Part 4 (1/2)
-- 3. THE PRINCIPLE OF CARNOT AND CLAUSIUS
The principle of Carnot, of a nature a.n.a.logous to the principle of the conservation of energy, has also a similar origin. It was first enunciated, like the last named, although prior to it in time, in consequence of considerations which deal only with heat and mechanical work. Like it, too, it has evolved, grown, and invaded the entire domain of physics. It may be interesting to examine rapidly the various phases of this evolution. The origin of the principle of Carnot is clearly determined, and it is very rare to be able to go back thus certainly to the source of a discovery. Sadi Carnot had, truth to say, no precursor. In his time heat engines were not yet very common, and no one had reflected much on their theory. He was doubtless the first to propound to himself certain questions, and certainly the first to solve them.
It is known how, in 1824, in his _Reflexions sur la puissance motrice du feu_, he endeavoured to prove that ”the motive power of heat is independent of the agents brought into play for its realization,” and that ”its quant.i.ty is fixed solely by the temperature of the bodies between which, in the last resort, the transport of caloric is effected”--at least in all engines in which ”the method of developing the motive power attains the perfection of which it is capable”; and this is, almost textually, one of the enunciations of the principle at the present day. Carnot perceived very clearly the great fact that, to produce work by heat, it is necessary to have at one's disposal a fall of temperature. On this point he expresses himself with perfect clearness: ”The motive power of a fall of water depends on its height and on the quant.i.ty of liquid; the motive power of heat depends also on the quant.i.ty of caloric employed, and on what might be called--in fact, what we shall call--the height of fall, that is to say, the difference in temperature of the bodies between which the exchange of caloric takes place.”
Starting with this idea, he endeavours to demonstrate, by a.s.sociating two engines capable of working in a reversible cycle, that the principle is founded on the impossibility of perpetual motion.
His memoir, now celebrated, did not produce any great sensation, and it had almost fallen into deep oblivion, which, in consequence of the discovery of the principle of equivalence, might have seemed perfectly justified. Written, in fact, on the hypothesis of the indestructibility of caloric, it was to be expected that this memoir should be condemned in the name of the new doctrine, that is, of the principle recently brought to light.
It was really making a new discovery to establish that Carnot's fundamental idea survived the destruction of the hypothesis on the nature of heat, on which he seemed to rely. As he no doubt himself perceived, his idea was quite independent of this hypothesis, since, as we have seen, he was led to surmise that heat could disappear; but his demonstrations needed to be recast and, in some points, modified.
It is to Clausius that was reserved the credit of rediscovering the principle, and of enunciating it in language conformable to the new doctrines, while giving it a much greater generality. The postulate arrived at by experimental induction, and which must be admitted without demonstration, is, according to Clausius, that in a series of transformations in which the final is identical with the initial stage, it is impossible for heat to pa.s.s from a colder to a warmer body unless some other accessory phenomenon occurs at the same time.
Still more correctly, perhaps, an enunciation can be given of the postulate which, in the main, is a.n.a.logous, by saying: A heat motor, which after a series of transformations returns to its initial state, can only furnish work if there exist at least two sources of heat, and if a certain quant.i.ty of heat is given to one of the sources, which can never be the hotter of the two. By the expression ”source of heat,” we mean a body exterior to the system and capable of furnis.h.i.+ng or withdrawing heat from it.
Starting with this principle, we arrive, as does Clausius, at the demonstration that the output of a reversible machine working between two given temperatures is greater than that of any non-reversible engine, and that it is the same for all reversible machines working between these two temperatures.
This is the very proposition of Carnot; but the proposition thus stated, while very useful for the theory of engines, does not yet present any very general interest. Clausius, however, drew from it much more important consequences. First, he showed that the principle conduces to the definition of an absolute scale of temperature; and then he was brought face to face with a new notion which allows a strong light to be thrown on the questions of physical equilibrium. I refer to entropy.
It is still rather difficult to strip entirely this very important notion of all a.n.a.lytical adornment. Many physicists hesitate to utilize it, and even look upon it with some distrust, because they see in it a purely mathematical function without any definite physical meaning. Perhaps they are here unduly severe, since they often admit too easily the objective existence of quant.i.ties which they cannot define. Thus, for instance, it is usual almost every day to speak of the heat possessed by a body. Yet no body in reality possesses a definite quant.i.ty of heat even relatively to any initial state; since starting from this point of departure, the quant.i.ties of heat it may have gained or lost vary with the road taken and even with the means employed to follow it. These expressions of heat gained or lost are, moreover, themselves evidently incorrect, for heat can no longer be considered as a sort of fluid pa.s.sing from one body to another.
The real reason which makes entropy somewhat mysterious is that this magnitude does not fall directly under the ken of any of our senses; but it possesses the true characteristic of a concrete physical magnitude, since it is, in principle at least, measurable. Various authors of thermodynamical researches, amongst whom M. Mouret should be particularly mentioned, have endeavoured to place this characteristic in evidence.
Consider an isothermal transformation. Instead of leaving the heat abandoned by the body subjected to the transformation--water condensing in a state of saturated vapour, for instance--to pa.s.s directly into an ice calorimeter, we can transmit this heat to the calorimeter by the intermediary of a reversible Carnot engine. The engine having absorbed this quant.i.ty of heat, will only give back to the ice a lesser quant.i.ty of heat; and the weight of the melted ice, inferior to that which might have been directly given back, will serve as a measure of the isothermal transformation thus effected. It can be easily shown that this measure is independent of the apparatus used.
It consequently becomes a numerical element characteristic of the body considered, and is called its entropy. Entropy, thus defined, is a variable which, like pressure or volume, might serve concurrently with another variable, such as pressure or volume, to define the state of a body.
It must be perfectly understood that this variable can change in an independent manner, and that it is, for instance, distinct from the change of temperature. It is also distinct from the change which consists in losses or gains of heat. In chemical reactions, for example, the entropy increases without the substances borrowing any heat. When a perfect gas dilates in a vacuum its entropy increases, and yet the temperature does not change, and the gas has neither been able to give nor receive heat. We thus come to conceive that a physical phenomenon cannot be considered known to us if the variation of entropy is not given, as are the variations of temperature and of pressure or the exchanges of heat. The change of entropy is, properly speaking, the most characteristic fact of a thermal change.
It is important, however, to remark that if we can thus easily define and measure the difference of entropy between two states of the same body, the value found depends on the state arbitrarily chosen as the zero point of entropy; but this is not a very serious difficulty, and is a.n.a.logous to that which occurs in the evaluation of other physical magnitudes--temperature, potential, etc.
A graver difficulty proceeds from its not being possible to define a difference, or an equality, of entropy between two bodies chemically different. We are unable, in fact, to pa.s.s by any means, reversible or not, from one to the other, so long as the trans.m.u.tation of matter is regarded as impossible; but it is well understood that it is nevertheless possible to compare the variations of entropy to which these two bodies are both of them individually subject.
Neither must we conceal from ourselves that the definition supposes, for a given body, the possibility of pa.s.sing from one state to another by a reversible transformation. Reversibility is an ideal and extreme case which cannot be realized, but which can be approximately attained in many circ.u.mstances. So with gases and with perfectly elastic bodies, we effect sensibly reversible transformations, and changes of physical state are practically reversible. The discoveries of Sainte-Claire Deville have brought many chemical phenomena into a similar category, and reactions such as solution, which used to be formerly the type of an irreversible phenomenon, may now often be effected by sensibly reversible means. Be that as it may, when once the definition is admitted, we arrive, by taking as a basis the principles set forth at the inception, at the demonstration of the celebrated theorem of Clausius: _The entropy of a thermally isolated system continues to increase incessantly._
It is very evident that the theorem can only be worth applying in cases where the entropy can be exactly defined; but, even when thus limited, the field still remains vast, and the harvest which we can there reap is very abundant.
Entropy appears, then, as a magnitude measuring in a certain way the evolution of a system, or, at least, as giving the direction of this evolution. This very important consequence certainly did not escape Clausius, since the very name of entropy, which he chose to designate this magnitude, itself signifies evolution. We have succeeded in defining this entropy by demonstrating, as has been said, a certain number of propositions which spring from the postulate of Clausius; it is, therefore, natural to suppose that this postulate itself contains _in potentia_ the very idea of a necessary evolution of physical systems. But as it was first enunciated, it contains it in a deeply hidden way.
No doubt we should make the principle of Carnot appear in an interesting light by endeavouring to disengage this fundamental idea, and by placing it, as it were, in large letters. Just as, in elementary geometry, we can replace the postulate of Euclid by other equivalent propositions, so the postulate of thermodynamics is not necessarily fixed, and it is instructive to try to give it the most general and suggestive character.
MM. Perrin and Langevin have made a successful attempt in this direction. M. Perrin enunciates the following principle: _An isolated system never pa.s.ses twice through the same state_. In this form, the principle affirms that there exists a necessary order in the succession of two phenomena; that evolution takes place in a determined direction. If you prefer it, it may be thus stated: _Of two converse transformations unaccompanied by any external effect, one only is possible_. For instance, two gases may diffuse themselves one in the other in constant volume, but they could not conversely separate themselves spontaneously.
Starting from the principle thus put forward, we make the logical deduction that one cannot hope to construct an engine which should work for an indefinite time by heating a hot source and by cooling a cold one. We thus come again into the route traced by Clausius, and from this point we may follow it strictly.
Whatever the point of view adopted, whether we regard the proposition of M. Perrin as the corollary of another experimental postulate, or whether we consider it as a truth which we admit _a priori_ and verify through its consequences, we are led to consider that in its entirety the principle of Carnot resolves itself into the idea that we cannot go back along the course of life, and that the evolution of a system must follow its necessary progress.
Clausius and Lord Kelvin have drawn from these considerations certain well-known consequences on the evolution of the Universe. Noticing that entropy is a property added to matter, they admit that there is in the world a total amount of entropy; and as all real changes which are produced in any system correspond to an increase of entropy, it may be said that the entropy of the world is continually increasing.
Thus the quant.i.ty of energy existing in the Universe remains constant, but transforms itself little by little into heat uniformly distributed at a temperature everywhere identical. In the end, therefore, there will be neither chemical phenomena nor manifestation of life; the world will still exist, but without motion, and, so to speak, dead.
These consequences must be admitted to be very doubtful; we cannot in any certain way apply to the Universe, which is not a finite system, a proposition demonstrated, and that not unreservedly, in the sharply limited case of a finite system. Herbert Spencer, moreover, in his book on _First Principles_, brings out with much force the idea that, even if the Universe came to an end, nothing would allow us to conclude that, once at rest, it would remain so indefinitely. We may recognise that the state in which we are began at the end of a former evolutionary period, and that the end of the existing era will mark the beginning of a new one.