Problem

Steam locomotives have an efficiency of 17.0$\%$ …

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Problem 30 Hard Difficulty

(a) What is the hot reservoir temperature of a Carnot engine that has an efficiency of 42.0$\%$ and a cold reservoir temperature of $27.0^{\circ} \mathrm{C} ?(\mathrm{b})$ What must the hot reservoir temperature be for a real heat engine that achieves 0.700 of the maximum efficiency, but still has an efficiency of 42.0$\%$ of (and a cold reservoir at $27.0^{\circ} \mathrm{C} ) ?$ (c) Does your answer
imply practical limits to the efficiency of car gasoline engines?

Answer

a) $T_{h}=244^{\circ} \mathrm{C}$
b) $T_{h}=477^{\circ} \mathrm{C}$
c) yes

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Physics 101 Mechanics

College Physics for AP® Courses

Chapter 15

Thermodynamics

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Video Transcript

this problem 30 Chapter 15 on were asked to find the hot reservoir temperature. Um And then we're also asked to find the hot reservoir temperature under some different conditions for efficiency. And and then we are asking the question, Does your answer imply practical limits to the efficiency of car gasoline's engines? And so the first thing we can do for part a a cz we have this equation here for our efficiency of a car knots engine on, and so we can go ahead and rearrange this to actually get our temperature of our heart reservoir. So th is gonna be equal to T C over one, minus the efficiency. And so from here, we can actually substitute some of our values. Since we have 2 27 degrees Celsius, we need to convert this thio, Calvin. But I'll go ahead and do that in this. So substitute our values that we have so th it's gonna be equal to uh, T. C. Which is gonna be 273 plus 27 which is our temperature in degree Celsius in this converts it two degrees, um, Kellen, And so now weaken, uh, divide by one minus 0.4 to 0, which is our efficiency on this is gonna come out too. 517 point to Kelvin, which could be converted into degree Celsius. So we'll just subtract 273.2 on this is gonna come out too th equaling 244 degrees Celsius s o. The temperature of the hot reservoir in this case is 244 degrees Celsius. Uh, and so now we need to go ahead and find our efficiency for the part B. And so, with part B, we can go ahead and do an extract expression for the efficiency which will be 0.7 times and see, uh, equaling 0.4 to 0. So now we can go ahead and divide by 0.7 on both sides. An n C. Is gonna be equal to 0.420 over 0.7, which is gonna be equal to 0.6. And now we can use this previous equation that we used to go ahead and solve for the hot water temperature. So th is gonna be equal to T. C, which is gonna be junior 73 plus 27 um, all over one minus 0.6. This is gonna equal up to 750 degrees Kelvin, which we can subtract 273 degrees from to go ahead and get 477 degrees Celsius. And that's for party. Now on a new page. Uh, part See, we can go ahead and look at on were asked the question. Does your answer implied practical limits to the efficiency of the car engines? And so we're actually gonna say yes on this is because the automobile engines cannot actually get too hot without overheating on, and thus their efficiency is actually gonna be limited.

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