Part 20

VI. THE DIFFERENCES OF THE ENERGIES AND THE LIMITS OF THE PRINCIPLE OF ENERGY.

Of every quantity of heat Q which does work in a reversible process (one unaccompanied by loss) between the absolute temperatures T₁, T₂, only the portion

(T₁-T₂)/T₁

is transformed into work, while the remainder is transferred to the lower temperature-level T₂. This transferred portion can, upon the reversal of the process, with the same expenditure of work, again be brought back to the level T₁. But if the process is not reversible, then more heat than in the foregoing case flows to the lower level, and the surplus can no longer be brought back to the higher level T₂ without some special expenditure. W. Thomson (1852), accordingly, drew attention to the fact, that in all non-reversible, that is, in all real thermal processes, quantities of heat are lost for mechanical work, and that accordingly a dissipation or waste of mechanical energy is taking place. In all cases, heat is only partially transformed into work, but frequently work is wholly transformed into heat. Hence, a tendency exists towards a diminution of the mechanical energy and towards an increase of the thermal energy of the world.

For a simple, closed cyclical process, accompanied by no loss, in which the quantity of heat Q₁ is taken from the level T₁, and the quantity Q₂ is deposited upon the level T₂, the following relation, agreeably to equation (2), exists,

-(Q₁/T₁) + (Q₂/T₂) = 0.

Similarly, for any number of compound reversible cycles Clausius finds the algebraical sum

Q/T = 0,

and supposing the temperature to change continuously,

Here the elements of the quantities of heat deducted from a given level are reckoned negative, and the elements imparted to it, positive. If the process is not reversible, then expression (4), which Clausius calls entropy, increases. In actual practice this is always the case, and Clausius finds himself led to the statement:

1. That the energy of the world remains constant.

2. That the entropy of the world tends toward a maximum.

Once we have noted the above-indicated conformity in the behavior of different energies, the peculiarity of thermal energy here mentioned must strike us. Whence is this peculiarity derived, for, generally every energy passes only partly into another form, which is also true of thermal energy? The explanation will be found in the following.

Every transformation of a special kind of energy A is accompanied with a fall of potential of that particular kind of energy, including heat. But whilst for the other kinds of energy a transformation and therefore a loss of energy on the part of the kind sinking in potential is connected with the fall of the potential, with heat the case is different. Heat can suffer a fall of potential without sustaining a loss of energy, at least according to the customary mode of estimation. If a weight sinks, it must create perforce kinetic energy, or heat, or some other form of energy. Also, an electrical charge cannot suffer a fall of potential without loss of energy, i. e., without transformation. But heat can pass with a fall of temperature to a body of greater capacity and the same thermal energy still be preserved, so long as we regard every quantity of heat as energy. This it is that gives to heat, besides its property of energy, in many cases the character of a material substance, or quantity.

If we look at the matter in an unprejudiced light, we must ask if there is any scientific sense or purpose in still considering as energy a quantity of heat that can no longer be transformed into mechanical work, (for example, the heat of a closed equably warmed material system). The principle of energy certainly plays in this case a wholly superfluous rôle, which is assigned to it only from habit. To maintain the principle of energy in the face of a knowledge of the dissipation or waste of mechanical energy, in the face of the increase of entropy is equivalent almost to the liberty which Black took when he regarded the heat of liquefaction as still present but latent. It is to be remarked further, that the expressions "energy of the world" and "entropy of the world" are slightly permeated with scholasticism. Energy and entropy are metrical notions. What meaning can there be in applying these notions to a case in which they are not applicable, in which their values are not determinable?

If we could really determine the entropy of the world it would represent a true, absolute measure of time. In this way is best seen the utter tautology of a statement that the entropy of the world increases with the time. Time, and the fact that certain changes take place only in a definite sense, are one and the same thing.

VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.

We are now prepared to answer the question, What are the sources of the principle of energy? All knowledge of nature is derived in the last instance from experience. In this sense they are right who look upon the principle of energy as a result of experience.

Experience teaches that the sense-elements [alpha beta gamma delta ...] into which the world may be decomposed, are subject to change. It tells us further, that certain of these elements are connected with other elements, so that they appear and disappear together; or, that the appearance of the elements of one class is connected with the disappearance of the elements of the other class. We will avoid here the notions of cause and effect because of their obscurity and equivocalness. The result of experience may be expressed as follows: The sensuous elements of the world ([alpha beta gamma delta ...]) show themselves to be interdependent. This interdependence is best represented by some such conception as is in geometry that of the mutual dependence of the sides and angles of a triangle, only much more varied and complex.

As an example, we may take a mass of gas enclosed in a cylinder and possessed of a definite volume ([alpha]), which we change by a pressure () on the piston, at the same time feeling the cylinder with our hand and receiving a sensation of heat ([gamma]). Increase of pressure diminishes the volume and increases the sensation of heat.

The various facts of experience are not in all respects alike. Their common sensuous elements are placed in relief by a process of abstraction and thus impressed upon the memory. In this way the expression is obtained of the features of agreement of extensive groups of facts. The simplest sentence which we can utter is, by the very nature of language, an abstraction of this kind. But account must also be taken of the differences of related facts. Facts may be so nearly related as to contain the same kind of a [alpha beta gamma ...], but the relation be such that the [alpha beta gamma ...] of the one differ from the [alpha beta gamma ...] of the other only by the number of equal parts into which they can be divided. Such being the case, if rules can be given for deducing from one another the numbers which are the measures of these [alpha beta gamma ...], then we possess in such rules the most general expression of a group of facts, as also that expression which corresponds to all its differences. This is the goal of quantitative investigation.

If this goal be reached what we have found is that between the [alpha beta gamma ...] of a group of facts, or better, between the numbers which are their measures, a number of equations exists. The simple fact of change brings it about that the number of these equations must be smaller than the number of the [alpha beta gamma ...]. If the former be smaller by one than the latter, then one portion of the [alpha beta gamma ...] is uniquely determined by the other portion.

The quest of relations of this last kind is the most important function of special experimental research, because we are enabled by it to complete in thought facts that are only partly given. It is self-evident that only experience can ascertain that between the [alpha beta gamma ...] relations exist and of what kind they are. Further, only experience can tell that the relations that exist between the [alpha beta gamma ...] are such that changes of them can be reversed. If this were not the fact all occasion for the enunciation of the principle of energy, as is easily seen, would be wanting. In experience, therefore, is buried the ultimate well-spring of all knowledge of nature, and consequently, in this sense, also the ultimate source of the principle of energy.

But this does not exclude the fact that the principle of energy has also a logical root, as will now be shown. Let us assume on the basis of experience that one group of sensuous elements [alpha beta gamma ...] determines uniquely another group [lambda mu nu ...]. Experience further teaches that changes of [alpha beta gamma ...] can be reversed. It is then a logical consequence of this observation, that every time that [alpha beta gamma ...] assume the same values this is also the case with [lambda mu nu ...]. Or, that purely periodical changes of [alpha beta gamma ...] can produce no permanent changes of [lambda mu nu ...]. If the group [lambda mu nu ...] is a mechanical group, then a perpetual motion is excluded.

It will be said that this is a vicious circle, which we will grant. But psychologically, the situation is essentially different, whether I think simply of the unique determination and reversibility of events, or whether I exclude a perpetual motion. The attention takes in the two cases different directions and diffuses light over different sides of the question, which logically of course are necessarily connected.

Surely that firm, logical setting of the thoughts noticeable in the great inquirers, Stevinus, Galileo, and the rest, which, consciously or instinctively, was supported by a fine feeling for the slightest contradictions, has no other purpose than to limit the bounds of thought and so exempt it from the possibility of error. In this, therefore, the logical root of the principle of excluded perpetual motion is given, namely, in that universal conviction which existed even before the development of mechanics and co-operated in that development.

It is perfectly natural that the principle of excluded perpetual motion should have been first developed in the simple domain of pure mechanics. Towards the transference of that principle into the domain of general physics the idea contributed much that all physical phenomena are mechanical phenomena. But the foregoing discussion shows how little essential this notion is. The issue really involved is the recognition of a general interconnexion of nature. This once established, we see with Carnot that it is indifferent whether the mechanical laws are broken directly or circuitously.

The principle of the excluded perpetual motion is very closely related to the modern principle of energy, but it is not identical with it, for the latter is to be deduced from the former only by means of a definite formal conception. As may be seen from the preceding exposition, the perpetual motion can be excluded without our employing or possessing the notion of work. The modern principle of energy results primarily from a substantial conception of work and of every change of physical condition which by being reversed produces work. The strong need of such a conception, which is by no means necessary, but in a formal sense is very convenient and lucid, is exhibited in the case of J. R. Mayer and Joule. It was before remarked that this conception was suggested to both inquirers by the observation that both the production of heat and the production of mechanical work were connected with an expenditure of substance. Mayer says: "Ex nihilo nil fit," and in another place, "The creation or destruction of a force (work) lies without the province of human activity." In Joule we find this passage: "It is manifestly absurd to suppose that the powers with which God has endowed matter can be destroyed."