A Carnot heat engine uses a hot reservoir consisting of a large a...

Question

A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work W is performed by the engine?

Accepted Answer

Welcome back everybody. We have a heat engine that is operating on the basis of a carnot cycle. We're told that it typically operates at 150°C. And we are told that it also operates uh in conjunction with a thermal reservoir or sorry, a large block of ice at 0°C. We are told that with the heat that comes off the engine melts the ice at a rate of 100 g per minute. And we want to find after 10 minutes, what is the work done by our engine? So the formula for work is just going to be as follows. It is going to be the absolute value of heat absorbed minus the absolute value of heat rejected. Now, in order to work with our variables here, want to make sure everything is in the correct units actually going to go over Our temperatures over to Calvin by adding to 73.15 to each of them, giving us a operating temperature of 423. Kelvin and a temperature of the ice block of 2 73.15 Calvin. Great. So in order to find our work, we need to find these two terms, both are absorbed and rejected heat. I'm going to start with the rejected heat here. The formula for this is the new negative of the mass of the ice. The total ice that melts times the latent heat of fusion. This is negative times. Well, what is the total mass of ice that melts? We're told that ice melts at a rate of 100 g per minute and it passes over, sorry, 10 minutes. So we're just simply going to do 100 times 10 to get our total mass. Now the latent heat of fusion for this state change is 334 times 10 to the third. When you plug this into your calculator, You get an answer of negative 3.34 times to the fifth jewels for our rejected heat. So what about our absorbed heat here? Well, since we are operating off of the basis of a carnot cycle, we are going to have that the ratio between our absorbed heat and our rejected heat is just going to be equal to the ratio. Let's see here. Oh my apologies. I actually got those two flipped around. Going to be the rejected heat over the absorbed heat. It's going to be equal to the ratio of the temperatures that they are at respectively. Wonderful. Now, in order to isolate our Q. H. Term, which is what we are trying to find. I'm going to multiply both sides by T. H. U. H over T. C. Let me put this over here as well. E H. H. Over. You'll see that the Q. H. S cancel out on the left and the temperatures cancel out on the right. Leaving us with that our heat absorbed is equal to the absolute value of the heat rejected times that ratio between those heats. Let's go ahead and plug in some numbers here we have that uh we found that our heat absorbed is going to be, sorry our heat rejected was negative. It will be the absolute value symbol negative 3.34 times 10 to the 5th times the ratio between our temperatures. So we have the operating temperature of 4 23. divided by 273.15. And this gives us a rejected heat, sorry, and absorbed heat of 5.17 times 10 to the fifth duels. Wonderful. So now we have our heat absorbed and our heat rejected. We are ready to go ahead and find the work done by our system. So once again, the formula is the absolute value of heat absorbed minus the absolute value of heat rejected that we have 5.17 times 10 to the fifth minus the absolute value of negative 3.34 times 10 to the fifth. Which when you plug into your calculator, you get the work done by the engine is 1.83 times 10 to the fifth jewels, which corresponds to our final answer choice of a Thank you all so much for watching. Hope this video helped. We will see you all in the next one

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