A window air conditioner has an average coefficient of performance of 2.4. In a futile attempt to cool a bedroom, this unit has been placed on the floor by the bed. During this attempt, 80,000 J of heat is removed from the air in front of the unit. Determine the net heat added to the room by operating the air conditioner in this manner.

Question

08-10-2023
Physics

High School

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A window air conditioner has an average coefficient of performance of 2.4. In a futile attempt to cool a bedroom, this unit has been placed on the floor by the bed. During this attempt, 80,000 J of heat is removed from the air in front of the unit. Determine the net heat added to the room by operating the air conditioner in this manner.

Answer :

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OliviaMariThompson OliviaMariThompson · Answer 1 · 2023-09-02T09:04:24-04:00

To determine the net heat added to the room by operating the air conditioner in this manner, we need to understand the concept of the Coefficient of Performance (COP) and how an air conditioner works.

The Coefficient of Performance (COP) for a cooling system is defined as:

[tex]COP = \frac{Q_c}{W}[/tex]

Where:

[tex]Q_c[/tex] is the amount of heat removed from the cold reservoir (the air in front of the unit, in this case, 80,000 J).
[tex]W[/tex] is the work input to the system.

From the COP equation, we can rearrange to find the work done by the air conditioning unit:

[tex]W = \frac{Q_c}{COP} = \frac{80000}{2.4} = 33333.33 \text{ J}[/tex]

The air conditioner operates by removing heat from the area around it and releasing it elsewhere, typically outside. However, since the air conditioner is placed on the floor by the bed, it effectively releases this heat back into the room along with the work done.

The net heat added to the room is the sum of the heat expelled back into the room and the work done by the air conditioner:

[tex]Q_{room} = Q_c + W = 80000 + 33333.33 = 113333.33 \text{ J}[/tex]

Therefore, the net heat added to the room by the operation of the air conditioner while placed inside is 113,333.33 J. This is because the attempt to cool the room is futile; the device ends up heating the room overall due to the energy expenditure.