PV diagrams - part 2: Isothermal, isometric, adiabatic processes

- All right, so last time we talked about isobaric processes This time let's talk about isothermal processes. Iso means constant, thermal, this is short for temperature so this is a process where the temperature remains constant. Or, in other words, T equals a constant, which we could also write, if temperature is constant, that means the change in the temperature, that means there is no change in the temperature, so the change in the temperature is just zero. Now, before we move on, let me show you one more important thing. Remember, we said previously, the internal energy of a gas is not equal to, but it's proportional to, the temperature of a gas. And so if the temperature doubles, the internal energy doubles. If the temperature doesn't change, the internal energy doesn't change. So for an isothermal process, not only is ΔT=0, but more importantly, in terms of the First Law of Thermodynamics, ΔU is also equal to 0. This is important, this is something you have to know. For an isothermal process, ΔU is 0. Now you might be confused here. You might say, "Hold on a minute, how can you have "a thermal process if there's no thermal, if there's "no change in the temperature at all?" Well, you can, stuff's gonna happen here. This doesn't mean nothing happens. Things are gonna happen, but they're gonna happen in such a way that there's no change in the temperature and there's no change in the internal energy, so what can we say? Well, let's look at the First Law of Thermodynamics. The First Law holds, for any thermal process in here, whether it's an isotherm, an isobar, any of them, so we can say that ΔU has got to equal Q, which is the heat that flows into a gas, plus W, which is the work done on the gas, and now we know, for an isothermal process the ΔU is just 0. So what does that mean? That means that Q + W have to add up to 0. So this means if you do, say, 300 joules of work by pushing this down, you do 300 joules of work. The only way the temperature's gonna remain constant is for 300 joules of heat to leave the gas, 300 joules of heat would have to leave. That would mean that Q is -300 joules, that way 300 joules and -300 joules add up to 0, you've got an isothermal process. But it's not enough for just the initial temperature to equal the final temperature. In order for this process to be truly isothermal, the temperature has to remain the same at every moment during the process. So every bit of energy you add has to immediately get taken away, or every bit of energy you take away has to immediately get added back in. There can't be a delay, otherwise you'd add this 300 joules, the temperature of the gas would increase, and then the heat would conduct outta here, you know, at its leisure, it would take some time, and then finally you'd reach the same temperature as before, that doesn't count. That's not an isothermal process. The gas has to be at the same temperature at all moments. So how do you do this? Well, just make sure you push down the piston, or if you're pulling it up, you do so very slowly. That way, the heat always has time to conduct out or in accordingly. If you make the process happen too fast, this heat, yanno it takes some time for heat to conduct through a container, and if you make the process happen too fast, this heat can't conduct out of the container or into the container fast enough, so you've gotta make the process happen very slow, add 1 joule per hour or per day, make it as slow as possible so that this heat always has time to conduct accordingly and maintain a constant temperature. Constant temperature with what? Well, just stick this whole thing in a thermal reservoir, this is how you could actually do it. Put the whole container in a tank of water that has a temperature of, say, 290K, a huge tank of water. Water doesn't change its temperature very easily since it has such a high specific heat, so if the tank is very large, this water's gonna maintain the same temperature, it's not gonna care about a little piston in here, but the gas in the piston is gonna try to maintain equilibrium with temperature of the water. So if you make this process happen very slow, if I push down the piston very slow, I'll add energy, but that energy's gonna get taken out, the temperature of the gas will remain the same if I do it slow enough, or I can pull up on the piston very slowly, then some heat has to enter into the gas so that it always maintains the same temperature with the outside environment, ensuring that it's an isothermal process. So what's an isothermal process look like on a PV diagram? Well, let's look at the Ideal Gas Law. The Ideal Gas Law says that P x V = NKT, at least the Boltzmann's Constant version of it does, so I want to know what the pressure is as a function of volume, let me just solve for pressure. Pressure = NKT/V. Now lookit. N number of molecules in here, that's a constant we're not letting any gas molecules in or out. K, that's Boltzmann's Constant, that number doesn't change. For an isothermal process, temperature's also a constant. Everything up here in the numerator's a constant, and we just have P, as a function is constant/V, so P just goes like 1/x. It's like having a function y = a constant over x, and we know what 1/x looks like, it looks like that. So on a PV diagram, an isothermal process is gonna look something like this, it's gonna curve like 1/x and it can be an isothermal expansion if volume increases or an isothermal compression if volume decreases. So the actual shape of the line drawn on a PV diagram for an isothermal process is sometimes called an isotherm and they look like that. Notice that we cannot find the work done by just saying Work is PΔV. Remember, that's how we found the work done by the gas in an isobaric process, but that was because we had a nice rectangle. The area underneath this graph is still gonna give us the work done, that's true. This definitely is the work done by the gas, but it's not a perfect rectangle so you can't use this formula, you'll have to know, you'll have to be given the heat and then you can figure out the work or given the work, you'll find the heat. There's not a really good way, unless you're gonna do calculus, to figure out the area underneath this curve. One more thing that you should definitely know. Because the NKT is a constant, right, all of this stuff is not changing for an isothermal process, that means P x V is also not changing. That's another thing that doesn't change. So T doesn't change, U, the internal energy, doesn't change, and P x V does not change as well because T isn't changing over here in the Ideal Gas Law. That means, if you take the pressure times the volume at any point along this isotherm, you'll get the same number so this volume here and this pressure right here, if I take those two and I multiply those two together, I'll get some number, and if I take the final one, this volume and this pressure, and I multiply those two numbers, I'll get the same number. I'll get the same result for P x V. I'll get the same result here, if I take these two, any P x V value along this line is gonna be the same because that number can't change, cause if it did that'd mean the temperature had to be changing and then you wouldn't have an isotherm. So that's the isothermal process. That's one of the four most common thermal processes. We've got two more to go. Let's talk about the isometric process. The first thing you should know is this is sometimes called isochoric, and it's also sometimes called isovolumetric. Why does it have three names? I don't know, but they all mean the same thing. Iso means constant, volumetric and choric and metric all refer to size or volume This means constant volume. How do you make sure that happens? Well, just don't let the piston move. The piston is the thing that regulates the volume. Well, that thing's shut, I dunno, keep that thing from moving and you'll have, no matter what else happens, an isometric, isochoric, or isovolumetric process, which all mean the same thing. Now, since the piston can't move, that means no work can be done. The gas can't do any work, the outside forces can't do any work, you can't do any work on the gas. No work can be done if this piston cannot be moved up or down. So W's always going to equal 0 for one of these isometric processes. That means if we write the First Law, the First Law of Thermodynamics is true for every process, says that ΔU = Q, the amount of heat that flows into or out of the gas, + W, except W is 0, so we have no work done. This is 0 and our First Law just becomes ΔU = Q or, in other words, for an isometric process, the only way to change the internal energy would be to add heat or to take heat away. So these isometric processes are actually pretty simple. What do they look like on a PV diagram? Well, the volume is staying constant. Pressure staying constant is a horizontal line, so volume staying constant is a vertical line. And if I add heat, I'll increase the pressure, and if I take heat away, I'll decrease the pressure, and this volume will remain the same, cause this piston is not allowed to move. Now remember that work is the area underneath the curve. Does that make sense over here? How much area is underneath this curve? There's no area underneath this curve. There's no area, you've just got this line here, that's not an area, that's infinitesimally thin and so that means there's no area, no area means no work is done, and that agrees with what we know about an isometric process All right, one more of the big four processes to go. Let's talk about the adiabatic process. This is one in which no heat is exchanged, so sometimes people hear that and they think, "Oh, that means that there's no change "in the temperature, right?" No, that is not right. This is definitely not what we're saying. No heat exchanged means that Q, our letter that we use to represent the heat, is 0. It means that no heat is allowed into the gas, no heat is allowed to flow out of the gas. These do not happen for an adiabatic process. And that does not mean that the temperature can't change. The temperature can change here because the piston can do work or the work can be done by the gas, but no heat can flow in or out. So you've gotta get good at delineating between the temperature and the heat. These are not the same thing. Temperature is kind of a measure of how much energy a gas has at a given moment. Q, the heat, is how much thermal energy is flowing into that gas or out of that gas. It doesn't represent how much energy the gas actually has, it's how much thermal energy you're adding or taking away. And for an adiabatic process, there is no thermal energy conducted in or out. What does that mean for the First Law? The First Law is true for every process, ΔU = Q + W, the work done on the gas, but for an adiabatic process, there is no heat, so that just means ΔU equals the work done on the gas, that's the only way you're gonna add energy to the gas is by doing work on the gas or allowing the gas to do work, then energy can be removed, but you can't add or take away energy thermally, conductively, through the walls of the container. It has to be done by the piston. So how do you do this? How do you make sure no heat gets conducted? One thing you should do is insulate this so that the heat does not easily conduct through the walls of this container, but that's not really good enough. You've gotta make sure no heat is exchanged, so you take this piston and you shove it down as fast as you can or you lift it up as fast as you can. It's the opposite of an isothermal process. There we wanted the process to happen slow so that the heat always had time to flow in or out. Here we want the process to happen so fast that the heat has no time to flow in or out. That way, we ensure that it's an adiabatic process and that Q is actually 0. So what does an adiabatic process look like on a PV diagram? It looks kind of like an isothermal process, it's just steeper. So this would be an adiabatic expansion, and these lines are sometimes called adiabats, and if you have an adiabatic compression, it would look like that. If you compare that to an isothermal process, say that started here, it would not get as far down. You can tell that that's an isothermal process because it's not as steep. So those are the four most common thermal processes you'll hear about when talking about PV diagrams, and each of them had something unique and special about them The isobaric process had constant pressure and you could find the work easily because it was a nice rectangle, which meant you could just do height times width to get the work done by the gas. There's the isothermal process where temperature is constant internal energy is constant, and the quantity P x V, pressure times volume, is also constant. There's the isometric process, also known as isochoric or isovolumetric, where the change in volume is 0, which meant, remember, that means no work can be done. The work was also 0 for an isometric process. And then there's the adiabatic process where no heat is allowed to flow into or out of the system.