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Chapter 19: Problem 38

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
The existence of a perfect heat engine and a perfect refrigerator would create a perpetual motion machine, which contradicts the Clausius statement of the second law of thermodynamics. This is because the work generated by the perfect heat engine would be used to power the perfect refrigerator, causing an endless energy cycle that is not possible according to energy conservation principles.

Step by step solution

01

Understand the Concept of a Perfect Heat Engine

A perfect heat engine is a hypothetical engine that converts all absorbed heat into work. This engine would operate in such a way that, given an amount of heat energy \(Q_h\) absorbed from a hot reservoir, it will convert this entire energy into work \(W\). The process is illustrated by this energy-flow diagram: \(Q_h -> W\). In this process, there is no waste heat \(Q_c\) released to a cold reservoir since all the heat is transformed into work.
02

Understand the Concept of a Perfect Refrigerator

A perfect refrigerator is a hypothetical cooling device that operates with 100% efficiency. It is meant to transfer heat from a cold body (or reservoir) to a hot body (or reservoir) without using any work or energy. If the perfect refrigerator were possible, given energy \(Q_c\) from a cold reservoir, it could transfer this entire energy to a hot reservoir \(Q_h\). The process is shown by the energy-flow diagram: \(Q_c -> Q_h\). There is no work input in this case as it violates the Clausius statement.
03

Show the Connection between the Perfect Heat Engine and the Perfect Refrigerator

If both the perfect heat engine and the perfect refrigerator could exist, one could use the work produced by the heat engine to operate the refrigerator. This would result in a perpetual motion machine, which violates the Clausius statement of the second law of thermodynamics. There would be a complete energy-flow cycle \(Q_h -> W -> Q_c -> Q_h\), where the heat released by the heat engine is absorbed by the refrigerator and put back into the hot reservoir, continuing the cycle indefinitely without any external energy input. This would result in a violation of the principle of energy conservation.

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Most popular questions from this chapter

The human body can be \(25 \%\) efficient at converting chemical energy of fuel to mechanical work. Can the body be considered a heat engine, operating on the temperature difference between body temperature and the environment?

Could you cool the kitchen by leaving the refrigerator open? Explain.

For a gas of six molecules confined to a box, find the probability that (a) all the molecules will be found on one side of the box and (b) half the molecules will be found on each side.

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An object's heat capacity is inversely proportional to its absolute temperature: \(C=C_{0}\left(T_{0} / T\right),\) where \(C_{0}\) and \(T_{0}\) are constants. Find the entropy change when the object is heated from \(T_{0}\) to \(T_{1}\)
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