## The Carnot Principle

This is one of the fundamental principles of thermodynamics. It says that there is no heat engine operating between two given reservoirs that can be more efficient than a Carnot heat engine operating between the same two reservoirs. To prove this statement, assume that the reverse is true, Fig. 2.2. Then since the Carnot engine is reversible, it can be reversed to operate as a refrigerator. Therefore, if the engine E is more efficient than the Carnot engine CE, the result becomes ('7e > nce).

Fig. 2.2 The graphical explanation of the Kelvin-Planck and Clausius statements of the second law.

This can also be written as

giving as the final result v qh qc ,

(Qc > Qh ) • If this is the case, then there is a net flow of energy in the form of heat from the cold to the hot reservoir without any consumption of work or other external effects. Such a result is impossible since it leads to the contradiction of the Clausius statement of the second law of thermodynamics. Thus the original assumption was wrong which says that the efficiency of an engine cannot be larger than that of a Carnot engine operating between the same two reservoirs.

A corollary to the Carnot principle is that all Carnot engines operating between the same two temperature reservoirs TH and TL have the same efficiency. The proof of this statement follows from the above. Assume that the first one is more efficient than the other. A contradiction will be observed. Then assume the other way around. The same contradiction will be obtained. The only possibility remaining, therefore, is naturally the equivalence of the two efficiencies.

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