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Figure 16.15 shows a $p V$ diagram for a heat engine thatuses 1.40 moles of an ideal diatomic gas. (a) How much heatgoes into this gas per cycle, and where in the cycle does itoccur? (b) How much heat is ejected by the gas per cycle, and where does it occur? (c) How much work does this engine do each cycle? (d) What is the thermal efficiency of the engine?

B

Physics 101 Mechanics

Chapter 16

The Second Law of Thermodynamics

Temperature and Heat

Thermal Properties of Matter

The First Law of Thermodynamics

Cornell University

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Simon Fraser University

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So let's do some preliminary work before we get started on the actual problem. A. B and CD, our constant pressure. So it's a constant be their constant pressure processes. So the work is equal to pressure Delta V for those constant pressure processes. This is the general formula for the work, and it still will be doing Percy This problem also, we know that that he is equal to in which is the mall's time, cpi times, Dr T. And this is for cost impression processes. This is just a journal for half with heat. But then now what I'm going to do is use the ideal yes, along, which is PV is equal to our tea, and I'm going to apply the differences to this. So what I mean by that is make it so that it's in our or indulge titty is equal to P Delta bi over our. So if this is the ideal gas law, I can make everything the change of tear, the change of the But in this case, I don't do change a p because we're in a constant pressure process. And so now I'm going to use this and I'm going to plug this back into here, gonna plug in Delta t back into here. And when I do that, I get Q is equal to CP. Never are times pressure time still divine, and this is we'll be using to find the heats for the constant pressure processes. Now one more piece of information we need is that for di atomic molecules, which is what we have in this problem CP is 7/2 are and so we'll see if we plug this in ours. Cancel. But we're not going to do that just yet. And so I'll just say here is this This whole page is constant pressure. Now let's look at constant volume. So in this case, we're talking about B, C and D A. These are the two lines on the graph were given that correspond to comes 1,000,000 processes. And so work is equal to zero for constantly in processes. And cue is equal to in cvi times delta t The difference here is this fee. Now we're going to do the same thing by using the ideal gas law and take the differences. Except in this case, we have built a p, not delta V. But regardless we play all that and we're going to get CV in an hour. I was very Delta P and there are two differences between this and the one on produce base one is this CV instead of C P. And the other is now. We have Delta P envy instead of pity and not to be. And so this is what we used to find the heat for cost and buying processes. Okay, now I can finally start with theatrical, so heat is going in for processes. Abie India. One of those is constant alignment, the others constant pressure. No, he's going in because he is getting larger. And so what's analyzed the process? Maybe so for a B, the heat going in is equal to C P. This is the constant pressure process over our a Times B Delta V, and so this is just equal to seven House because wherever they are, they're going to cancel Pressure's five atmospheres. I want to convert this in the pascal's what the conversion factor, which is this guy? 10 to the fifth Pascal's per one atmosphere. I'll be using this conversion factor multiple times the problem and then times Delta V, which is 4.0 times 10 sending a three meter cubed, and this is equal to 7,091. Jules. Now we have to analyze D A, which is a constant volume process. And so this is going to be able to see V over r P Earth. What's its constant volume? So the daughter p in this case, we have five has C P R C. V is five. Half our and CPS seven house are the Irish cancels? Still, a change in P is four atmospheres. We're gonna multiply by this conversion factor here, converted into Pascal's. And then I have my volume, which is 2.0 times 10 to the third. This is equal to 2,000 and 26 tools, and so in total Q H is the sum of these two, which is 9,120. Jules, I'm going to do pardon me very similarly, Heatley's for processes BC and CD. And so we look at the process bc You are just going to plug in this form again. This is a constant building process. I'm going to use this form of it being Delta P and just doing the same thing, playing in the same thing. The volume changes and Delta P changes because really and different process. But but the process of doing this is the exact same. I get negative 6,000 and 78 Jules for that. That's what you should get. And it's negative because Delta P is is negative because the pressure is decreasing. Sing for this process. So that explains the negative there for CD. Same thing. We're just going to find the heat using the same sort of formula, except it's a constant pressure process now. That's why I have that simply here and the daughter being served the Delta P. This will give you 1,400 and 18 jewels. And again, it's negative because he is decreasing for this process, something them together will give us QC, which is negative 75 100 jewels. And that's the answer to be for Park Si. We want the work of all four processes and the total work. So for a B, it is a constant pressure process, or call that if you have a constant pressure process, the work is simply P Delta V. So this is five atmospheres. That's what he is. I'm going to convert it by that same commercial factor, which converts it from atmospheres to Pascal's. And then I need Delta V, which is four times tented in any three for this process to be. And this gives 2,026 Jules now for BC. Work is equal to zero, since that's a cost constant volume process and for constant William processes work. Zero for CT. We're going to do the same thing as we did here, except for different numbers, and we'll get that work is equal to negative 405 jewels, and it's negative, since Delta V is negative, and then for D A. Again, it's a constant volume process. So work zero. Adding these two work together gives us a total work of 16 20. Jules. That's the answer to see for Bardi. We want to figure out what the efficiency is, and the efficiency is defined to be the work or the total heat. But we already know what these two values are, so we can just put them in work was 16 20 Jules and Q. H was 9,120 Jules, and so this gives 17 point point a percent, which is the efficiency

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