FIGURE P21.46 shows a Carnot heat engine driving a Carnot refrigerator.

a. Determine Q₂, Q₃ and Q₄.

Question

FIGURE P21.46 shows a Carnot heat engine driving a Carnot refrigerator.

a. Determine Q₂, Q₃ and Q₄.

Accepted Answer

Hello, fellow physicists today, we're gonna 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 in the combined system, in which a carnot engine powers a car, no refrigerator, the heat absorbed from the hot reservoir. Q one is 1200 Jews. The hot reservoir has a temperature of 800 Kelvin while the cold reservoir has a temperature of 500 Kelvin shown in the figure, determine the heat transfer values Q three and Q four for this interconnected arrangement. So that's our end goal is to determine the heat transfer values for Q three and Q four. OK. So we're given some multiple choice answers for both Q three and Q four and they're both in the same units of jewels. Let's read them off to see what our final answer pair might be. A is 1408 B is 1804 C is 1207 D is 1550. OK. So we have our diagram here to the right of the multiple choice answers in the red rectangle. It represents the hot reservoir and the blue rectangle represents the cold reservoir. The hot reservoir has a temperature of 800. Kelvin. The cold reservoir has a temperature of 500 Kelvin and it shows arrows going down. So from the hot reservoir down to the car heat engine and down to the cold reservoir shows the transfer of heat and Q one equals 1200 jewels. And it also shows with an arrow going to the right, you know the work between the Carno heat engine and the Carno refrigerator. And for the direction of heat transfer or the for the car, no refrigerator, it goes from the cold reservoir up to the car, no refrigerator and up to the heat or the hot reservoir, I should say. So Q four is facing from the cold reservoir up to the car, no refrigerator and then up to Q three, up to the hot reservoir. OK. So first off, let us recall the equation for the thermal efficiency of a Carno engine. So the efficiency of a Carno engine states that A is equal to one minus the temperature of the cold reservoir divided by the temperature of the hot reservoir is equal to the work out divided by Q one. OK. We also need to recall that a car, no refrigerator is driven by the output of the heat engine. Thus the work done by the engine workout is equal to the work input of the refrigerator work in. So therefore w out so the work out, so the work output is equal to the work input, which is equal to the work. Thus, work is equal to Q one multiplied by one minus the temperature of the cold reservoir divided by the temperature of the hot reservoir. OK. So now we can plug in our known variables to so solve for the work. So the work is equal to 1200 Jews multiplied by one minus 500. Kelvin divided by 800. Kelvin means it's the cold reservoir temperature divided by the hot reservoir. Temperature is equal to 450 drools when you plug it into a calculator. Now, we must recall the coefficient of performance for the refrigerator and that states that OK. CK Carne is equal to the temperature of the cold reservoir divided by the temperature of the hot reservoir divided by the temper sorry, the temperature of the cold reservoir divided by the temperature of the hot reservoir minus the temperature of the cold reservoir is equal to QC divided by the work input is equal to Q four divided by the work. Thus, we can write that the temperature of the cold reservoir divided by the temperature of the hot reservoir minus the temperature of the cold reservoir is equal to Q four divided by the work which we can now take this equation and rearrange it to isolate and solve for Q four. So when we do that, using a little bit of algebra, we get that Q four is equal to the work multiplied by the temperature of the cold reservoir divided by the temperature of the hot reservoir minus the temperature of the cold reservoir. OK. So at this stage, we can plug in our known variables to solve for Q four. So let's do that. So Q four is equal to 450 jewels multiplied by 500 Calvin divided by 800 Kelvin minus 500 Alvin, which is equal to when we plug it into a calculator, 750 yours, which is one answer that we are looking for. So we found Q four. Now we need to find Q three. So let's work on that. Now, shall we? So now we must recall and use the heat transfer equation and that states that Q three is equal to Q four plus the work input. So we also need to quickly remember and note that the work input is equal to the work. So we can substitute this into our equation this little fact. So Q three equals Q four plus the work. So now we need to plug in our known variables to solve for Q three. So Q three is equal to 750 Jews, which is the value. We just found a second ago plus the work which was 450 jewels. So when we plug into a calculator, we should get 1200 yours. And that is our answer for Q three ray, we did it. So let's go look at our multiple choice answers to see what the correct answer should be. So looking at the multiple choice answers, it looks like C is the correct answer. So Q three is equal to jewels and Q four is 750 jewels. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

FIGURE P21.46 shows a Carnot heat engine driving a Carnot refrigerator. a. Determine Q₂, Q₃ and Q₄.

Verified Solution

Key Concepts

Carnot Cycle

Heat Transfer (Q)

Coefficient of Performance (COP)