What are Wₒᵤₜ and QH and (b) the thermal efficiency for the heat engine shown in FIGURE EX21.14?

Video transcript

Hey, everyone. So this problem has to do with heat engines and PV diagrams. Let's see what it's asking us. We have a gasoline powered engine that follows the PV trajectory shown in the figure below. We're asked to calculate the useful work or W sub out two, the heat extracted from the hot reservoir or Q of H and three. The thermal efficiency denoted by Ada. Our multiple choice answers are a 37.5 jewels, 230 jewels and 6.75%. B 37.5 jewels, 278 jewels and 13.5% C 52.5 jewels, 230 jewels and 6.75% or D 52.5 jewels, 27.8 jewels and 13.5%. Our PV diagram as pressure on the Y axis and volume on the X axis. And we'll be talking through each of those points through the rest of this problem. So the first thing we need to do is recall that our extracted heat or Q sub H is equal to our work. W plus our exhaust heat or Q sub C and our efficiency or our A is equal to our work divided by Q sub H our exhaust heat. We can see in the diagram is the heat that is exiting the closed area of the triangle. So we have from the B to C step of the cycle, we have 60 jewels exhausted and then from the C to a step, we have 180 jewels. So Q of C is 60 jewels plus 180 jewels which equals 240 jewels. Our work we can recall is equal to the area enclosed. So the area of a triangle is one half multiplied by the base, multiplied by the height. And so have one half, our base will be our along our X axis. So our volume goes from 50 and that's units of 10 to the negative 5 m cubed and it goes from 50 to 20. So 30 times 10 to the negative 5 m cubed is our base and then our height we can see from the diagram goes from 3.75 to 1.25. So that's 2.5 and that's in units of 10 to the five pascals. So 2.5 times 10 to the five mascots plug this in. And we come up with a work of 37.5 jewels. And then lastly, you can find our use of H to be able to solve for our efficiency. So Q sub H as we said before was work plus our exhaust heat or Q sub C. So we have 240 jewels plus 37.5 jewels which holds 277 0.5 jewels rounded to the whole integer, 278 jolts. And then excuse me, our efficiency work. 37.5 jewels divided by 277.5 jewels is equal to 0.135 or 13.5%. And so now we have the answers to part one, part two and part three of our problem. When we look at our multiple choice answers, it aligns with answer choice B so B is the correct answer for this problem. So that's all we have for this one. We'll see you in the next video.