 A gas within a piston-cylinder assembly undergoes a thermodynamic cycle consisting of three processes. From one to two, we have compression, where pressure times volume is constant, from P1, then V1 to V2, what they described, delta U. From two to three, we have an isochoric pressure change until P3 is equal to P1, and from three to one, we have an isobaric adiabatic process. There are no significant changes in kinetic or potential energy. With that in mind, we are to determine the following. First, the network of the cycle in kilojoules, and then the type of cycle, power, refrigeration, or heat pump. I will start by populating what I know. At state 1, I know a pressure and a volume, and those were 1 bar and 2 cubic meters. At state 2, I know the volume is 0.2. At state 3, I know the pressure is the same as state 1. Then I will look at my process descriptions. I was told that the process from 1 to 2 had a compression process where the pressure times volume was constant. Therefore, I could say that P1 times V1 is going to equal P2 times V2. So if I need it, I could calculate P2 by knowing P1, V1, and V2. And just in the interest of trying to complete pressure and volume at all three state points, I will calculate that, despite the fact that I don't know for sure that it's going to be useful yet. So I'm going to take P1 times V1 divided by V2. So I'm going to say 1 bar multiplied by 2 divided by 0.2 is 10. Therefore, P2 is 10 bar. Then I was told that the process from 3 to 1 is isobaric. Therefore, P3 is equal to P1, which is something that I had already written down because it was duplicated here. I will point out that that means I could have had just isochoric pressure change as a description from 2 to 3, and that wouldn't have changed anything about what I know so far. Lastly, I know that the process from 2 to 3 is isochoric, which means that I have a constant volume from 2 to 3, which means V3 is equal to V2. So I know pressure and volume at all three state points. Furthermore, I know how pressure and volume are related for all three state points. And the reason that that's important is because what I care about here is boundary work. The boundary work is the integral of pressure with respect to volume. That's the work associated with a piston-cylinder assembly. So as the volume is changing, there's going to be some work associated with that change in volume. So first of all, we have a compression process from 1 to 2. We know that because it was described as a compression process, but we also know that because the volume goes from 2 cubic meters to 0.2 cubic meters. And then the volume doesn't change from 2 to 3, and then it expands again from 3 to 1. So I'm going to have a boundary work from 1 to 2, and then I'm going to have no boundary work from 2 to 3, and then I'm going to have a boundary work again from 3 to 1. The integral for the process from 3 to 1 is going to be relatively straightforward. It's just going to be pressure times change in volume. That's going to be V1 minus V3, note. Because the change in volume, our delta V here, is end minus beginning. It's a process from 3 to 1, so I read it as 1 minus 3. The pressure from 1 to 2 changes, and it changes linearly with volume. So I could say that pressure since P2 V2 is equal to P1 V1, I can say the pressure at some point as a function of volume is going to be the initial pressure times the initial volume divided by the volume at that point. So pressure at state 2 was the initial pressure times the initial volume divided by V2, and since the initial pressure and volume are described as our state 1 properties, I'm going to say V1 over V. Then when I integrate this function with respect to volume, that's the integral of P1 times the quantity V1 over V with respect to V, I am saying pressure 1 comes out, V1 comes out, because they're constants. So I have P1 times V1. So I'm saying I really have the integral of 1 over volume d volume, which is P1 times V1 times the natural log of volume. And this was a definite integral from state 1 to state 2. So this is evaluated from 1 to 2. So I'm saying P1 times V1 times the natural log at V2 minus the natural log of V1. That would be inside of a quantity, that entire thing is multiplied by P1 V1. And then because of my logarithm rules, I can write that as natural log of V2 over V1. And this is all chaos, so I'm going to rearrange this to hopefully make this easier to follow. So I'm going to write this down here, and I'm going to say this is over here, and then this all comes back for boundary work. Hopefully that is somewhat intelligible. The point being that the pressure from 1 to 2 changes as a function of volume. And since I know how it changes, I can describe the boundary work from 1 to 2. So that's going to be the pressure at state 1, which was 1 bar, times the volume at state 1, which is 2 cubic meters, and then I'm multiplying by the natural log of V2 over V1, which is 0.2 divided by 2. And I want an answer in kilojoules, because I'm eventually going to be using these three work quantities, whatever I get from 1 to 2 and 0 and whatever I get from 3 to 1, in order to come up with a network. So I'm going to describe this in kilojoules. A kilojoule is 1,000 joules, and a joule is a Newton times a meter. And then from my conversion factor sheet, the beginning of my textbook, I can find the relation between bar and Newtons per square meter. I know 1 bar is equal to 10 to the 5th Newton's per square meter. So I can say 1 bar, 10 to the 5th Newton's per square meter. And now bar cancels bar, joules cancels joules, Newtons cancels Newtons, meters and square meters, cancels cubic meters, and the natural log of a unitless proportion is unitless, so I'm left with kilojoules. I'm going to create yet more room here. Therefore, the boundary work from 1 to 2 is 1 times 2 times the natural log of natural log. Come on, calculator. You can do it down there. Natural log of, what's that? 0.2 divided by 2 times 10 to the 5th. Those are nowhere near the right symbol's calculator. 10 to the 5th. And then I'm dividing that entire quantity by 1,000. So I get negative 460 or 60.517 kilojoules. Negative boundary work describes a work input, remember? So I could say that the process from 1 to 2 has a work input of 460.517 kilojoules. And because we know delta U from 1 to 2, we could use that to describe the amount of heat transfer occurring as well. But let's leave that aside until we are pretty confident that we need it. Then the boundary work from 2 to 3 was 0. How did I know that? Because the process from 2 to 3 is isochoric. No change in volume means no moving boundary, which means no work associated with the moving boundary. Then for 3 to 1, I had P3 or P1, both of which are 1 bar still, multiplied by the difference in volume between 1 and 3. 3 is equal to 2, which is 0.2, and V1 was 2. Therefore, I'm taking 2 minus 0.2. And then my goal here is going to be to try to write that in kilojoules again. So I'm going to do the same unit conversions as I had in the previous calculation. So kilojoules of 1,000 joules, joules of Newton meter, and a bar is 10 to the 5th newtons per square meter. It cancels joules, newtons cancels newtons, meters and square meters cancels cubic meters, bar cancels bar, leaving me with kilojoules. And I recognize that 2 minus 0.2 is going to yield a positive quantity, so this will be a workout term, which makes sense. We're likely going to have a cycle with our work in and a workout. It's unlikely that it would just be work in one direction. So 1 times the quantity 2 minus 0.2 times 10 to the 5th, all divided by 1,000, and I get 180. So the boundary work occurring from 3 to 1 is 180. The boundary work occurring from 2 to 3 is 0. The boundary work occurring from 1 to 2 is negative 460.517. Therefore, this process has a work in from 1 to 2, and then no work from 2 to 3, and a workout from 3 to 1. And because there's more work in than there is workout, that means that this is a cycle where the network is in the inward direction. That means that this is not going to be a power cycle. This is either going to be a refrigeration cycle that is a refrigeration cycle used for refrigeration or a heat pump. In order to know more about which type of cycle I have, I would have to know what the intention is. Does that make sense? I don't actually know what the goal of this cycle is, and that's the thing that makes the difference between a refrigeration cycle for cooling or a refrigeration cycle used for heating. It's the same device, but I don't know which end we care about right now. This is like saying, I know that I have a shop vac, and whether I'm using the shop vac as a vacuum or a blower depends on the application. So far, with this description, all I can say with confidence is that I have either a refrigeration or a heat pump cycle. So my answer to Part B is going to be, it is either refrigeration or heat pump. We cannot clarify further without knowing what the goal of the device is. But I can answer Part A. Part A is network in is equal to work in minus workout. So I would take all the work inputs for the cycle, of which there's only one, 460.517. Because we are flipping the sign, we're calling an input now. I'm going to write that as a positive, 460.517, minus all the workouts of which there are only one, and it is 180. Meaning that I have a network in the input direction of 280.517. And my answer to Part B is, I cannot answer further without knowing what the intention of the device is. I would have to know if it was actually being used for heating or cooling, in order to know if it's a refrigeration cycle for cooling or a heat pump. Either way though, it would be a refrigeration cycle.