Work done by isothermic process

Let's start with our classic system that I keep using over and over again. And that's because it tends to be very useful for instruction. It also tends to be the system that is most covered in classrooms. So hopefully it'll be productive for you and your school work. So I have this container. It has a movable piston on top, or kind of a movable ceiling. Well, of course, inside of my system I have a bunch of molecules or atoms bouncing around, creating some type of pressure on the system. So let's say it has some pressure, P1. This volume right here, let's call that V1. And let's say it also has some temperature that it starts off with. Everything is in equilibrium. Remember these are macro states. The only way I can even tell you what the volume, or the pressure, or the temperature is, is if the system is in equilibrium, if everything in it is uniform. The temperature is consistent throughout. Fair enough. And in order to keep it placed down, I have to put some rocks on top. And I've done this in multiple processes so far. And, of course, I'm doing these little pebbles, because I'm going to remove them slowly. Because I want to approximate a quasi-static process. Or I want to approximate a system that's always close enough to equilibrium that I'm cool with defining our macro states, our pressure, our temperature, or our volume. Let me write V for volume. Now, in this video, I'm going to study what's called an isothermic process. And really what it just means is, I'm going to keep the temperature the same. Iso- just means the same. You probably remember when we studied the periodic table. Isotopes, those are the same element just with different mass numbers. So this is the same temperature we're going to run our process. So my question is, how can we do that? Because as I remove pebbles, what's going to happen? If I just did this, without any-- if it was completely isolated from the world. And actually I'll add in a word right here. If it was an adiabatic process-- If it was adiabatic-- fancy word. All that means is, completely isolated from the world. So no heat is going into or out of this system. If this was the case, what would happen as I released or I took away some of these little particles? Let me copy and paste it. Well, let me just redraw it actually. So I have my one wall. I have another wall. I have another wall. As I release a couple of pebbles, one at a time, my volume is going to increase. So I'm going to have a slightly higher volume. My volume's going to go up. I have fewer pebbles here now. And since I have the same number of molecules, they're going to bump into this less. So pressure is going to go down. Volume is going to go up. And if I was adiabatic, if I had no extra heat being added to the system, what do I know is going to happen to the temperature? Well, think about it this way. Some work was done, right? Our old ceiling was maybe some place around here. We pushed it up with some force for some distance. So we did work. And so we changed some kinetic energy, or we transferred some kinetic energy, out of the system. That's essentially what the work did. That kinetic energy was turned into work. And temperature is just a macro measure of average kinetic energy. In fact, we just-- well, I won't go into it. But in the last video, the kind of proof you want-- if you didn't watch it because you didn't want to go through the math, which is completely fair enough because it normally wouldn't be done in an intro chemistry class-- I showed that the internal energy is equal to the total kinetic energy, which was equal to 3/2 times the number of moles, times R, times temperature. So temperature is just, by some scaling factor, a measure of kinetic energy. Now, when I do some work, it's essentially a transfer of kinetic energy. And I can't replace that energy with some heat because it's adiabatic. There's no heat going into or out of the system. So in that situation, the kinetic energy of the system went down. The average kinetic energy of the system went down. So the temperature would've also gone down. And actually, just as a bonus point, what happened to the internal energy? Well, the internal energy is the total kinetic energy of the system. And I could even right down the original formula. Change in internal energy is equal to change-- let me not do that, because I said I shouldn't-- is equal to heat added to the system, minus work done by the system. This is work done by the system. That's why we're subtracting it. Now, it's adiabatic, so there's no heat added to the system. So the change in internal energy is equal to the minus the work done by the system. Well, in this situation, the system did do work. It pushed this piston up by some distance with some force. So, your delta U is negative. It's less than 0. So U went down, and that makes sense. If temperature changed, then the internal energy is going to change. And for our simple system, where internal energy is represented by the kinetic energy of these molecules, that's always going to be the case. If temperature doesn't change, internal energy won't change. If temperature goes up, internal energy goes up. If temperature goes down, internal energy goes down. And, of course, they're not the same thing, though. The difference between internal energy and temperature is the scaling factor, 3/2 times the number of molecules, times our ideal gas constant. So fair enough. I went through this whole exercise just to show you that if I was completely isolated, and if I removed a couple of these pebbles, that my temperature is going to go down. Now, I told you already that I want to do an isothermic process. So I want to do this process while keeping the temperature the same. So how can I do that? Well, what I'm going to do is I'm going to place my system on top of what we'll call, a reservoir. So a reservoir, you can kind of view as an infinitely large amount of something that is the temperature that we started off with. So this reservoir is T1. So even though if, I took two relative things of comparable size. That says, temperature A. This is temperature B. And I've put them next to each other. They're going to average out to A plus B over 2, whatever their temperatures are. But if B is massive-- if A is just a speck of particle-- let's say it's iron dust-- while B is the Eiffel Tower, then essentially B's temperature will not change a lot. A will just become B's temperature. Now, a reservoir is theoretically infinitely large. It's an infinitely large object. So if something is next to a reservoir and is given enough time, it'll always assume the heat of the reservoir, or the temperature of the reservoir. So what's going to happen? So this is adiabatic, but now I'm actually putting it next to a reservoir. So this isn't going to happen. The adiabatic situation isn't going to happen. Now, I'm going to have a situation where I'm going to stay the same temperature. So what's that going to look like on the PV diagram? So let me draw the PV diagram. This is my pressure. This is my volume. So this is my starting point right here. And what I'm saying is, if I'm doing an isothermic process, so I just keep removing these pebbles. So I start at this state right here. Let me copy and paste it since I've done so much art already here. So I'm going from there to here where I'm removing a couple of the particles. So let's say I've removed a couple of them over here. And because of that, I've increased the volume. So let's say the volume, it's not there anymore. Let's say it's a little bit higher. Let's say the volume is-- just for the sake of our discussion-- let's say the volume has expanded a little bit, because I've remove some particles, the little pebbles on the top keeping it down. So it's like the adiabatic process, but instead of the temperature going down, my temperature stays at T1. My temperature's at T1 the entire time, because I'm next to this theoretical thing called a reservoir. So because of that, I will travel along what we'll call an isotherm. So this is my first state. When I'm done, I might end up some place over here. And so this is state 2. So this is state 2, this is state 1. What I'm claiming is that my path along this is going to be on some type of a rectangular hyperbola, or at least part of it. If I were to add rocks to it and compress it, I claim that my PV diagram would go like this. If I were to keep removing rocks from this diagram, I claim that my PV diagram would keep going like that. And so what's the intuition? That if I keep the temperature constant, that I'm essentially moving along this hyperbola. Well, let's just take out the ideal gas formula. Let me box off all this stuff over here. If I just take the ideal gas formula, PV is equal to nRT. If T is constant-- we know that R is a constant, it's the ideal gas constant. We know that we're not changing the number of moles of particles. Then that means that PV is equal to some constant. This whole thing is equal to some constant. And then, if we wanted to write P as a function of V, we would just write P is equal to K over V. Now, this might not look 100% familiar to you, but if I wrote it in algebraic terms, if I told you to graph y is equal to 1 over x, what does that look like? That's a rectangular hyperbola. That looks like this. And this is the y-axis, that's the x-axis, at least in this quadrant it looks like this. It also looks like that in the third quadrant, but we won't worry about that too much. So whenever you hold temperature constant, you're on some rectangular hyperbola like this, like an isotherm. Now, if the temperature was a different temperature, if it was a lower temperature, you'd be on a different isotherm. So you would be on an isotherm that looked like this, maybe over here. It would also be a rectangular hyperbola but at a lower state. Why is that? Because, if you're at a lower temperature, for any volume, you should have a lower pressure and that works out. That's why this is some temperature T2, that is lower than T1. So I want to do a couple of things in this video. I inadvertently explained to you what an adiabatic process is, and why the temperature would naturally go down on its own if you didn't have this reservoir here. But the whole reason why I even thought about doing this video is because I wanted you get comfortable with this idea of, one, that a reservoir will keep you in kind of an isothermic state. It will keep the temperature the same. And that if you keep the temperature the same, that you will travel along this isotherm, these rectangular hyperbolas. And that each temperature has associated with it an isotherm. So if you take that, let's just do one more step. And let's think about the actual work we did by traveling from this state to this state. Or if you just want to think of it in visual terms, from removing our pebbles slowly and slowly with this reservoir down here the whole time, from this state to this state. Where our volume has increased, our pressure has-- So our volume has increased. Our pressure has gone down. but our temperature has stayed the same the entire time. So, several videos ago, we learned that the work done is the area under this graph. It's the area under that graph. Or, if we want to do in calculus terms-- and I'm about to break into calculus, so if you don't want to see calculus cover your eyes or ears. It would be the integral. And the rest of this video will be a little bit mathy, and I guess I should make that statement on the title of the video. But if I want to calculate what this area is, I can now do this. The isotherm assumption makes our math a little bit easier. Because we know that PV is equal to nRT, ideal gas law. Or we could say P, if we divide both sides by V, is equal to nRT, divided by V. So there we have it. We have P as a function of V. This function right here, this graph right here, is this. We could write P as a function of v is equal to nRT over V. So if we want to figure out the area under the curve, we just integrate this function from our starting, our V1 to our ending point, to our V2. So what is that going to be? Well, we're going to integrate from V1 to V2. Actually, that shouldn't be an equal. The work is going to be the integral from V1 to V2, times our function, P as a function of V, times dV. We're summing up all of the little rectangles here. We did that a couple of videos ago. So what's P as a function of V? So work done is equal to, from V1 to V2, nRT over V, times dV. Now, this is our simplifying assumption. We said we're sitting on top of a reservoir. That this reservoir keeps our temperature the same the whole time. And we're going to learn in a second, it's doing that by transferring heat into the system. And we're going to calculate how much heat is transferred into the system. So, if we look at this right here, temperature-- since we're assuming we're on an isotherm, is a constant. n and R are definitely constants. So we can rewrite this integral as the integral from V1 to V2 of 1 over V, dV. And then we could put the nRT out here. I should have done that first. nRT, it's just a constant term. Now, what's the antiderivative of 1 over V? It's the natural log of V. So our work is equal to nRT times the natural log-- this is the antiderivative-- of V, evaluated at V2 minus it evaluated at V1. So that is equal to nRT times, evaluated at V2, so the natural log of V2. Minus the natural log of V1. Now we know from logarithm properties is the same thing as nRT times the natural log of V2 over V1. So there you have it, we actually calculated the real value. If we know our starting volume and our finishing volume, we can actually figure out the work done in this isothermic process. The work done in this isothermic process is the area under this. And we figured out what it was. By pushing up that piston, it was nRT. These are the number of moles we have, ideal gas constant. Our temperature that we're sitting on. It would be T1 in this case. And the natural log of our finishing volume divided by our starting volume. Now, let me ask a follow-up question. How much heat was put into the system by this isotherm here? It put in heat to keep the temperature up, otherwise the temperature would have gone down, right? Heat was going into the system the entire time. How much was it? Well, since it's an isotherm, since the temperature did not change, what do we know about the internal energy? Did the internal energy change? The temperature not changing told us that the kinetic energy didn't change. If the kinetic energy didn't change, then the internal energy did not change. And we know that the change in internal energy is equal to the heat put into the system minus the work done by the system. Now, if this is 0-- so we know that this didn't change, because the temperature didn't change. So that means 0 is equal to Q minus W, or that Q is equal to W. So this is the work done by the system. You'll end up getting something in joules. And this is also equal to the heat put into the system. It's also equal to Q. So when you look at this, if we were to just draw this part of the curve-- let me redraw it just to make things neat. I want to give you a little bit of the convention of what people in the thermodynamics world tend to do. I'll make a neat drawing here. We started here, at state 1. And we moved along this rectangular hyperbola, which is an isotherm, to state 2. And now we calculated the area under this, which is the work done, which was this value right here. Let me write it there. It's nRT natural log of V2 over V1. This is V2. This is V1. This whole axis, remember, was the V-axis, volume axis. This axis here was the pressure axis. And the convention is that because we did work, but we were constant temperature, so our internal energy didn't change, we had to add energy to the system to make up for the work we did. So some heat must have been added to the system. And then what they do is they just put this little downward arrow and they write a Q right there. So some heat was put into the system during this isothermic process right there. And the value of that Q is equivalent to the work we did. We put the exact amount of heat into the system as the work that was performed. And because of that, our internal energy didn't change. Or you could say our temperature didn't change. Or you could go the other way. Because our temperature didn't change, these two things have to be equal. Anyway, I want to leave you there. Hopefully I gave you a little bit more of an intuition of how PV diagrams work, a little bit more intuition behind what isotherms and adiabatic mean. And the most important thing is, once we get a little bit mathy, this result can be useful for coming up with other interesting things about a lot of these thermal systems that we're dealing with. See you in the next video.