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Chapter 19: Problem 44

Use energy-flow diagrams to show that the existence of a perfect heat engine would permit the construction of a perfect refrigerator, thus violating the Clausius statement of the second law.

Short Answer

Expert verified
In a perfect heat engine, all heat 'Q' is converted into work 'W', and in a perfect refrigerator, heat 'Q' can be moved from a cooler body to a hotter one without work. Thus, combining the two would allow for a cyclic process which would fully recycle energy, implying the existence of a perfect refrigerator which violates the second law of thermodynamics, particularly the Clausius statement. The Clausius statement states that it is impossible to build a device that, in a cycle, transfers heat from a cooler body to a hotter one without requiring work.

Step by step solution

01

Defining Terms

First, it is essential to define and understand the main terms. A perfect heat engine is a hypothetical engine which converts heat completely into work without any loss of energy. It operates between two reservoirs at different temperatures. A perfect refrigerator, on the other hand, is a hypothetical cooling mechanism that works without any energy input, which means it can move energy from a cooler body to a hotter one without any work. The Clausius statement then states it is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a cooler body to a hotter one.
02

Analyze Energy Flow for Perfect Heat Engine

Based on the definition, a perfect heat engine absorbs heat energy 'Q' from the high-temperature reservoir, converts it entirely into work 'W', and releases no heat into the low-temperature reservoir. The energy-flow diagram would show that the heat 'Q' from the high-temperature reservoir is completely converted into work 'W', and no energy is dissipated or lost.
03

Analyze Energy Flow for Perfect Refrigerator

For a perfect refrigerator, it absorbs heat 'Q' from the low-temperature reservoir and transfer it to the high-temperature reservoir without any work input. The energy-flow diagram would then show heat 'Q' being transferred from a cooler body to a hotter one without energy (work 'W') input.
04

Connection Between Perfect Heat Engine and Perfect Refrigerator

Under ideal conditions, a complete recycling of the heat 'Q' and work 'W' between the heat engine and refrigerator in a cyclic process would be possible, making energy transfer fully reversible. However, this perfect recycling of energy would violate the Clausius statement of the second law of thermodynamics.
05

Violation of the Clausius Statement

According to the Clausius statement, it is impossible for a device operating in a cycle to produce no other effect than the transfer of heat from a cooler body to a hotter one. The above loop violates this statement because heat 'Q' can be transferred from a low-temperature reservoir to a high-temperature reservoir without work (energy). This set-up thus implies the existence of a perfect refrigerator which is a violation of the second law of thermodynamics.

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Most popular questions from this chapter

In an adiabatic free expansion, \(6.36\) mol of ideal gas at \(305 \mathrm{~K}\) expands 15 -fold in volume. How much energy becomes unavailable to do work?

Your power company claims that electric heat is \(100 \%\) efficient. Discuss.

Find the COP of a reversible refrigerator operating between \(0^{\circ} \mathrm{C}\) and \(35^{\circ} \mathrm{C}\).

A gas cylinder with interior volume \(11.5 \mathrm{~L}\) holds compressed air at \(16.8 \mathrm{MPa}\). The cylinder is joined by a hose of negligible volume to a second cylinder that's been pumped down to vacuum. When the valve on the full cylinder is opened, the air expands to fill the entire system, all the while maintaining a constant temperature. If the energy that becomes unavailable to do work is \(169 \mathrm{~kJ}\), what's the volume of the second cylinder?

A cylinder holds \(n\) mol of compressed gas at temperature \(T\). It's connected by a hose of negligible volume to an identical cylinder that's been pumped down to vacuum. When the valve on the full cylinder is opened, the gas expands to fill the entire system while maintaining the constant temperature \(T\). Find an expression for the energy that becomes unavailable to do work as a result of this process.
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