A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320 320 320 K and 270 270 270 K. If the refrigerator completes 165 165 cycles each minute, what power input is required to operate it?

Hey everyone in this problem. We have a freezer operating on the basis of the carnot cycle. It has a constant temperature of 255 talbot In each cycle. The freezer is going to discard 600 joules of heat into the kitchen. The kitchen has a temperature of 300 Calvin. The Freezer completes 220 cycles per minute. And we're asked to find the power p required to maintain its constant temperature in the freezer. Alright, so we're looking for power Now let's recall. Power is equal to the work. We're divided by time. You know? We have this information about per time. Okay, we have this 220 cycles per minute. So we have this per time information. So let's see if we can figure out the work. All right, well, recall that work. We're talking about a carnot cycle can be written as Q. H. What? Q. C. Alright. Where Q. H. Is that quantity of heat delivered to the high temperature reservoir? And Qc. Is the quantity of heat I'm coming out of the cold reservoir. Alright, so let's think about what we've been given. Okay, we have the temperature in the freezer, 255 Calvin. Okay so we have a. T. C. So the cold reservoir 255 Calvin. Okay. And we have 300 Calvin in the kitchen, The temperature of our our high temperature reservoir will be 300 Calvin. And we're also given that the freezer discards the engine discards 600 joules of heat to the kitchen. Okay so this is the quantity of heat being delivered to the kitchen or to the high temperature reservoir. So this is going to be the quantity Q. H. Okay this is gonna be 600 jules. Alright so we have all that information. So we have Q. H. But we don't have QC. In order to determine the work. We need to find QC. Let's recall in a carnot cycle, we can relate the ratio of the quantities of heat to the ratio of temperature. Okay So we have quantity of heat, the cold reservoir divided by the quantity of heat, high temperature reservoir. And I shouldn't say in the reservoir those um quantity of heats that are being exchanged. Okay? Is equal to the ratio of the temperatures. Okay great. We know three of the four. So we can find the fourth. So we have QC. Which we're trying to find to use in our work calculation Divided by Q. h. jewels is equal to minus T. C minus 255 Calvin, Divided by Th 300 Calvin. And this is gonna give us a Q. C. We do the division and we multiply the unit of Calvin will cancel and we're gonna be left with -510 jules. Okay? Alright so we found the quantity of heat here Q. C. Now we can go ahead and find that work. Okay so our work again. Q. H. Plus Q C. In this case Q. H. is 600 jewels. QC is -510 jewels. And we get a work of 90 jewels. Alright great. We're one step closer. Okay again we're trying to find power in order to find power. We need to do work divided by time. We found the work we have some information about per time. Now we can use our power equation. So again power is equal to work divided by time. In this case we have 90 jewels and then our time is going to be Well we're giving cycles per minute. So we have this per minute is our time. I wanna go ahead and convert this into seconds or standard unit. And so we get 90 jewels Many Times 220, divided by 60 seconds. Which gives us a power P. Of 330 watts. So we go back up to our answer choices. We found a power p of 330 watts. So we see that that's gonna be answer. A thanks everyone for watching. I hope this video helped see you in the next one

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Problem 17b

Textbook Question

A Carnot refrigerator is operated between two heat reservoirs at temperatures of $320$ K and $270$ K. If the refrigerator completes $165$ cycles each minute, what power input is required to operate it?

Verified step by step guidance

Understand the concept of a Carnot refrigerator: A Carnot refrigerator is a theoretical model that operates on the Carnot cycle, which is the most efficient cycle possible for a heat engine or refrigerator. It transfers heat from a colder reservoir to a hotter reservoir.

Identify the temperatures of the heat reservoirs: The refrigerator operates between two reservoirs at temperatures $ T_H = 320 \text{ K} $ and $ T_C = 270 \text{ K} $.

Calculate the coefficient of performance (COP) for the Carnot refrigerator using the formula: $ \text{COP} = \frac{T_C}{T_H - T_C} $. Substitute the given temperatures into this formula.

Determine the work input per cycle using the relationship between COP and work: $ \text{COP} = \frac{Q_C}{W} $, where $ Q_C $ is the heat extracted from the cold reservoir and $ W $ is the work input. Rearrange to find $ W = \frac{Q_C}{\text{COP}} $.

Calculate the power input required: Since the refrigerator completes 165 cycles per minute, multiply the work input per cycle by the number of cycles per minute to find the total work per minute. Convert this to power by dividing by 60 seconds to get the power in watts.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that provides the maximum possible efficiency for a heat engine or refrigerator operating between two temperatures. It consists of two isothermal processes and two adiabatic processes. For a refrigerator, it describes the idealized process of extracting heat from a cold reservoir and expelling it to a hot reservoir.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of a refrigerator's efficiency, defined as the ratio of the heat removed from the cold reservoir to the work input required. For a Carnot refrigerator, the COP is given by the formula COP = T_c / (T_h - T_c), where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively. This helps determine the work needed for a given heat transfer.

Power and Energy

Power is the rate at which work is done or energy is transferred, measured in watts (W). In the context of a refrigerator, power input refers to the energy required per unit time to operate the refrigerator. Given the number of cycles per minute and the work per cycle, the power input can be calculated to determine the energy consumption of the refrigerator.

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A Carnot refrigerator is operated between two heat reservoirs at temperatures of 320320320 K and 270270270 K. If the refrigerator completes 165165 cycles each minute, what power input is required to operate it?

Key Concepts

Carnot Cycle

Coefficient of Performance (COP)

Power and Energy

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A Carnot refrigerator is operated between two heat reservoirs at temperatures of $320$ K and $270$ K. If the refrigerator completes $165$ cycles each minute, what power input is required to operate it?