# Pressure-Volume Diagrams: Cycle

by Jared Rovny

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00:01 Now we're ready that we have the basics of the pressure volume diagram to discuss a cycle.

00:07 So, a cycle is any change in your pressure and volume in your diagram that starts at one point, undergo some possibly very complicated behavior, and then goes back to the original starting point -- the original pressure and volume.

00:20 The really important thing about a cycle is that because pressure and volume are state variables, and because the energy is also a state variable, if I know a particular point on this graph that I know I've returned to the same amount of energy of my system if the same amount of material is in my system at the same temperature.

00:41 So we can trace our path along the cycle.

00:45 In cycles, by the way, there are many examples of this cycles just so we can have something concrete in our head as we go through this.

00:51 One kind of cycle that was very important to the historical development of curves like this one, is an engine.

00:58 An engine is undergoing the same kind of process over and over again to maybe turn a wheel, like in your car.

01:04 And this may be igniting some sort of a substance and then maybe compressing again with a piston, and it's going to keep undergoing the same sort of change, increasing and changing the pressure or decreasing and changing the volume, and so that will be an example of a cycle.

01:17 But we also have cycles much closer to us, the action of your heart.

01:21 As it changes, the pressure and the volume of the heart to pump blood to the system, is also cyclic.

01:27 It's also changing the pressure, changing the volume, and then returning to its original state to repeat the same process over and over again.

01:34 So this cyclic processes where you have pressure and volume in a system changing and going back to where it started and doing the same thing over and over again are very, very common.

01:43 So again, let's follow a particular system that's doing this.

01:47 Let's suppose that in the graph we have here.

01:49 During the forward process, we can have a change where we're going down the line, as we just discussed.

01:56 Finding the area under this line, we already said, was the work done during this process because we have the pressure times the change in the volume.

02:05 We then are going to go to the next line, which is this horizontal line which is going backwards and we do have to be very careful about the sign here.

02:13 Because this line is moving to the left, meaning that the given pressure is decreasing our volume, We have a negative work done by the system, or to say it another way, work is being done on our system.

02:25 Which again, you can think about very physically to help it stay in your mind.

02:29 If you have a given pressure of your system and the volume is decreasing, that means there must be a force acting while the walls of your system are closing in and that force times the distance means that work is being done outside onto your system.

02:43 So work again is being done on the system in this case as this line moves right to left.

02:49 But again, the amount of the work we can still find is the area under the curve but we will considered this to be a negative work done even though we can find the area just by multiplying the delta V times the pressure that this is at, the P value and the P delta V.

03:04 Finally, we have what we said was as an isochoric change, which is the line going vertically up back to our starting point.

03:10 So that line doesn't contribute at all to the work because there is no P delta V.

03:14 Or in other words there is no area unto that vertical line; We couldn't find that.

03:18 So there is no last work contributions to be found.

03:21 So the total work done during the course of this whole process will be this positive green area which remember, originally extended all the way to the axis, that volume-labeled axis, minus this red area, because again that was a negative amount of work.

03:36 So when we put the entire green area together with the entire red area, one being positive and one being negative, that red area will simply subtract out or cancel out the part of the green area that was originally below this pressure-volume curve.

03:50 Meaning, that the total work done by our system is equal to the enclosed area by the curve created by our cycle.

04:00 And that is the key expression.

04:02 It's very important especially for an exam setting to be able to look for, or look at a cycle like this one, a PV curve where we have some process no matter how complicated it is, and know that to find the work done in total during this process, you simply need to somehow find that area, the area enclosed by this curve.

04:21 It will be equal to the work done for the reasons that we just discussed, we should always able to use some geometry of your system.

04:27 Sometimes it won't be a curve like this.

04:29 Maybe it will be very geometric in straight line so you can find the areas of triangles or rectangles.

04:33 But finding the area enclosed by this curve no matter what those shapes will give you the amount of work done by your system.

04:41 One last thing to say about this: Since we went through a complete cycle and we're saying that the energy of our system is conserved, We know that across this whole cycle, work was done which means that heat must have been added to the system.

04:57 Because otherwise, how could we keep the energy the same? We know that the change in the energy has to do with both the work done by your system and the heat energy added to your system.

05:06 So if the energy doesn't change and we know there was some work done the only way to keep this equation consistent going back to the energy equation we wrote before would be to add a heat energy to our system.

05:17 One other way of saying this is that if your system is doing work, it's causing its environment to change somehow but it's keeping its energy constant, somehow it must have received some outside contribution.

05:29 And in this case the contribution was heat contribution.

05:31 So, heat was added to our system in ordered to again keep this energy, this delta U, zero, since we went through an entire cycle.

05:40 So in this case if we have the delta U is equal to zero, meaning that we added heat while we did work, we can simply rearrange this equation and see that the amount of work that our system did has to be equal to the amount of energy or the heat energy it received.

05:56 So an engine or any cycle like this, like an engine or even your heart, is exactly doing this.

06:01 It's taking in heat energy and then turning that into some useful work, some change in its environment, maybe pushing blood to our system or making your car go forward.

06:11 We can also change this and say that we want the other way.

06:15 Suppose that this graph as I've drawn it is exactly the opposite.

06:18 The only thing I've changed is the arrows.

06:20 We did have the lines going one way and then tracing back and then going up.

06:24 Now we have it reversed; we're saying that in this long curved line we are actually going up the line, up to that higher pressure.

06:31 If this is the case, as we already discussed, we have the area under this curve which is going backwards will now be a negative quantity because we're moving in the negative change in volume direction.

06:42 We could use all the exact same arguments for everything else including that the area under the curve below it to the axis will be cancelled up, positive to negative.

06:50 And so in this case, the exact opposite thing is happening -- work is now being done on your system.

06:57 We would say that this area now enclosed by the lines of your graph is now a negative work.

07:03 Again, just following the logic of moving the lines backwards, creating areas that are negative, we have work done on your system rather than by your system.

07:12 Again, since this is a cycle, this means that the total energy is conserved.

07:17 So by the same logic that we just applied, this means heat was lost from our system rather than added to our system.

07:24 And all other same logic will apply otherwise to the system we just described with the energy change being zero and work and heat being exchanged.

07:31 So this would be the exact same for this system as well.

07:36 So this completes our introduction to pressure-volume diagrams as well as some different ways to transfer heat either into or out of a system.

07:42 And again, this pressure volume diagrams are very, very commonly discussed especially in an exam setting.

07:48 So I highly recommend you go over many pressure-volume diagram examples. Try those example problems, and use the principles that we just discussed to really keep a physical picture in your mind of what's happening, as maybe a gas is expanding or some work is being done on a system.

08:02 Because the more physical the example that you have in you head as you picture these things, the more accurate you can be sure you are as you go through a very complex or potentially long example problem on an exam.

08:13 So we have few more topics in thermodynamics coming up and until then, thanks for listening.

The lecture Pressure-Volume Diagrams: Cycle by Jared Rovny is from the course Thermodynamics and Thermochemistry.

### Included Quiz Questions

1. In a cycle, if positive work is done by the system heat must be removed from the system in order to have ΔU = 0.
2. In a cycle the system returns to its original pressure and volume.
3. The total heat energy taken from the system is the area enclosed by the pressure-volume curve if the cycle direction is counter-clockwise.
4. During a backward process in a cycle, positive work is done on the system.
5. The net work done by the system in a cycle is positive if the cycle direction is clockwise and is given by the area enclosed by the pressure-volume curve.
1. Q = W
2. Q > W
3. Q < W
4. Q.W ≤ 0 and |Q| = |W|
5. Q = W = 0
1. If the cycle direction is clockwise A is the net work done by the system otherwise if the cycle direction is counter-clockwise -A is the net work done by the system.
2. If the cycle direction is clockwise -A is the net work done by the system otherwise if the cycle direction is counter-clockwise A is the net work done by the system.
3. The net work done by the system is the work done by the system when the volume of the system is expanding.
4. The net work done by the system is A.
5. The net work done by the system is the work done by the system when the volume of the system is expanding plus the work done on the system when the volume of the system is contracting.

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