Problem

A heat engine takes 0.350 mol of an ideal diatomi…

08:44

Problem 39 Easy Difficulty

.. You are designing a Carnot engine that has 2 mol of $\mathrm{CO}_{2}$
as its working substance; the gas may be treated as ideal. The
gas is to have a maximum temperature of $527^{\circ} \mathrm{C}$ and a maximum pressure of 5.00 atm. With a heat input of 400 $\mathrm{J}$ per cycle, you want 300 $\mathrm{J}$ of useful work. (a) Find the temperature
of the cold reservoir. (b) For how many cycles must this engine
run to melt completely a 10.0 -kg block of ice originally at
$0.0^{\circ} \mathrm{C},$ using only the heat rejected by the engine?

Answer

(a) $T_{C}=-73.0^{\circ} \mathrm{C}$
(b) The number of cycles must the engine run to melt mass 10.0 $\mathrm{kg}$ of ice is $334 \times 10^{2} \mathrm{Cycles}$

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Chapter 16

The Second Law of Thermodynamics

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Video Transcript

So for this problem, we're talking about signing a Carnot engine and a You want to figure t obvious Re read the question. So you have carnal engine. Two moles of carbon dioxide, the gas students, ideal gases to have a mass temperature of forty seven degrees Celsius. Max Pressure five atmospheres. He input Cue hot A four hundred rolls for cycle you once three hundred jewels of useful work. Avon, the temperature, the whole reservoir B How many cycles must stand and run to completely melted? Lilia ten kilogram. What? Nice. Originally at Sarah greasy using the on ly the he rejected by the engine. Okay, so I'm just going on ly right down the quantities that ended up being useful. I worked out the problem, So t hot is five twenty seven degrees c. Um, if you want to seventy three that's a hundred Calvin. So liketo always just convert to avoid making mistake of using Celsius in the problem. So again we were these problems. You have to identify the engine problems won't identify you. Hike You called. It's a heat engine. Is refrigerator. Yeah, in this case for saying he that goes into it is for hundred jewels. Let me just double check the wording he employed. Yes, four hundred, Jules. And the work is three hundred. That's kind of what, um and first, the goal is basically to find the key. Was the temperature Yes, the temperature. So tea cold equals question mark. And actually, before I address that, I'm just going to add a few more kind of things to our box of given quantities. So we later we'll have to use. It's not well, Yeah, The mass of the block of ice is ten kilograms and then lay in heat of fusion is forty one ninety Jules per kilogram. And I think that's all that I actually done. Uh, go ahead and box that away. So to find he called Wake your saree can't have a few of our givens work tio high and tea hot. And you know, it's a carnot engine. So we know there's a relationship between, uh, the the heat and the temperature basically. So I'm just going to use the equation that work over cue. Hot is E, but then for a car, know that same easy bullet tc over t age, and that's those are just the formulas from, like, the the end of the chapter. And if you saw this algebraic clear for T C, that's our goal, Theun. Maybe I'll get what I got, which is tea Hot times one minus Work over Hugh Hot. And if you just like plugging all these numbers So this tea hot is eight hundred k work three hundred Q hod four hundred. Then you should got its two hundred Calvin, which is actually quite cold seventy three below zero in Celsius. So I guess my suspicions of the engine where I think founded because to get that much work out, you really you really need to have a cold reservoir. Uh, okay, so for pride be he wanna melt the ice with the rejected heat. So it just as a side note, it seems like this engine must be doing something else, and it just kind of inside job is heating. This this ice, remember, kind of wondering what is this engine during heating ice? But I don't know. Ah, sort of like a side question. So basically, it's if it's heating it with its rejected he want to solve for, um, we wanted basically we want to get how much heat is rejected per cycle and figure out how much heat is needed to Teo to melt the ice and then multiply of the heat per cycle. Times the number times multiply the heat. Perseid Gold Times Some number to get the total he needed. So we need to find that multiplication factor. So first of all, let's get the qc per cycle on DH that comes from thie equation that relates Q. You because you could get in a few ways. I chose to obtain this by using this equation, which, yeah, it comes from the fact that the engine is car now. And so if you just plug in the four hundred TC is ah two hundred, go ahead and box that on and then th is eight hundred. Then maybe you'LL get what I got, which is It's a hundred jewels for cycle. Nice round to juror on dso So great. So now you know it's getting a hundred euros per cycle. How many? How many of these cycles doing it to melt it And I even just put per cycle here so qc tonal how much heat is needed to melt a block of ice. If you just need to melt it not changes temperature. And that's just the mass of the ice. Times that late. He infusion. And so, uh, yeah, eso if you calculate that you simas ten. Olof is for you one ninety. Uh oh. Lf is not forty one ninety. Oops. That's the heat capacity. Me a race that, um Oh, what? Okay, my eyes. That's not working. Okay. Not sure what happened here. So good to discard this. And I think I must have made some kind of texts from a box or something. Um, thank you. Continue racing thiss of him. Okay, that number is I should three point three four times ten to the five. Okay, If you use that number anymore by by ten, then you'LL get three point three four times ten to sixteen. You need calculator for that. And if we have one hundred joules per cycle and you need that money drools and, um, then the number of cycles is gonna be given by three point three four times ten to the sixth. Jules, divided by hundred drools per cycle Patrol's cancel. And then you got three point three four times ten before cycle's. Nick. No, fox that up? What kind of funny box? Oh,

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