A heat pump has a coefficient of performance of 5.0. If the heat pump absorbs 40.0 cal of heat from the cold outdoors in each cycle, what is the amount of heat expelled to the warm indoors?

Short Answer

Expert verified
Answer: The heat pump expels 50.0 cal of heat to the warm indoors in each cycle.

Step by step solution

01

Write down the formula for the coefficient of performance

The formula for the coefficient of performance of a heat pump is given by: COP = Q_out / W_in Where COP is the coefficient of performance, Q_out is the heat expelled to the warm indoors and W_in is the work input.
02

Express the work input in terms of the heat absorbed and expelled

The heat absorbed from the cold outdoors Q_in and the work input are used to produce the heat expelled to the warm indoors. Therefore, we can write the relationship between Q_in, W_in, and Q_out as: Q_out = Q_in + W_in
03

Substitute the formula for Q_out in terms of Q_in and W_in into the formula for COP

Substitute the expression for Q_out from the previous step into the formula for COP: COP = (Q_in + W_in) / W_in
04

Solve for W_in

Rearrange the equation in order to solve for the work input: W_in = Q_in / (COP - 1)
05

Plug in the given values of COP and Q_in

The given COP is 5.0 and the heat absorbed from the cold outdoors (Q_in) in each cycle is 40.0 cal. Plug these values into the equation for W_in: W_in = 40.0 cal / (5.0 - 1)
06

Calculate the work input (W_in)

Now calculate the work input: W_in = 40.0 cal / 4 = 10.0 cal
07

Calculate the heat expelled to the warm indoors (Q_out)

Use the relationship between Q_in, W_in, and Q_out to find the heat expelled to the warm indoors: Q_out = Q_in + W_in = 40.0 cal + 10.0 cal = 50.0 cal The heat pump expels 50.0 cal of heat to the warm indoors in each cycle.

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