 Alright, so we have an understanding of what a heat engine is. A heat engine is a process that converts heat into work, specifically by absorbing some amount of heat from a hot temperature reservoir, using some of that heat to do work, and having some leftover heat that it rejects to a cold temperature reservoir. Several times when talking about heat engines, we've talked about the need for those processes to be reversible. So if in fact they're reversible, let's think about what happens if we actually reverse them. And if we change the direction of each of these arrows. So remember that the arrows indicate whether energy is flowing into the system, or absorbing heat, or whether it's flowing out of the system, we're rejecting heat, or we're doing work. So if I redraw this figure, just reversing each of these arrows, the hot heat process is flowing from the system to the surroundings. The work is being done on the system rather than by the system. And at the cold temperature, heat is flowing out of the surroundings into the system. I've just reversed all those arrows. Let's think about what that means. If this process involves absorbing some heat at a hot temperature, using some of it to do work, and letting the rest of it flow down to the cold temperature, let me point out that it's no accident that I've drawn this figure with the hot temperature on top and the cold temperature on bottom. It's a very natural process for heat to flow from the hot temperature to the cold temperature. In fact, when Carnot came up with this idea of heat engines and this way to describe them, he was thinking of something like a waterfall, where it's very natural for water to flow from a high position to a low position, and you can make use of the natural tendency for water to flow downhill to turn a water wheel, to get some work out of the process as water flows from high to low. Same thing here. Heat will naturally flow from a hot temperature to a cold temperature. We can make use of that natural tendency to do some work along the way. When we reverse the process, what these arrows describe is we're absorbing some amount of heat from a cold body, from a cold reservoir. We're going to put some amount of heat up into a hotter temperature. So I've got a net amount of heat flow from a cold temperature to a hot temperature. That's not a natural thing to happen. It doesn't normally happen that heat flows from a cold thing to a hot thing. So we have to actually do some work. It costs us some energy in order to pump the heat from the cold temperature up to the hot temperature just as it would cost us some energy to pump water from a low level up to a high level. So in analogy with that statement, this process is called a heat pump. And it's just the exact reverse of a heat engine. Some examples of heat pumps would be things like a refrigerator. So that's exactly what we expect a refrigerator to do. We think of a refrigerator as a way to cool down our food, but we have a cold inside a refrigerator and we need to remove some heat from that cold place and we end up putting that heat in a hotter place. So when you bring your groceries home and you put them in the refrigerator, you want to remove some of the heat that's keeping those groceries at room temperature. You want to cool them down. So you're removing heat from the cold surroundings, costs you some electricity to do that, and then you dump that excess heat out the back of the refrigerator. So that's exactly what a refrigerator is doing. It's also exactly what an air conditioner does. If you have a window air conditioner in the window of a room, it's removing heat from the cool inside of your house, cooling it down further and dumping that excess heat out the window to a place that's probably warmer. Another example, and this one is perhaps the one to keep in your mind for the rest of this video because it's the best illustration and also the most confusing illustration of this idea of a heat pump. So it's worth getting the details correct. What we call a heat pump in a central air HVAC system for a house, for example, is an example of a heat pump. In the summer, your heat pump works the same way as an air conditioner. It's hot outside, it's cool inside, you want to keep it cool so you want to remove some heat from the inside and dump it out to the warmer outside. So that's exactly what this heat pump is doing. Counter-intuitively, what your heat pump does in the winter, when you warm up your house, you know, it's cold outside, warm inside, you want to make sure it stays warm inside. So the way the heat pump works in the winter is it uses exactly this process. It extracts some heat from the outdoors, from the cold winter outdoor air. It extracts some heat from the cold air and dumps it into your house. So your house is like the place where the waste heat goes and you're using that waste heat to heat up your house. So heat pump, when you run it in the winter time, it's removing heat from the outdoors, dumping it into your house. So heat pumps, what we want to say about heat pumps is, oh yes, their efficiency. How do we talk about the efficiency of a heat pump? Efficiency of a heat engine we've talked about before. It's related to this ratio of temperatures, T cold to T hot. The efficiency of a heat pump, we don't actually end up calling it efficiency. So we're going to use a different term. We're going to use beta. This term beta, we call it coefficient of performance, and it will become clear in just a minute why I'm not going to call that an efficiency. The coefficient of performance or COP for heat pump, that's still philosophically going to be the amount of benefit energetically we get from the heat pump relative to how much it costs us to run the heat pump. The definition of what the benefit is, what the cost is, depends on whether we're using it to heat something up or cool something down, but the basic idea is still the same. So let's think about, let's say first, when we're using the heat pump to heat something up, when we're running this heat pump in the winter and our benefit is the heating effect that we get from the heat pump. You're using your heat pump to heat your house. So the benefit you get from this heat pump is you pay some electricity to do some work to pump in this unnatural direction, remove some heat from the cold outdoors, and pump it into your warmer house. So the benefit you're getting is this Q sub H. The cost doesn't cost you anything to remove the heat from the outside. The cost to you is the amount of work that you have to do. The electricity you have to pay to make this heat pump run. So that's the cost. Let me make sure I have my signs right. Q H is a negative number, so I want to make that negative on top. So the benefit to us is a positive amount of heat, so that's negative of this negative quantity. And work is already a positive number, so I'm going to leave that just the way it is. So the way I can make sense out of this is the same thing we did before. Whether I look at this diagram or whether I look at this diagram, the net change in energy of the system has to be zero if I combine all these arrows together. So if I add up total amount of heat and total amount of work, that has to be zero. So that, let's say, W is going to be minus Q H minus Q C. So that's what I'll put in the denominator here. We've got minus Q H on top, minus Q H minus Q C on bottom. And that's too many negative signs, so let me just get rid of all of them at the same time. So Q H over Q H plus Q C. I want eventually to turn those Q's into temperatures the way we did when we were talking about the energy, I mean the efficiency of a heat pump. Remember, we used the fact that the ratio of the heats is the negative of the ratio of the temperatures. So maybe not the shortest, but the way I can guarantee I won't make any mistakes. Let's divide through top and bottom by Q sub H. So Q H becomes one, Q H becomes one, Q C becomes Q C over Q H. Then I've got this ratio of Q C over Q H, which I can write as negative T C over T H. And now that I've done that, I don't like the way it looks with the fraction inside a fraction. So let me go through and re-multiply by T H in this case. And what I'll get is, so one becomes T H, one becomes T H, and T C over T H becomes T C. So the efficiency, sorry not the efficiency, the coefficient of performance for a heat pump being run in heating mode is this ratio, hot temperature divided by hot minus cold. So similar to efficiency of a heat engine in the sense that it involves just the ratio of temperatures or difference in temperatures, but it's a different combination of those. We'll plug some numbers in in just a minute and see what kind of values we get for this coefficient of performance. But before we do that, let's run through the same calculation real quick. If we're using our heat pump in cooling mode rather than in heating mode. So if we're using heat pump as an air conditioner, as a refrigerator, where the benefit we get to us is how much heat it removes from the inside of the refrigerator or the inside of the house when we're using our air conditioner or our heat pump. So in this case, the benefit that we get isn't the amount of waste heat that's dumped out the back of the refrigerator or outside the house when you're cooling your house. The benefit we get is how much heat we extract from the inside of the house. So the benefit we get is Qc, which is a positive number. I want to divide that by the electricity cost or the amount of work I have to do in order to pump that heat uphill to the higher temperature reservoir. So again, the algebra is going to be almost the same, but not quite. The work in the denominator is minus Qh minus Qc. Let's again flip the signs. I'm going to make these positive and that one becomes negative. So I've got, if I divide through in this case, again by Qh, I'll get minus Qc over Qh over 1 plus Qc over Qh. Now I can use the fact that the ratio of heats is equal to the negative the ratio of the temperatures. So that'll be negative becomes positive Qc over Qh. The denominator I've got 1 minus Qc over Qh. And then if I get rid of the fractions inside the fractions, multiplying by Qh gives me Qc over Qh minus Qc. Notice that this result for cooling is not the same as this result for heating. Heating the coefficient of performance is hot temperature over the difference in temperatures. When I'm cooling, the coefficient of performance is cold temperature over the difference in temperatures. So slightly different equations depending on whether we're heating or we're cooling. Just to make sure that concept sticks a little better, let's work an example. Let's say we are using a heat pump in the winter to heat up our house. So let's say the outdoor temperature in the winter is 5 degrees Celsius or if you prefer Fahrenheit, that's 41 degrees Fahrenheit. So 273 plus 5 is 278 Kelvin. And we're keeping the inside of our house at a comfortable 68 Fahrenheit or 20 Celsius. So 293 Kelvin. And we'd like to know how efficient we can expect a heat pump to be under those circumstances. So we're using the heat pump in heating mode so the equation we need is hot temperature over the difference in temperatures. So our hot temperature is 293 Kelvin, temperature inside our house. The cold temperature is hot minus cold, 293 minus 78. So that's 293 divided by 15 Kelvin. And mathematically that works out to be 19.5. So our coefficient of performance for a heat pump run on a day with these temperatures is 19.5. So the first thing we notice here explains why I didn't want to call this an efficiency. This coefficient of performance is typically very often larger than one for a heat pump. So if we were calling it efficiency, if we said the efficiency was greater than 100%, that wouldn't be wrong. It's just a little bit confusing. It's not what we think of efficiency as being. If heat's flowing downhill and we siphon off a little bit of that to do work, the efficiency is just how much of it we can manage to siphon off. In this case, the amount of energy we have to supply to pump energy uphill to get it from the cold temperature to the hot temperature in the form of heat. Turns out that's a pretty nice ratio. So I want to dump 19.5 joules worth of heat into my house. I've only got to pay for one joule worth of electricity and that will be enough to pay to suck 18.5 joules out of the outside and dump it into my house. So coefficient of performance is commonly greater than one because heat engines are inefficient and the consequence of that fact is heat pumps tend to be quite efficient. What that means is if you've ever paid much attention to the energy efficiency of your heat pump, you may know that heat pumps are a much more energy efficient way to heat your house than a heater, a gas heater, an electric heater. If you run a gas heater or more specifically an electric heater, every joule of energy you supply in the form of gas or in the form of electricity gets turned into heat. So that electric heater may be 100% efficient. You might turn all the electricity into heat, but it can't compete with a heat pump, which is generating 20 times as much heat in this case as the amount of electricity that you provide for it. So that's an interesting fact to know about heat pumps from an everyday point of view, but from a thermodynamic point of view, really the important thing we've learned is that a heat pump is really just the reverse of a heat engine. Instead of letting heat flow downhill and using that to do some work, we're pumping heat uphill and supplying work in order to do that.