New Physics and Its Evolution — Principles - Part 2

§ 3. THE PRINCIPLE OF CARNOT AND CLAUSIUS

The principle of Carnot, of a nature analogous 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 _Réflexions 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 quantity 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 quantity of liquid; the motive power of heat depends also on the quantity 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 associating 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 pass 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 analogous, 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 quantity 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 furnishing 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 analytical 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 quantities 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 quantity of heat even relatively to any initial state; since starting from this point of departure, the quantities 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 passing 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 pass 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 quantity of heat, will only give back to the ice a lesser quantity 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 analogous 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 pass by any means, reversible or not, from one to the other, so long as the transmutation 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 passing 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 circumstances.

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 passes 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 quantity 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.

Like an elastic and mobile object which, thrown into the air, attains by degrees the summit of its course, then possesses a zero velocity and is for a moment in equilibrium, and then falls on touching the ground to rebound, so the world should be subjected to huge oscillations which first bring it to a maximum of entropy till the moment when there should be produced a slow evolution in the contrary direction bringing it back to the state from which it started.

Thus, in the infinity of time, the life of the Universe proceeds without real stop.

This conception is, moreover, in accordance with the view certain physicists take of the principle of Carnot.

We shall see, for example, that in the kinetic theory we are led to admit that, after waiting sufficiently long, we can witness the return of the various states through which a mass of gas, for example, has passed in its series of transformations.

If we keep to the present era, evolution has a fixed direction--that which leads to an increase of entropy; and it is possible to enquire, in any given system to what physical manifestations this increase corresponds.

We note that kinetic, potential, electrical, and chemical forms of energy have a great tendency to transform themselves into calorific energy.

A chemical reaction, for example, gives out energy;

but if the reaction is not produced under very special conditions, this energy immediately passes into the calorific form.

This is so true, that chemists currently speak of the heat given out by reactions instead of regarding the energy disengaged in general.

In all these transformations the calorific energy obtained has not, from a practical point of view, the same value at which it started.

One cannot, in fact, according to the principle of Carnot, transform it integrally into mechanical energy, since the heat possessed by a body can only yield work on condition that a part of it falls on a body with a lower temperature.

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_.

It would thus reintroduce an element of differentiation of which it seems very difficult to give a mechanical explanation.

Certain philosophers and physicists see in this fact a reason which condemns _a priori_ all attempts made to give a mechanical explanation of the principle of Carnot.

It is right, however, not to exaggerate the importance that should be attributed to the phrase degraded energy.

If the heat is not equivalent to the work, if heat at 99° is not equivalent to heat at 100°, that means that we cannot in practice construct an engine which shall transform all this heat into work, or that, for the same cold source, the output is greater when the temperature of the hot source is higher; but if it were possible that this cold source had itself the temperature of absolute zero, the whole heat would reappear in the form of work.

The case here considered is an ideal and extreme case, and we naturally cannot realize it; but this consideration suffices to make it plain that the classification of energies is a little arbitrary and depends more, perhaps, on the conditions in which mankind lives than on the inmost nature of things.

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.

Helmholtz, in a justly celebrated theory, endeavoured to fit the principle of Carnot into the principle of least action; but the difficulties regarding the mechanical interpretation of the irreversibility of physical phenomena remain entire.

Looking at the question, however, from the point of view at which the partisans of the kinetic theories of matter place themselves, the principle is viewed in a new aspect.

Gibbs and afterwards Boltzmann and Professor Planck have put forward some very interesting ideas on this subject.

By following the route they have traced, we come to consider the principle as pointing out to us that a given system tends towards the configuration presented by the maximum probability, and, numerically, the entropy would even be the logarithm of this probability.

Thus two different gaseous masses, enclosed in two separate receptacles which have just been placed in communication, diffuse themselves one through the other, and it is highly improbable that, in their mutual shocks, both kinds of molecules should take a distribution of velocities which reduce them by a spontaneous phenomenon to the initial state.

We should have to wait a very long time for so extraordinary a concourse of circumstances, but, in strictness, it would not be impossible.

The principle would only be a law of probability.

Yet this probability is all the greater the more considerable is the number of molecules itself.

In the phenomena habitually dealt with, this number is such that, practically, the variation of entropy in a constant sense takes, so to speak, the character of absolute certainty.

But there may be exceptional cases where the complexity of the system becomes insufficient for the application of the principle of Carnot;-- as in the case of the curious movements of small particles suspended in a liquid which are known by the name of Brownian movements and can be observed under the microscope.

The agitation here really seems, as M.

Gouy has remarked, to be produced and continued indefinitely, regardless of any difference in temperature; and we seem to witness the incessant motion, in an isothermal medium, of the particles which constitute matter.

Perhaps, however, we find ourselves already in conditions where the too great simplicity of the distribution of the molecules deprives the principle of its value.

M.

Lippmann has in the same way shown that, on the kinetic hypothesis, it is possible to construct such mechanisms that we can so take cognizance of molecular movements that _vis viva_ can be taken from them.

The mechanisms of M. Lippmann are not, like the celebrated apparatus at one time devised by Maxwell, purely hypothetical.

They do not suppose a partition with a hole impossible to be bored through matter where the molecular spaces would be larger than the hole itself.

They have finite dimensions.

Thus M. Lippmann considers a vase full of oxygen at a constant temperature.

In the interior of this vase is placed a small copper ring, and the whole is set in a magnetic field.

The oxygen molecules are, as we know, magnetic, and when passing through the interior of the ring they produce in this ring an induced current.

During this time, it is true, other molecules emerge from the space enclosed by the circuit; but the two effects do not counterbalance each other, and the resulting current is maintained.

There is elevation of temperature in the circuit in accordance with Joule's law; and this phenomenon, under such conditions, is incompatible with the principle of Carnot.

It is possible--and that, I think, is M. Lippmann's idea--to draw from his very ingenious criticism an objection to the kinetic theory, if we admit the absolute value of the principle; but we may also suppose that here again we are in presence of a system where the prescribed conditions diminish the complexity and render it, consequently, less probable that the evolution is always effected in the same direction.