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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 ideal gas is heated at a constant pressure of $2.0 \times 10^{5} \mathrm{Pa}\( from a temperature of \)-73^{\circ} \mathrm{C}$ to a temperature of \(+27^{\circ} \mathrm{C}\). The initial volume of the gas is $0.10 \mathrm{m}^{3} .\( The heat energy supplied to the gas in this process is \)25 \mathrm{kJ} .$ What is the increase in internal energy of the gas?
List these in order of increasing entropy: (a) 1 mol of water at $20^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)20^{\circ} \mathrm{C}$ in separate containers; (b) a mixture of 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C} ;\) (c) 0.5 mol of water at \(20^{\circ} \mathrm{C}\) and 0.5 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers; (d) a mixture of 1 mol of water at $30^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)30^{\circ} \mathrm{C}$.
An air conditioner whose coefficient of performance is 2.00 removes $1.73 \times 10^{8} \mathrm{J}$ of heat from a room per day. How much does it cost to run the air conditioning unit per day if electricity costs \(\$ 0.10\) per kilowatt-hour? (Note that 1 kilowatt-hour \(=3.6 \times 10^{6} \mathrm{J} .\) )
An inventor proposes a heat engine to propel a ship, using the temperature difference between the water at the surface and the water \(10 \mathrm{m}\) below the surface as the two reservoirs. If these temperatures are \(15.0^{\circ} \mathrm{C}\) and \(10.0^{\circ} \mathrm{C},\) respectively, what is the maximum possible efficiency of the engine?
How much heat does a heat pump with a coefficient of performance of 3.0 deliver when supplied with \(1.00 \mathrm{kJ}\) of electricity?
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