Carnot's theorem

“ This line of reasoning does not appear quite satisfactory, because it is tacitly assumed, in the reasoning by which Carnot’s principle was established, as a result of universal experience, that a number of bodies within the same impervious enclosure, which contains no source of heat, will ultimately acquire the same temperature, and that difference of temperature is required to produce flow of heat. ”

“ When I speak thus, you no doubt think of radium, that grand revolutionist of the present time, and in fact I shall come back to it presently; but there is something else. It is not alone the conservation of energy which is in question; all the other principles are equally in danger, as we shall see in passing them successively in review.
Carnot's Principle.—Let us commence with the principle of Carnot. This is the only one which does not present itself as an immediate consequence of the hypothesis of central forces ”

“ Assuming the specific heat to be also independent of the volume, he shows that the function F′t should be constant. But this assumption is inconsistent with the caloric theory of latent heat of expansion, which requires the specific heat to be a function of the volume. It appears in fact impossible to reconcile Carnot’s principle with the caloric theory on any simple assumptions. As Carnot remarks: “The main principles on which the theory of heat rests require most careful examination. Many experimental facts appear almost inexplicable in the present state of this theory.” ”

“ That is not impossible; that is only infinitely improbable. The chances are that we should wait a long time for the concourse of circumstances which would permit a retrogradation; but sooner or later they will occur, after years whose number it would take millions of figures to write. These reservations, however, all remained theoretic; they were not very disquieting, and Carnot's principle retained all its practical value. But here the scene changes. The biologist, armed with his microscope, long ago noticed in his preparations irregular movements [Pg 305] of little particles in suspension ”

“ For those who take this point of view, Carnot's principle is only an imperfect principle, a sort of concession to the infirmity of our senses; it is because our eyes are too gross that we do not distinguish the elements of the blend; it is because our hands are too gross that we can not force them to separate; the imaginary demon of Maxwell, who is able to sort the molecules one by one, could well constrain the world to return backward. ”

“ Carnot, Leonard Sadi, son of Nicolas, founder of thermo-dynamics; in his "Reflexions sur la Puissance du Feu" enunciates the principle of Reversibility, considered the most important contribution to physical science since the time of Newton (1796-1832) . See Dr. Knott's "Physics."
Carnot, Marie François, civil engineer and statesman, born at Limoges, nephew of the preceding; Finance Minister in 1879 and 1887; became President in 1887; was assassinated at Lyons by an anarchist in 1894.
Carnot, Nicolas, French mathematician and engineer, born at Nolay, in Burgundy ”

“ Thus appears the idea that energies which exchange with each other and correspond to equal quantities have not the same qualitative value. Form has its importance, and there are persons who prefer a golden louis to four pieces of five francs. The principle of Carnot would thus lead us to consider a certain classification of energies, and would show us that, in the transformations possible, these energies always tend to a sort of diminution of quality—that is, to a degradation. ”

“ In fact, the attempts which have often been made to refer the principle of Carnot to mechanics have not given convincing results. It has nearly always been necessary to introduce into the attempt some new hypothesis independent of the fundamental hypotheses of ordinary mechanics, and equivalent, in reality, to one of the postulates on which the ordinary exposition of the second law of thermodynamics is founded. ”

“ The condition (a) is fulfilled because, by theorem E, a part of can only have one of the cardinal numbers , and therefore, by theorem A, cannot be equivalent to the aggregate . The condition (b) is fulfilled because itself is a part of .
Proof of C.— Let be a cardinal number which is less than . Because of the condition (b) of §2, there is a part of with the cardinal number . By theorem E, a part of can only have one of the cardinal numbers . Thus is equal to one of the cardinal numbers . By theorem B, none of these is greater than . ”

“ A partial concentration may be brought in a system, but only at the expense of greater dissipation, and the sum total is always an increase in dissipation." [26] Through the labors of Joule and Kelvin, Maxwell and Boltzmann, Gibbs and Helmholtz, Carnot's simple generalization about heat engines has been elevated to the dignity of an irrevocable law of nature, a principle of scientific determinism, giving one of the most complete and satisfactory answers that man can furnish to the great question: How does any event in the material universe come to pass? ”

“ Prediction in other Sciences.
Generations, or even centuries, may elapse before mankind are in possession of a mathematical theory of the constitution of matter as complete as the theory of gravitation. Nevertheless, mathematical physicists have in recent years acquired a hold of some of the relations of the physical forces, and the proof is found in anticipations of curious phenomena which had never been observed. Professor James Thomson deduced from Carnot’s theory of heat that the application of pressure would lower the melting-point of ice. ”

“ It is clear no scientist can subscribe to this conclusion. The laws of nature bind the antecedent to the consequent in such a way that the antecedent is as well determined by the consequent as the consequent by the antecedent. But whence came the error of this philosopher? We know that in virtue of Carnot's principle physical phenomena are irreversible and the world tends toward uniformity. When two bodies of different temperature come in contact, the warmer gives up heat to the colder ”

“ In the second part of his paper he introduces Carnot’s principle, which he quotes as follows: “The performance of work is equivalent to a transference of heat from a hot to a cold body without the quantity of heat being thereby diminished.” This is not Carnot’s way of stating his principle (see § 15) , but has the effect of exaggerating the importance of Clapeyron’s unnecessary assumption. ”

“ Heat equilibrium in an enclosure of constant temperature such as is here postulated by Prevost, has often been regarded as a consequence of Carnot’s principle. Since difference of temperature is required for transforming heat into work, no work could be obtained from heat in such a system, and no spontaneous changes of temperature can take place, as any such changes might be utilized for the production of work. ”

“ The second fundamental principle is that of the transformations of equilibrium, or of the conditions of reversibility, or again, Carnot’s principle. This principle, which first assumed a concrete form in thermodynamics, has been very widely extended. It has reached a degree of generality such that contemporary theoretical physicists such as Lord Kelvin, Le Châtelier, etc., consider it the universal law of physical, mechanical, and chemical equilibrium. ”

“ Originality of Carnot's principle that fall of temperature necessary for production of work by heat— Clausius' postulate that heat cannot pass from cold to hot body without accessory phenomena—Entropy result of this—Definition of entropy—Entropy tends to increase incessantly—A magnitude which measures evolution of system—Clausius' and Kelvin's deduction that heat end of all energy in Universe—Objection to this— Carnot's principle not necessarily referable to mechanics —Brownian movements—Lippmann's objection to kinetic hypothesis. ”

“ Carnot’s principle contains, as was shown by G. Robin, d’Alembert’s principle of virtual velocities, and according to physicists of to-day, as we have just remarked, it contains the laws peculiar to physico-chemical equilibrium. The application of this principle gives us the differential equations from which are derived numerical relations between the different energies, or the different modalities of universal energy. ”

“ To establish a doctrine of energetics that shall form a sufficient foundation for a theory of the trend of chemical and physical change, we have, therefore, to impart precision to this motion of available energy.
Carnot’s Principle: Entropy.—The whole subject is involved in the new principle contributed to theoretical physics by Sadi Carnot in 1824, in which the far-reaching modern conception of cyclic processes was first scientifically developed. ”

“ It can be held, in fact, that this suggestion does not fall short of a demonstration, on the basis of Carnot’s principle, and independent of special molecular theory, that in all cases where the molecules of a component, whether it be of a gas or of a solution, are outside each other’s spheres of influence, the available energy, so far as regards dilution, must have a common form, and the physical constants must therefore be the known gas-constants. ”

“ Wherein do these permanently electrified molecules differ from Coulomb's electric molecules? It is true that in the electrons the electricity is supported by a little, a very little matter; in other words, they have a mass (and yet this is now contested) ; but Coulomb did not deny mass to his fluids, or, if he did, it was only with reluctance. It would be rash to affirm that the belief in electrons will not again suffer eclipse; it was none the less curious to note this unexpected resurrection.
But the most striking example is Carnot's principle. Carnot set it up starting from false hypotheses ”

“ Clapeyron gave some of its main features in an analytical form, and he also employed Watt's diagram for the exhibition of others. Even this, however, failed to call attention properly to the extremely novel processes of Carnot, and it was reserved for Sir W. Thomson (in 1848, and more at length in 1849) to point out to scientific men their full value. His papers on Carnot's treatise, follow ing closely after the splendid experimental researches of Colding and Joule, secured for the dynamical theory of heat its position as a recognized branch of science. ”

“ After it had been proved that heat must disappear if mechanical work was to be done at its expense, Carnot's principle could no longer be regarded as a complete expression of the facts. Its improved form was first given, in 1850, by Clausius, whom Thomson followed in 1851. It runs thus: "If a quantity of heat Q' is transformed into work in a reversible process, another quantity of heat Q of the absolute [55] temperature T1 is lowered to the absolute temperature T2." ”

“ The irreversible phenomena are much more difficult to bring into line; but they, too, are being co-ordinated and tend to come into the unity. The light which illuminates them comes from Carnot's principle. For a long time thermo-dynamics was confined to the study of the dilatations of bodies and of their change of state. For some time past it has been growing bolder, and has considerably extended its domain. We owe to it the theories of the voltaic cell and of their thermo-electric phenomena; there is not a corner in physics which it has not explored, and it has even attacked chemistry itself. ”

“ In p. 30 some conclusions drawn by Carnot from his general reasoning were noticed; according to which it appears, that if the value of μ for any temperature is known, certain information may be derived with reference to the saturated vapor of any liquid whatever, and, with reference to any gaseous mass, without the necessity of experimenting upon the specific medium considered. Nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning. ”

“ Professor Tait has said:"Without this work of Carnot, the modern theory of energy, and especially that branch of it which is at present by far the most important in practice, the dynamical theory of heat, could never have attained in so few years its now enormous development. Carnot 's claims to recognition are of an exceedingly high order, because they depend not merely upon his method, which is one of startling novelty and originality and is not confined to the subject of heat ”

“ Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before b and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. ”

“ The peculiar merit of Carnot's reasoning consists in the idea of bringing the body back to its initial state as to temperature, density and molecular condition, after a cycle of operations, before making any assertion as to the amount of heat-energy gained or lost. This he accomplished by causing the working substance (a gas in Carnot's case) to pass through two isothermal changes, the first at a higher and the second at a lower temperature, alternated with two adiabatic [2] changes by which the temperature of the working substance is allowed to fall and then raised again. ”