At what cold-reservoir temperature (in ℃) would a Carnot engine with a hot-reservoir temperature of 427℃ have an efficiency of 60%?

Video transcript

Hello, fellow physicists today, we're going to solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given a Carnot Engine with a hot reservoir temperature of Calvin and an efficiency of 45% find the temperature of the cold reservoir expressed in degrees Celsius. So that's our end goal is to find the temperature of the cold reservoir in degrees Celsius. OK. We're also given some multiple choice answers. They're all in degrees Celsius. Let's read them off to see what our final answer might be. A is 75. B is 30 C, is 330 and D is 57. OK. So first off, let us recall the equation for the efficiency of a Carnot engine. So N car is equal to one minus the temperature of the cold reservoir divided by the temperature of the hot reservoir. OK. So now we can plug in all of our known variables. So let's note that we're given the efficiency as 45% but we need to convert the percent to a decimal. So all we need to do is just take 45 and divide it by 100. So we'll get 0.45 is equal to one minus the temperature of the cold reservoir, which is our end goal. That's the value we're trying to find divided by the temperature of the hot reservoir which was 600 Kelvin. OK. So now we need to use algebra to rearrange this equation to isolate and solve for the temperature of the cold reservoir. So when we do that, using a lot of algebra, we get that the temperature of the cold reservoir is equal to 330 Kelvin. But we need to convert Kelvin to degrees Celsius. So all we have to do to do that is we take Calvin and we subtract 273 Calvin. And when we do that, when we plug into a calculator, we should get 57 degrees Celsius, which is our final answer for the temperature of the cold reservoir. So looking at our multiple choice answers, the correct answer is D 57 degrees Celsius. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.