# Compute work in Carnot Refrigerator

• duran9987
In summary, to return 3000 J of energy to the outside by running an ideal refrigerator, approximately 68.82 J of electrical energy must be supplied to the refrigerator. This can be achieved by using the efficiency equation and setting Q_h to 3000 J. However, the interpretation of the problem may vary and using 3000 J for Q_c may also be a valid approach.
duran9987

## Homework Statement

The temperature inside the room ##T_I= 25 C ## and the temperature outside the house is ##T_O = 32C##. The temperature difference causes energy to flow into the room (by conduction through the walls and window glass) at the rate 3,000 J/s. To return this energy to the outside by running an ideal refrigerator, how much electrical energy must be supplied to the refrigerator (to perform the external work)?

## Homework Equations

##Q_h=W_{in}+Q_c##, Energy Conservation
##Efficiency = 1 - T_c/T_h=(Q_h-Q_c)/Q_h##

## The Attempt at a Solution

My thinking is that we have to put in work so that the refrigerator takes energy from the cold reservoir and dumps 3000 Joules onto the hot reservoir. This leads me to set ##Q_h=3000## and use the efficiency to find ##Q_C## and eventually solve for ##W_{in}##.

##Efficiency = 1 - (298.15K/305.15K) = 0.02264##

##0.02264 = (3000-Q_c)/3000 ⇒ Q_c = 2931.19J##

##W_{in} = 3000J-2931.19J = 68.82J##

Is my approach correct? seems like a small amount of energy needed to dump 3000J back to the outside. Any feedback (or more efficient methods) would be greatly appreciated.

Your approach looks pretty good. However, are you sure that 3000 J should be used for ##Q_h## rather than for ##Q_C##?

duran9987
TSny said:
Your approach looks pretty good. However, are you sure that 3000 J should be used for ##Q_h## rather than for ##Q_C##?

I chose ##Q_h## because the problem wants me to return 3000 joules of energy to the outside. If I set ##Q_c## to 3000J I would be returning more energy (Including the energy from the work, which wouldn't be wrong i guess since I technically am returning the incoming 3000 joules plus some extra). I guess either way I can't be wrong if the math checks out.

Perhaps it's open to interpretation. But, it seems to me you want to keep the room from continually warming up. So, for every 3000 J that enters the room from outside, you need to remove 3000 J from the room. You will then be returning more than 3000 J to the outside. (You will also have to position the refrigerator such that the back of the refrigerator is outside the house while the front of the refrigerator is inside the house with the door of the refrigerator kept open!)

duran9987

## 1. What is a Carnot Refrigerator?

A Carnot Refrigerator is a type of refrigeration system that operates on the Carnot cycle, a theoretical thermodynamic cycle that is the most efficient cycle possible for converting heat into work. It consists of four reversible processes: isothermal compression, adiabatic compression, isothermal expansion, and adiabatic expansion.

## 2. How does a Carnot Refrigerator work?

A Carnot Refrigerator works by using a refrigerant to absorb heat from a cold reservoir and release it to a hot reservoir. This is achieved through the four reversible processes of the Carnot cycle, which allow the refrigerant to undergo phase changes and transfer heat efficiently between the two reservoirs.

## 3. What is the efficiency of a Carnot Refrigerator?

The efficiency of a Carnot Refrigerator is given by the Carnot efficiency formula, which is the temperature difference between the hot and cold reservoirs divided by the temperature of the hot reservoir. This means that the efficiency of a Carnot Refrigerator increases as the temperature difference increases, and it is the maximum possible efficiency for any refrigeration system.

## 4. How is work computed in a Carnot Refrigerator?

The work in a Carnot Refrigerator is computed by using the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In a Carnot Refrigerator, the work is done during the adiabatic processes, where the refrigerant is compressed or expanded without any heat transfer.

## 5. What are the applications of a Carnot Refrigerator?

Carnot Refrigerators have various applications in industries such as food and beverage, pharmaceuticals, and air conditioning. They are also used in scientific research to study thermodynamic processes and in the development of more efficient refrigeration systems.

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