With the aid of these new concepts he proceeded, in one of the most brilliant feats of the imagination in the history of science, to show that with a heat reservoir at a given temperature from which heat could be taken by a working substance, and with another reservoir at a lower temperature into which it could reject heat, the greatest possible amount of mechanical work can be obtained when all the processes undergone by the working substance are reversible.
The argument runs, in brief, as follows: Suppose we have an engine whose working substance, e.g., steam, takes an amount of heat (H) from a hot reservoir (such as a steam boiler), does an amount of work (W) on an external load, rejects an amount of heat (h) to a cooler reservoir, and is then returned to its original condition, so that the process can be repeated indefinitely. If all these operations can be imagined to be performed reversibly the engine will, when the work (W) is done upon it in each cycle by an outside agency, run in the reverse direction, taking the heat (h) from the cooler and delivering the heat (H) to the hotter reservoir in each cycle.
Thus if any engine (reversible or nonreversible) operating between the same two reservoirs can be supposed to be able to perform the same amount of work (W) on a smaller amount of heat than (H), i.e., can be more efficient, it could be made to drive the reversible engine backward; and since the combined engines would in each cycle of operation return to the hot reservoir a larger amount of heat than the driver takes from it, we would have heat continuously transferred from a cooler to a hotter body by a self-contained system receiving no external aid. As such a result is contrary to all experience, it must be concluded that no engine can be more efficient than a reversible engine when they both work through the same range of temperature; or alternatively, that the maximum amount of work from a given supply of heat is obtainable only by means of a reversible engine. This conclusion is known as "Carnot's principle." It should be noted that the argument on which it is based is independent of the nature of the working substance or of the particular cycle of operations which it follows. The only things essential to the conclusion are first, the existence of different temperatures for the reservoirs; second, the reversible nature of the operations undergone by the working substance; and third, the denial of the possibility that heat can be transferred from a colder to a hotter body by any unaided self-acting contrivance.
This third essential element of the argument is known as the "Second Law of Thermodynamics." Although inherent in the derivation of Carnot's principle, it was not specifically formulated by him as a law. Not until more than twenty-five years later did Clausius and William Thomson (Lord Kelvin) formulate it as one of the foundation stones of the science of thermodynamics. The two other essentials of Carnot's argument -- the concept of reversibility and the consequence that mechanical work cannot be obtained from heat in the absence of the existence of a temperature difference -- are also equally fundamental in the science. But although Carnot must be regarded as the creator of the new science, he was unable to take the next step: the quantitative determination of the magnitude of the ideal efficiency. This was because of the confusion then existing in the scientific world as to the nature of heat. There is some evidence in the Reflexions itself, and still more in his posthumous papers published many years later by his brother, that Carnot was dubious as to the soundness of the then widely accepted "caloric" or materialistic theory of heat and was inclined toward the kinetic theory of its nature.
However, the man who first took this second step, and through whom most physicists learned of Carnot's work, Benoit Pierre Emile Clapeyron,6 was a firm adherent of the idea that heat was a material "fluid.". "Memoire sur la puissance de la chaleur," Journal de l'ecole polytechnique. He put Carnot's principle in the mathematical form that the efficiency of a reversible engine was a function solely of the temperatures of the two reservoirs (Carnot's function), and proceeded to determine its value by the consideration of a particular reversible cycle applicable to the change of state of a substance (liquid to vapor or solid to liquid). Unfortunately, although his result was correct, in deriving his expression he assumed (as was natural on the caloric hypothesis) that the heat taken from the hot reservoir (H) was the same as that rejected to the cold reservoir (h), the work done being accomplished by the fall in temperature in a manner analogous to that done by a water wheel through the fall of the water because of the difference in height of the intake and the outflow.
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