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A heat engine takes 0.350 mol of an ideal diatomic gasaround the cycle shown in the $p V$ diagram of Figure 16.18 .Process 1$\rightarrow 2$ is at constant volume, process 2$\rightarrow 3$ is adiabatic, and process 3$\rightarrow 1$ is at a constant pressure of 1.00 atm.The value of $\gamma$ for this gas is 1.40 (a) Find the pressure andvolume at points $1,2,$ and $3 .$ (b) Calculate $Q, W,$ and $\Delta U$ foreach of the three processes. (c) Find the net work done by the gas in the cycle. (d) Find the net heat flow into the engine in one cycle. (e) What is thethermal efficiency of the engine'? How does this efficiency compare with that of a Carnot-cycle engine operating between the same minimum and maximumtemperatures $T_{1}$ and $T_{2} ?$

(a) Click to see the results in Table 1.0(b) Click to see the results in Table 2.0(c) $W_{\text {net}}=226 \mathrm{J}$(d) $Q_{\text {net}}=227 \mathrm{J}$(e) $e=10.35 \%$ and for Carnot engine ecarnot $=50.0 \%$

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 Washington

McMaster University

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party is asking us to find the pressure climate each point along the cycle. And so let's solve it for position one position one were given that the pressure is equal to one atmosphere and the volume do you want? We can get by using the ideal gas law by the ideal gas law this is equal to and our tea one overpay one. And we know in his 10.35 moles we know our is point 08 to 06 leader atmospheres. Her mole Kellen and T one is 300 killers and the one I just stayed. It was one atmosphere. And so when you do this al u get that Gey one is equal to a client. 62 leaders. And so this covers 0.1. Now the two is equal to be one since wanted to is a constant volume process. And so me too is equal to 8.62 leaders and that now we can use the ideal gas law specifically, we're going to use a different form of it. If you want over to Team one is equal to pee too over tea too. This essentially rises taking the ideal gas a lot, too, and the ideal gas a lot one and dividing them and then using this fact here and so you can drive this relationship. It's fairly straightforward and allows us to calculate what Peter says. Since this is true, P two is equal to t to times people in over Team one. And then now I complied him. Let P one is. It's one atmosphere. T one is 300 Kelvin, and tea, too, is 600. And so when you do this out, you get that P two is two atmospheres, and so that completes 0.2. A 0.3 we can read off B three is It's one atmosphere again, and we can get these reading by using the item gasoline and R T three over p three. And so we can plug in everything like we did in the first part, and we get 14.1 leaders and that completes party in Part B. We want to analyze every single branch of the process and determine what w q and Delta you are. And so let's start with the process wanted to and wanted to do. The work is going to be zero since Delta V is equal to zero and work is equal to pressure times. Delta Bi now, since the work is your own by the first law of thermodynamics that he must be equal to the change of internal energy and the heats can be determined by in CVI Delta T for constant blind process. Now just plugging in 0.35 moles. CVI is five Halfs our side five pass and I multiplied by Oregon. She's eight point 315 jewels Permal Kelvin in the 19 Delta T, which is 492 killed him. This is the Delta team along this path, so t to minus t one. And then when we do this, L it looks like we're going to get 2,183 jewels. And so this is what Q is. It's also adult to us, and so that completes the one, too to process. So now we have 2 to 3. So 123 we know it's automatic, so Q zero, which means that the work is equal to negative Delta U. This is by the first law of thermodynamics again, And don't you is equal to end CV, Delta T. And so we can plug in everything we know about the process to 23 What? Lt is what CV is and what the malls are. And when we do that, we get negative. 786 Jules. And by this, the first author on dynamics, the work must be positive. So 108 6 tools for this process. And now I can move on to 321 and 321 We're going to calculate the work, but using pressure Delta bi along the process. But P Delta V is also equal to in our delta t and we know it. Delta T is a long process. We could calculate using this if you wanted, but I'm going to go ahead and plug in numbers into this. It's 0.35 moles. Times are, which is a 0.315 times delta t which for this is Hold on. Let me find it in my notes. Negative 1 92 going. And so this equals negative 1,955 jewels. Uh, okay. Sorry. I'm reading my no strong. This is actually equal to native 559 jewels. So now we can determine what Q is from this formula in CPD dull titty. And we're just gonna blufgan things like we did in the previous parts. And we're going to get negative 1,955. And it's negative because Delta t for this process is negative. Delta. You can be found from the first off there on dynamics, which says that the Delta is Q mice W. And we found out what Q is here. And we found out what don't use there so we can put those in and get negative 1,396 Jules And that completes party course. He is asking what the total work is, and all we do is take work. 122 less work 223 plus work 321 And when we had these up, we get 227 jewels. Party is asking with a total heat and put it in this so cue. Total is equal to Q. Wanted to rescue 23 Rescue 321 Look you wanted Teo is 2,183 Q 2,183 2 to 3 to zero because it's automatic there and Q 321 is needed in 1,955. So if you put those in, you get 228. Jules, you see, they're pretty much identical. They should be identical. The reason they're not is because of some rounding errors that I had well, actually doing the calculations. Now let's move on to party. Party were capturing the efficiency, which is the total work over he imported. But he was on ly inputted daring steps wanted to. So really, you're gonna play in queue on the two there and we do that. We had to 27 jewels, just the total work over 2,183 jewels. And this gives 10.4%. So now we need to compare this to a Carnot engine operating between the same temperature extremes. And so hee carnal you were called the efficiency of the Carnot engine is one minus tc over th and we know both these numbers. It's your honor killing and six are killing. Well, 300 over six, hundreds 60.5. So 1915 is just 19150.5. So this is 50%. So the efficiency of the engine depicted is significantly worsen. The Carnot engine and the Carnot engine is as efficient and Internet there is, and that completes.

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