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Chapter 5: Problem 4

An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes \(1.70 \times 10^{3} \mathrm{W}\) of electrical power and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 how much heat can be extracted from the house in a day?

Short Answer

Expert verified
The air conditioner can extract \(4.74 \times 10^8 \mathrm{J}\) of heat from the house in a day.

Step by step solution

01

Understanding the Coefficient of Performance (COP)

The coefficient of performance (COP) of a refrigeration cycle is the ratio of the heat removed from the cold reservoir (the inside of the house) to the work done by the cycle (the electrical power consumed). In our case, the COP is given as 3.30. Mathematically, COP can be written as: \[COP = \frac{Q_C}{W}\] Where: - COP is the coefficient of performance - \(Q_C\) is the heat extracted from the cold reservoir (inside of the house) - W is the work done by the cycle (electrical power consumption) We are also given the electrical power consumption, \(W = 1.70 \times 10^{3}\mathrm{W}\).
02

Calculate the heat extracted from the cold reservoir per second

Using the COP formula, we can solve for the heat extracted from the cold reservoir per second, \(Q_C\): \[Q_C = COP \times W\] We're given the COP as 3.30, so: \[Q_C = 3.30 \times (1.70 \times 10^{3}\mathrm{W})\] Calculate the value of \(Q_C\): \[Q_C = 5610\mathrm{W}\]
03

Calculate the heat extracted in a day

Now that we have the heat extracted per second, \(Q_C\), we can calculate the amount of heat extracted in an entire day. There are 84600 seconds in a day, so to find the total heat extracted, \(Q_{total}\), simply multiply \(Q_C\) by the number of seconds in a day: \[Q_{total} = Q_C \times 84600\] Substitute the value of \(Q_C\) from step 2: \[Q_{total} = 5610\mathrm{W} \times 84600\] Calculate the value of \(Q_{total}\): \[Q_{total} = 4.74 \times 10^8 \mathrm{J}\] So, the air conditioner can extract \(4.74 \times 10^8 \mathrm{J}\) of heat from the house in a day.

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Most popular questions from this chapter

From the following data, derive the absolute entropy of crystalline glycine at \(T=300 .\) K. You can perform the integration numerically using either a spreadsheet program or a curve-fitting routine and a graphing calculator (see Example Problem 5.9). $$\begin{array}{rc} T(\mathbf{K}) & C_{P, m}\left(\mathbf{J} \mathbf{K}^{-1} \mathbf{m o l}^{-1}\right) \\ \hline 10 . & 0.30 \\ 20 . & 2.4 \\ 30 . & 7.0 \\ 40 . & 13.0 \\ 60 . & 25.1 \\ 80 . & 35.2 \\ 100 . & 43.2 \\ 120 . & 50.0 \\ 140 . & 56.0 \\ 160 . & 61.6 \\ 180 . & 67.0 \\ 200 . & 72.2 \\ 220 . & 77.4 \\ 240 . & 82.8 \\ 260 . & 88.4 \\ 280 . & 94.0 \\ 300 . & 99.7 \end{array}$$

Calculate \(\Delta S\) if the temperature of \(2.50 \mathrm{mol}\) of an ideal gas with \(C_{V}=5 / 2 R\) is increased from 160 . to \(675 \mathrm{K}\) under conditions of \((\mathrm{a})\) constant pressure and (b) constant volume.

The interior of a refrigerator is typically held at \(36^{\circ} \mathrm{F}\) and the interior of a freezer is typically held at \(0.00^{\circ} \mathrm{F}\) If the room temperature is \(65^{\circ} \mathrm{F}\), by what factor is it more expensive to extract the same amount of heat from the freezer than from the refrigerator? Assume that the theoretical limit for the performance of a reversible refrigerator is valid in this case.

3.75 moles of an ideal gas with \(C_{V, m}=3 / 2 R\) undergoes the transformations described in the following list from an initial state described by \(T=298 \mathrm{K}\) and \(P=4.50\) bar. Calculate \(q, w, \Delta U, \Delta H,\) and \(\Delta S\) for each process. a. The gas undergoes a reversible adiabatic expansion until the final pressure is one third its initial value. b. The gas undergoes an adiabatic expansion against a constant external pressure of 1.50 bar until the final pressure is one third its initial value. c. The gas undergoes an expansion against a constant external pressure of zero bar until the final pressure is equal to one third of its initial value.

Calculate \(\Delta S, \Delta S_{\text {total}},\) and \(\Delta S_{\text {surroundings}}\) when the volume of \(150 .\) g of \(\mathrm{CO}\) initially at \(273 \mathrm{K}\) and 1.00 bar increases by a factor of two in (a) an adiabatic reversible expansion, (b) an expansion against \(P_{\text {external}}=0,\) and (c) an isothermal reversible expansion. Take \(C_{P, m}\) to be constant at the value \(29.14 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\) and assume ideal gas behavior. State whether each process is spontaneous. The temperature of the surroundings is \(273 \mathrm{K}\)
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