 All right, so we understand now the Carnot cycle, which I've drawn a little sketch of over here. The idea behind the Carnot cycle is I do a reversible isothermal expansion. So this is an isotherm. Here's another isotherm. Isothermal expansion followed by a reversible adiabatic expansion, and then an isothermal followed by an adiabatic compression. In the process of going around that cycle, we've managed to turn some heat into work somehow. So we're describing that another way. There's some heat associated with this process, the reversible isothermal expansion. Both of these expansion steps do some work. The gas expands, pushes back on the atmosphere, does some work on the atmosphere. That work is paid for either by absorbing heat from the surroundings or by using some of the energy of the gas itself when it cools down in the adiabatic expansion. And then to get back where I've started, I need to... The cold reversible isothermal expansion step has some heat associated with it of the opposite sign, so I have to give back some of that heat I absorbed from the atmosphere, from the surroundings. And then the adiabatic compression restores some of that energy to the gas as well. In the process, the net area inside these four sides of this diagram, the work is a negative number, but the area of this shape right here describes the amount of work, the net amount of work done by the gas during this PV process. So the details of that for an ideal gas or for some other type of gas involve some equations, but schematically, let me describe that a slightly different way that's a little bit simpler. So there's some surrounding, some atmosphere that's initially at our hot temperature, T-hot. If we absorb some heat from those surroundings, so here's... This circle right here will be the system. I'm going to absorb some heat from the surroundings. I'm going to use that heat to do some work. So this process of converting heat into work is the main thing we're interested in. That's called a heat engine. Engine for reasons that will become clear in just a second. So absorb some heat, do some work, but I don't get to keep all of that energy in the form of heat that I've absorbed and use all of it to do some work. I have to give some of it back. So that's this Q-cold. And when I give it back, I'm doing that at the colder temperature. The arrows on this diagram are meant to indicate the signs of Q or W or the direction of the energy flows into or out of the system. When Q is positive, energy is flowing into the system. When W or Q is negative, the system is either doing work on the surroundings or energy is flowing out of the system in the form of heat. So describing this diagram, what I mean to say is there's a heat reservoir. I absorb some amount of heat from the reservoir. I use some fraction of that heat to do some work, some fraction of the energy that I've absorbed in the form of heat. I use some fraction of that to do some work. And the rest of it gets paid back to the surroundings, but at this cold temperature. I dump that heat into this cold temperature reservoir. So this is just a diagram illustrating what's going on over here without having to worry so much about the details of the PV processes involved. Notice that this can capture the details without specifically talking about whether it's isothermal or adiabatic or what the details of the process are. In fact, a heat engine, so this process illustrates what we call a heat engine. Examples of a heat engine could be something like the Carnot cycle that we've spent a fair amount of time talking about, isothermal and adiabatic expansions and compressions. But it could also refer to other types of cycles as well, such as the cycle that goes on inside a steam engine. That's a very old-fashioned technology these days, but that's the technology that Carnot was interested in when he started studying this process. More modern process is a combustion engine. In each of these two processes, the goal is still to convert heat into useful work. Either heat, when we burn coal and boil water and generate steam, we use that steam to do the work, or we burn gasoline and cause the pistons in an engine to go up and down, and that is what we use to do the useful work in a car engine, for example. But at its core, both of those processes just involve generating heat and using some fraction of that heat to do some work. We don't get to use all the heat. Some of it gets rejected out the tailpipe or in some other way to a colder temperature reservoir. That's the basic idea of a heat engine. What's interesting, once we've started talking about engines and using heat to do work, is how efficient that process is. What is the efficiency that we have of turning heat into work? We're going to use this symbol eta for the efficiency. For the efficiency of a heat engine, you can think of that as how much benefit you get out of the engine relative to how much it costs you as inputs to that process. For this heat engine, the benefit that we get out of it is the useful work. Whether it's the car or the locomotive, or whatever it is, the work that we derive from that process, that's the benefit that we get. The benefit is W, but we need to keep in mind W remembers a negative number. That's from the system's point of view, it is lost energy. W is a negative number. The amount of work that we get out of the process being outside the system is negative W that converts W into a positive number. The cost it takes to generate that much work, to have the system generate that much work, is Q sub H. Again, whether it's a car engine or locomotive, steam engine, or some other process, we need to heat something up to the hot temperature and that's the source of the heat that's provided to the system in this step. So that's either the gasoline or the coal. Somehow, the cost of generating this heat is our cost. So heat from the hot step is a cost that needs to be paid by us as a cost of doing that work. We can then recall from our discussion, our more quantitative discussion of the Carnot cycle, what the network for the whole process was. That was NR T-hot minus T-cold log V2 over V1 for the Carnot cycle. We had minus NR T-cold minus T-cold. So negative W, I just removed the negative sign when I write that down. The heat for the hot step in that process, remember the hot step was this reversible isothermal expansion. So the heat associated with that step we had was NR T-hot log V2 over V1. So that ratio is our efficiency. How much work we get out of the process relative to the heat it costs us to generate that work. There's a lot of cancellation. The N's and R's cancel. The log terms cancel. So what we're left with is T-hot minus T-cold over T-hot. Or perhaps an easier way to think about that, T-hot over T-hot is one. And T-cold over T-hot, I can't do anything with that one necessarily. So I'll just write that as T-cold over T-hot. So the efficiency of this process, the fraction of the input heat that I get out in the form of work, is one minus this ratio of the temperatures. That'll make a little more sense probably if we use a specific example. So let's say, just because the numbers are fairly easy, let's take the case of a steam engine. Let's say we're running a steam engine, so we're getting our work out of boiling water. So in a steam engine, the hot temperature is the temperature of boiling water, 100 degrees Celsius or 373 Kelvin if it's not a very high pressure boiler. So that's the hot temperature. And let's say we're running our steam engine at ordinary conditions where the environmental temperature is something like 298 Kelvin. The efficiency, let's do it instead as using this form, T-hot minus T-cold over T-hot. Or 373 minus 298 over T-hot. So 75 Kelvin in the numerator divided by 373. That works out to be 0.20. So numerically that's just an example. Intuitively what that means is if we run a steam engine by generating heat at a temperature of 373, use some of that to do work, we only get back 20% of the input heat in the form of work. So that's sort of a limit on how efficient our steam engine can be. We can only get back 20% of the energy that we put in. So that steam engine is not terribly efficient and that's because our operating temperature, our room temperature 298 Kelvin is a pretty large fraction of the operating temperature of 373 Kelvin. So in general these heat engines tend not to be very efficient as in this example. All we've done so far is illustrate that that's true for the case of an ideal gas doing a Carnot cycle. But it turns out these equations about the efficiency of a heat engine are a little more general than just for a Carnot cycle on an ideal gas. So that's our next step is to talk about the efficiency of heat engines more generally.