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Chapter 15: Problem 30

A heat pump is used to heat a house with an interior temperature of \(20.0^{\circ} \mathrm{C} .\) On a chilly day with an outdoor temperature of \(-10.0^{\circ} \mathrm{C},\) what is the minimum work that the pump requires in order to deliver \(1.0 \mathrm{kJ}\) of heat to the house?

Short Answer

Expert verified
Answer: The minimum work required by the heat pump is approximately 0.102 kJ.

Step by step solution

01

Convert temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin. To do this, we add 273.15 to the Celsius temperature: \(T_{hot} = 20.0^{\circ} \mathrm{C} + 273.15 = 293.15 \mathrm{K}\) \(T_{cold} = -10.0^{\circ} \mathrm{C} + 273.15 = 263.15 \mathrm{K}\)
02

Calculate the coefficient of performance (COP)

Next, we use the formula for the COP for a heat pump in terms of the hot and cold temperatures: $$COP = \frac{T_{hot}}{T_{hot} - T_{cold}}$$ Plugging in our values, we get: $$COP = \frac{293.15 \mathrm{K}}{293.15 \mathrm{K} - 263.15 \mathrm{K}} = \frac{293.15}{30} = 9.77$$
03

Calculate the work done by the heat pump

Now that we have the coefficient of performance, we can use it to find the minimum work required by the heat pump to deliver 1.0 kJ of heat to the house. We rearrange the COP formula to solve for the work: $$W = \frac{Q_{hot}}{COP}$$ Plugging in our values, we get: $$W = \frac{1.0 \mathrm{kJ}}{9.77} \approx 0.102 \mathrm{kJ}$$
04

State the final answer

Thus, the minimum work that the heat pump requires in order to deliver \(1.0 \mathrm{kJ}\) of heat to the house on a chilly day with an outdoor temperature of \(-10.0^{\circ} \mathrm{C}\) is approximately \(0.102 \mathrm{kJ}\).

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Most popular questions from this chapter

An electric power station generates steam at \(500.0^{\circ} \mathrm{C}\) and condenses it with river water at \(27^{\circ} \mathrm{C} .\) By how much would its theoretical maximum efficiency decrease if it had to switch to cooling towers that condense the steam at \(47^{\circ} \mathrm{C} ?\)
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