Page:Elementary Text-book of Physics (Anthony, 1897).djvu/261

§ 231] susceptible of immediate demonstration, and is accepted as a general principle for reasons similar to those which determine the acceptance of Newton's laws of motion as statements of general truths respecting motion.

230. Efaciency of a Reversible Engine.— Carnot's proposition, which is now to be proved, asserts that the efficiency of all reversible engines is the same. To show this, let us suppose a reversible engine $$A$$ and a non-reversible engine $$B,$$ working between the same source and the same refrigerator, and let us assume that the efficiency of the non-reversible engine $$B$$ is greater than that of the reversible engine $$A.$$ Let the engine $$B$$ work forward, so as to do work $$W$$ and give to the refrigerator the quantity of heat $$h_{B};$$ it will therefore take from the source the quantity of heat $$H_{B} = W + h_{B}.$$ Let the work $$W$$ be expended in driving the reversible engine $$A$$ backward. The engine $$A$$ will take from the refrigerator the quantity of heat $$h_{A}$$ and give to the source the quantity of heat $$H_{A} = W + h_{A}.$$ Now, by hypothesis, the efficiency of the non-reversible engine $$B$$ is the greater, so that $$\frac{W}{H_{B}} > \frac{W}{H_{A}},$$ and therefore $$H_{A} > H_{B},$$ and also $$h_{A} > h_{B}.$$ The result of these combined operations is that no work is done by the engines, and that the source receives heat while the refrigerator loses heat. This conclusion is contrary to Clausius's principle and must be rejected, as inconsistent with the operations of Nature. We conclude, therefore, that no engine can have an efficiency greater than that of the reversible engine. It follows as a corollary that the efficiency of all reversible engines is the same.

231. Absolute Scale of Temperatures.—Since the efficiency of all reversible engines is the same and is a maximum, it is manifestly indifferent what material is used in the working body; indeed, since the demonstration just given does not involve as essential the particular mode of doing work assumed in the construction of the diagram, it is also indifferent in what way the working body changes its dimensions and does work. The work done depends only on the heat received from the source and on the