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Chapter 19: Problem 59

In an alternative universe, you've got the impossible: an infinite heat reservoir, containing infinite energy at temperature \(T_{\mathrm{h}} .\) But you've only got a finite cool reservoir, with initial temperature \(T_{\mathrm{c} 0}\) and heat capacity \(C .\) Find an expression for the maximum work you can extract if you operate an engine between these two reservoirs.

Short Answer

Expert verified
The maximum work that can be extracted from an engine operating between an infinite hot reservoir and a finite cool reservoir is given by the expression \(W_{\mathrm{max}}=-C\cdot(T_{\mathrm{h}} - T_{\mathrm{c} 0})\).

Step by step solution

01

Applying the First Law of Thermodynamics

According to the first law of thermodynamics - energy conservation - we know that the heat gained \(Q_{\mathrm{h}} = Q_{\mathrm{c}} + W\), where \(Q_{\mathrm{h}}\) is the heat gained from the hot reservoir, \(Q_{\mathrm{c}}\) is the heat given to the cold reservoir and \(W\) is the work extracted. Since \(Q_{\mathrm{h}}\) is infinite, the primary constraint here is the heat capacity of the cold reservoir.
02

Understanding the Second Law of Thermodynamics

The second law of thermodynamics states that heat cannot flow spontaneously from a colder location to a hotter location, or, equivalently, heat at a given temperature cannot be converted into work with 100% efficiency.
03

Calculate Qc (Heat capacity of cold reservoir)

The heat transferred, \(Q_{\mathrm{c}}\), to a system that make its temperature change from \(T_{\mathrm{c} 0}\) to \(T_{\mathrm{h}}\) is given by \(Q_{\mathrm{c}}=C\cdot(T_{\mathrm{h}} - T_{\mathrm{c} 0})\).
04

Substitute \(Qc\) in the equation for \(Q_{\mathrm{h}}\)

Substitute the expression for \(Q_{\mathrm{c}}\) into the equation \(Q_{\mathrm{h}} = Q_{\mathrm{c}} + W\). This yields the expression for the maximum work as \(W = Q_{\mathrm{h}} - C\cdot(T_{\mathrm{h}} - T_{\mathrm{c} 0})\). Since the \(Q_{\mathrm{h}}\) is infinite, we can disregard it in the equation.
05

Final expression for maximum work

Hence, the final expression for the maximum work extractable is \(W_{\mathrm{max}}=-C\cdot(T_{\mathrm{h}} - T_{\mathrm{c} 0})\). This means that the work obtained is the heat required to increase the temperature of the cold reservoir from initial temperature \(T_{\mathrm{c} 0}\) up to the temperature of the hot reservoir \(T_{\mathrm{h}}\). The negative sign indicates that energy is taken out of the system in the form of work.

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Most popular questions from this chapter

Refrigerators remain among the greatest consumers of electrical energy in most homes, although mandated efficiency standards have decreased their energy consumption by some \(80 \%\) in the past four decades. In the course of a day, one kitchen refrigerator removes \(30 \mathrm{MJ}\) of energy from its contents, in the process consuming \(10 \mathrm{MJ}\) of electrical energy. The electricity comes from a \(40 \%\) efficient coal-fired power plant. The electrical energy a. is used to run the light bulb inside the refrigerator. b. wouldn't be necessary if the refrigerator had enough insulation. c. retains its high-quality status after the refrigerator has used it. d. ends up as waste heat rejected to the kitchen environment.

Why doesn't the evolution of human civilization violate the second law of thermodynamics?

A power plant extracts energy from steam at \(250^{\circ} \mathrm{C}\) and delivers 800 MW of electric power. It discharges waste heat to a river at \(30^{\circ} \mathrm{C} .\) The plant's overall efficiency is \(28 \% .\) (a) How does this efficiency compare with the maximum possible at these temperatures? (b) Find the rate of waste-heat discharge to the river. (c) How many houses, each requiring \(18 \mathrm{kW}\) of heating power, could be heated with the waste heat from this plant?

Refrigerators remain among the greatest consumers of electrical energy in most homes, although mandated efficiency standards have decreased their energy consumption by some \(80 \%\) in the past four decades. In the course of a day, one kitchen refrigerator removes \(30 \mathrm{MJ}\) of energy from its contents, in the process consuming \(10 \mathrm{MJ}\) of electrical energy. The electricity comes from a \(40 \%\) efficient coal-fired power plant. The total energy rejected to the surrounding kitchen during the course of the day is a. \(10 \mathrm{MJ}\). b. \(30 \mathrm{MJ}\). c. \(40 \mathrm{MJ}\). d. \(75 \mathrm{MJ}\).

The maximum steam temperature in a nuclear power plant is \(570 \mathrm{K}\) The plant rejects heat to a river whose temperature is \(0^{\circ} \mathrm{C}\) in the winter and \(25^{\circ} \mathrm{C}\) in the summer. What are the maximum possible efficiencies for the plant during these seasons?
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