ideal about the system in order for that to be true? Well, it means that the actual movement of our piston, of this movable ceiling, that it shouldn't have any friction. Because if some of the heat is lost to friction, then when we go back, we would have lost some of our heat. Some heat would have been destroyed, just going from one state forward and back. So the assumption that we have to make in order for the Carnot cycle to be reversible is that it's frictionless. So the Carnot heat engine, this theoretical engine, is a frictionless engine, which is theoretically impossible. To be completely frictionless. To be-- but we'll talk more about that in the future. So if you have a completely frictionless engine, and it's quasistatic, it's also reversible. So if we want to do it reversible, what does that mean? It means I could start in this state, my state A that I've labelled before, but instead of going around that way, I could go around the other way. So what I could do first, is I could adiabatically expand first-- so maybe let me redraw it, so I do it the other way. So I could reverse this reaction. And it would happen the exact same way. And that's an artifact of that I'm always in equilibrium, and that my system is frictionless, that I don't lose energy just going back and forth. So I could start at state A here, and then I could adiabatically contract. Adiabatic contraction would look something like this, and it'll get to that state. Then I can isothermically expand. So I'm going like this. And as I isothermically expand-- so I'm going like this, I'm all along an isotherm-- I'd doing some isothermic expansion-- so in this case, I'm doing work, but I'm doing work isothermically, right? At some cold isotherm. Let's call it T2, right? Just like this was T2. So in this case, if I'm expanding, and I'm staying at T2, and I'm sitting on top of my T2 reservoir, heat is coming. This area under the curve, the work I'm doing, is the heat added. This is Q2, and that is given to me by my T2 reservoir. So everything is going in reverse. That's the whole idea. Then I adiabatically contract, like that, and then I isothermically contract, like that, to get back where I started. When I isothermically contract, what's happening? Work is being done to me, so now all of this area over here will be negative. And in order to keep my temperature constant, I have to release heat. So I'm releasing heat, but I'm doing it at a high temperature. So I'm releasing it into my T1 reservoir. So it's the exact same thing as it happened before, but since when I go in a reverse direction, some work is being applied. So now, when you look at it this way, when you figure out all of the areas, the area in here will actually be negative. And the reason why I'm saying that is because the positive work values are going to be this. This is going to be the positive, what I'm doing in blue right here. And the negative work values are going to be all of this stuff. So if you wanted to figure out the total work done, it's going to be negative. So what's happening, if I run the Carnot cycle in reverse-- so I'll call it the Carnot refrigerator. No, that's not what I wanted to do. I'll call it Carnot reverse. But it's handy that R also stands for refrigerator. This is the Carnot engine. It does work by using heat, by taking advantage of the heat difference between this hot-- you could view this as the T hot and the T cold. Now, a reverse Carnot engine, or maybe you call it a Carnot refrigerator, does the opposite. That's exactly what I just drew over here. What it does is, it starts with a cold body-- I'll call that T cold, or T2-- it takes some smaller amount of heat from the cold body. Some work has to be input into the system in order to do this. And then it puts more heat-- you can kind of view it as a combination of this work and this heat taken from the cold body-- and it gives it to the warm body. Sorry. This is Q2, and it gives it Q1. So everything just happens completely in reverse. And that's just a byproduct of, this is reversible. So I can just go and I can do, if this is the way we went before, when we're an engine, if we want to be a refrigerator, we go the other direction, and everything just gets reversed. And I want you to really understand that this is doable. That there's nothing wrong with this. You might say, doesn't this defy the second law of thermodynamics? We're taking heat from a cold body to a warm body? And my answer will be the same thing I said on my entropy videos. I said, well, no. We're applying some work. This is a refrigerator. So some work has to be done in order to do this. And whatever object that is doing the work-- it may be some, in the case of your refrigerator, it's a compressor. That is adding more entropy to the universe than the entropy that's being destroyed by our refrigerator. So this does not defy the second law of thermodynamics. Now, I want to make another point about the Carnot engine. Let me take the reverse Carnot engine. Let me call it the Carnot refrigerator. So if we take that-- and this is really just more math than anything else. If we're taking in Q2, adding in some work, and producing Q1, we can scale this up arbitrarily. If we take x times Q2 in, and we put in x times W in, then we're going to put x times Q1 into our top reservoir. And that makes sense, because these are just arbitrary numbers. For example, if we have two Carnot engines in parallel, you can just kind of view that whole thing as two Carnot engines doing it together, so all of these would be 2s. If we had three Carnot engines doing it together, all of those would be three, you could just view them as one collective engine. Now, with that said, I think I've laid the framework for at least the ideas that will let us show that the Carnot engine is the most efficient engine that's able to be produced. And given that the Carnot engine's efficiency is this-- and we're going to prove that it's the most efficient engine-- this becomes the upper bound on efficiency for any engine that anybody can or will ever make. And I'll kind of do the crowning touch in the next video.