PRINCIPLES
§ 1. THE PRINCIPLES OF PHYSICS
Facts conscientiously observed lead by induction to the enunciation of
a certain number of laws or general hypotheses which are the
principles already referred to. These principal hypotheses are, in the
eyes of a physicist, legitimate generalizations, the consequences of
which we shall be able at once to check by the experiments from which
they issue.
Among the principles almost universally adopted until lately figure
prominently those of mechanics--such as the principle of relativity,
and the principle of the equality of action and reaction. We will not
detail nor discuss them here, but later on we shall have an
opportunity of pointing out how recent theories on the phenomena of
electricity have shaken the confidence of physicists in them and have
led certain scholars to doubt their absolute value.
The principle of Lavoisier, or principle of the conservation of mass,
presents itself under two different aspects according to whether mass
is looked upon as the coefficient of the inertia of matter or as the
factor which intervenes in the phenomena of universal attraction, and
particularly in gravitation. We shall see when we treat of these
theories, how we have been led to suppose that inertia depended on
velocity and even on direction. If this conception were exact, the
principle of the invariability of mass would naturally be destroyed.
Considered as a factor of attraction, is mass really indestructible?
A few years ago such a question would have seemed singularly
audacious. And yet the law of Lavoisier is so far from self-evident
that for centuries it escaped the notice of physicists and chemists.
But its great apparent simplicity and its high character of
generality, when enunciated at the end of the eighteenth century,
rapidly gave it such an authority that no one was able to any longer
dispute it unless he desired the reputation of an oddity inclined to
paradoxical ideas.
It is important, however, to remark that, under fallacious
metaphysical appearances, we are in reality using empty words
when we repeat the aphorism, "Nothing can be lost, nothing can be
created," and deduce from it the indestructibility of matter. This
indestructibility, in truth, is an experimental fact, and the
principle depends on experiment. It may even seem, at first sight,
more singular than not that the weight of a bodily system in a given
place, or the quotient of this weight by that of the standard
mass--that is to say, the mass of these bodies--remains invariable,
both when the temperature changes and when chemical reagents cause the
original materials to disappear and to be replaced by new ones. We may
certainly consider that in a chemical phenomenon annihilations and
creations of matter are really produced; but the experimental law
teaches us that there is compensation in certain respects.
The discovery of the radioactive bodies has, in some sort, rendered
popular the speculations of physicists on the phenomena of the
disaggregation of matter. We shall have to seek the exact meaning
which ought to be given to the experiments on the emanation of these
bodies, and to discover whether these experiments really imperil the
law of Lavoisier.
For some years different experimenters have also effected many very
precise measurements of the weight of divers bodies both before and
after chemical reactions between these bodies. Two highly experienced
and cautious physicists, Professors Landolt and Heydweiller, have not
hesitated to announce the sensational result that in certain
circumstances the weight is no longer the same after as before the
reaction. In particular, the weight of a solution of salts of copper
in water is not the exact sum of the joint weights of the salt and the
water. Such experiments are evidently very delicate; they have been
disputed, and they cannot be considered as sufficient for conviction.
It follows nevertheless that it is no longer forbidden to regard the
law of Lavoisier as only an approximate law; according to Sandford and
Ray, this approximation would be about 1/2,400,000. This is also the
result reached by Professor Poynting in experiments regarding the
possible action of temperature on the weight of a body; and if this be
really so, we may reassure ourselves, and from the point of view of
practical application may continue to look upon matter as
indestructible.
The principles of physics, by imposing certain conditions on
phenomena, limit after a fashion the field of the possible. Among
these principles is one which, notwithstanding its importance when
compared with that of universally known principles, is less familiar
to some people. This is the principle of symmetry, more or less
conscious applications of which can, no doubt, be found in various
works and even in the conceptions of Copernican astronomers, but which
was generalized and clearly enunciated for the first time by the late
M. Curie. This illustrious physicist pointed out the advantage of
introducing into the study of physical phenomena the considerations on
symmetry familiar to crystallographers; for a phenomenon to take
place, it is necessary that a certain dissymmetry should previously
exist in the medium in which this phenomenon occurs. A body, for
instance, may be animated with a certain linear velocity or a speed of
rotation; it may be compressed, or twisted; it may be placed in an
electric or in a magnetic field; it may be affected by an electric
current or by one of heat; it may be traversed by a ray of light
either ordinary or polarized rectilineally or circularly, etc.:--in
each case a certain minimum and characteristic dissymmetry is
necessary at every point of the body in question.
This consideration enables us to foresee that certain phenomena which
might be imagined _a priori_ cannot exist. Thus, for instance, it is
impossible that an electric field, a magnitude directed and not
superposable on its image in a mirror perpendicular to its direction,
could be created at right angles to the plane of symmetry of the
medium; while it would be possible to create a magnetic field under
the same conditions.
This consideration thus leads us to the discovery of new phenomena;
but it must be understood that it cannot of itself give us absolutely
precise notions as to the nature of these phenomena, nor disclose
their order of magnitude.
§ 2. THE PRINCIPLE OF THE CONSERVATION OF ENERGY
Dominating not physics alone, but nearly every other science, the
principle of the conservation of energy is justly considered as the
grandest conquest of contemporary thought. It shows us in a powerful
light the most diverse questions; it introduces order into the most
varied studies; it leads to a clear and coherent interpretation of
phenomena which, without it, appear to have no connexion with each
other; and it supplies precise and exact numerical relations between
the magnitudes which enter into these phenomena.
The boldest minds have an instinctive confidence in it, and it is the
principle which has most stoutly resisted that assault which the
daring of a few theorists has lately directed to the overthrow of the
general principles of physics. At every new discovery, the first
thought of physicists is to find out how it accords with the principle
of the conservation of energy. The application of the principle,
moreover, never fails to give valuable hints on the new phenomenon,
and often even suggests a complementary discovery. Up till now it
seems never to have received a check, even the extraordinary
properties of radium not seriously contradicting it; also the general
form in which it is enunciated gives it such a suppleness that it is
no doubt very difficult to overthrow.
I do not claim to set forth here the complete history of this
principle, but I will endeavour to show with what pains it was born,
how it was kept back in its early days and then obstructed in its
development by the unfavourable conditions of the surroundings in
which it appeared. It first of all came, in fact, to oppose itself to
the reigning theories; but, little by little, it acted on these
theories, and they were modified under its pressure; then, in their
turn, these theories reacted on it and changed its primitive form.
It had to be made less wide in order to fit into the classic frame,
and was absorbed by mechanics; and if it thus became less general, it
gained in precision what it lost in extent. When once definitely
admitted and classed, as it were, in the official domain of science,
it endeavoured to burst its bonds and return to a more independent and
larger life. The history of this principle is similar to that of all
evolutions.
It is well known that the conservation of energy was, at first,
regarded from the point of view of the reciprocal transformations
between heat and work, and that the principle received its first clear
enunciation in the particular case of the principle of equivalence. It
is, therefore, rightly considered that the scholars who were the first
to doubt the material nature of caloric were the precursors of R.
Mayer; their ideas, however, were the same as those of the celebrated
German doctor, for they sought especially to demonstrate that heat was
a mode of motion.
Without going back to early and isolated attempts like those of Daniel
Bernoulli, who, in his hydrodynamics, propounded the basis of the
kinetic theory of gases, or the researches of Boyle on friction, we
may recall, to show how it was propounded in former times, a rather
forgotten page of the _Mémoire sur la Chaleur_, published in 1780 by
Lavoisier and Laplace: "Other physicists," they wrote, after setting
out the theory of caloric, "think that heat is nothing but the result
of the insensible vibrations of matter.... In the system we are now
examining, heat is the _vis viva_ resulting from the insensible
movements of the molecules of a body; it is the sum of the products of
the mass of each molecule by the square of its velocity.... We shall
not decide between the two preceding hypotheses; several phenomena
seem to support the last mentioned--for instance, that of the heat
produced by the friction of two solid bodies. But there are others
which are more simply explained by the first, and perhaps they both
operate at once." Most of the physicists of that period, however, did
not share the prudent doubts of Lavoisier and Laplace. They admitted,
without hesitation, the first hypothesis; and, four years after the
appearance of the _Mémoire sur la Chaleur_, Sigaud de Lafond, a
professor of physics of great reputation, wrote: "Pure Fire, free from
all state of combination, seems to be an assembly of particles of a
simple, homogeneous, and absolutely unalterable matter, and all the
properties of this element indicate that these particles are
infinitely small and free, that they have no sensible cohesion, and
that they are moved in every possible direction by a continual and
rapid motion which is essential to them.... The extreme tenacity and
the surprising mobility of its molecules are manifestly shown by the
ease with which it penetrates into the most compact bodies and by its
tendency to put itself in equilibrium throughout all bodies near to
it."
It must be acknowledged, however, that the idea of Lavoisier and
Laplace was rather vague and even inexact on one important point. They
admitted it to be evident that "all variations of heat, whether real
or apparent, undergone by a bodily system when changing its state, are
produced in inverse order when the system passes back to its original
state." This phrase is the very denial of equivalence where these
changes of state are accompanied by external work.
Laplace, moreover, himself became later a very convinced partisan of
the hypothesis of the material nature of caloric, and his immense
authority, so fortunate in other respects for the development of
science, was certainly in this case the cause of the retardation of
progress.
The names of Young, Rumford, Davy, are often quoted among those
physicists who, at the commencement of the nineteenth century, caught
sight of the new truths as to the nature of heat. To these names is
very properly added that of Sadi Carnot. A note found among his papers
unquestionably proves that, before 1830, ideas had occurred to him
from which it resulted that in producing work an equivalent amount of
heat was destroyed. But the year 1842 is particularly memorable in the
history of science as the year in which Jules Robert Mayer succeeded,
by an entirely personal effort, in really enunciating the principle of
the conservation of energy. Chemists recall with just pride that the
_Remarques sur les forces de la nature animée_, contemptuously
rejected by all the journals of physics, were received and published
in the _Annalen_ of Liebig. We ought never to forget this example,
which shows with what difficulty a new idea contrary to the classic
theories of the period succeeds in coming to the front; but
extenuating circumstances may be urged on behalf of the physicists.
Robert Mayer had a rather insufficient mathematical education, and his
Memoirs, the _Remarques_, as well as the ulterior publications,
_Mémoire sur le mouvement organique et la nutrition_ and the
_Matériaux pour la dynamique du ciel_, contain, side by side with very
profound ideas, evident errors in mechanics. Thus it often happens
that discoveries put forward in a somewhat vague manner by adventurous
minds not overburdened by the heavy baggage of scientific erudition,
who audaciously press forward in advance of their time, fall into
quite intelligible oblivion until rediscovered, clarified, and put
into shape by slower but surer seekers. This was the case with the
ideas of Mayer. They were not understood at first sight, not only on
account of their originality, but also because they were couched in
incorrect language.
Mayer was, however, endowed with a singular strength of thought; he
expressed in a rather confused manner a principle which, for him, had
a generality greater than mechanics itself, and so his discovery was
in advance not only of his own time but of half the century. He may
justly be considered the founder of modern energetics.
Freed from the obscurities which prevented its being clearly
perceived, his idea stands out to-day in all its imposing simplicity.
Yet it must be acknowledged that if it was somewhat denaturalised by
those who endeavoured to adapt it to the theories of mechanics, and if
it at first lost its sublime stamp of generality, it thus became
firmly fixed and consolidated on a more stable basis.
The efforts of Helmholtz, Clausius, and Lord Kelvin to introduce the
principle of the conservation of energy into mechanics, were far from
useless. These illustrious physicists succeeded in giving a more
precise form to its numerous applications; and their attempts thus
contributed, by reaction, to give a fresh impulse to mechanics, and
allowed it to be linked to a more general order of facts. If
energetics has not been able to be included in mechanics, it seems
indeed that the attempt to include mechanics in energetics was not in
vain.
In the middle of the last century, the explanation of all natural
phenomena seemed more and more referable to the case of central
forces. Everywhere it was thought that reciprocal actions between
material points could be perceived, these points being attracted or
repelled by each other with an intensity depending only on their
distance or their mass. If, to a system thus composed, the laws of the
classical mechanics are applied, it is shown that half the sum of the
product of the masses by the square of the velocities, to which is
added the work which might be accomplished by the forces to which the
system would be subject if it returned from its actual to its initial
position, is a sum constant in quantity.
This sum, which is the mechanical energy of the system, is therefore
an invariable quantity in all the states to which it may be brought by
the interaction of its various parts, and the word energy well
expresses a capital property of this quantity. For if two systems are
connected in such a way that any change produced in the one
necessarily brings about a change in the other, there can be no
variation in the characteristic quantity of the second except so far
as the characteristic quantity of the first itself varies--on
condition, of course, that the connexions are made in such a manner as
to introduce no new force. It will thus be seen that this quantity
well expresses the capacity possessed by a system for modifying the
state of a neighbouring system to which we may suppose it connected.
Now this theorem of pure mechanics was found wanting every time
friction took place--that is to say, in all really observable cases.
The more perceptible the friction, the more considerable the
difference; but, in addition, a new phenomenon always appeared and
heat was produced. By experiments which are now classic, it became
established that the quantity of heat thus created independently of
the nature of the bodies is always (provided no other phenomena
intervene) proportional to the energy which has disappeared.
Reciprocally, also, heat may disappear, and we always find a constant
relation between the quantities of heat and work which mutually
replace each other.
It is quite clear that such experiments do not prove that heat is
work. We might just as well say that work is heat. It is making a
gratuitous hypothesis to admit this reduction of heat to mechanism;
but this hypothesis was so seductive, and so much in conformity with
the desire of nearly all physicists to arrive at some sort of unity in
nature, that they made it with eagerness and became unreservedly
convinced that heat was an active internal force.
Their error was not in admitting this hypothesis; it was a legitimate
one since it has proved very fruitful. But some of them committed the
fault of forgetting that it was an hypothesis, and considered it a
demonstrated truth. Moreover, they were thus brought to see in
phenomena nothing but these two particular forms of energy which in
their minds were easily identified with each other.
From the outset, however, it became manifest that the principle is
applicable to cases where heat plays only a parasitical part. There
were thus discovered, by translating the principle of equivalence,
numerical relations between the magnitudes of electricity, for
instance, and the magnitudes of mechanics. Heat was a sort of variable
intermediary convenient for calculation, but introduced in a
roundabout way and destined to disappear in the final result.
Verdet, who, in lectures which have rightly remained celebrated,
defined with remarkable clearness the new theories, said, in 1862:
"Electrical phenomena are always accompanied by calorific
manifestations, of which the study belongs to the mechanical theory of
heat. This study, moreover, will not only have the effect of making
known to us interesting facts in electricity, but will throw some
light on the phenomena of electricity themselves."
The eminent professor was thus expressing the general opinion of his
contemporaries, but he certainly seemed to have felt in advance that
the new theory was about to penetrate more deeply into the inmost
nature of things. Three years previously, Rankine also had put forth
some very remarkable ideas the full meaning of which was not at first
well understood. He it was who comprehended the utility of employing a
more inclusive term, and invented the phrase energetics. He also
endeavoured to create a new doctrine of which rational mechanics
should be only a particular case; and he showed that it was possible
to abandon the ideas of atoms and central forces, and to construct a
more general system by substituting for the ordinary consideration of
forces that of the energy which exists in all bodies, partly in an
actual, partly in a potential state.
By giving more precision to the conceptions of Rankine, the physicists
of the end of the nineteenth century were brought to consider that in
all physical phenomena there occur apparitions and disappearances
which are balanced by various energies. It is natural, however, to
suppose that these equivalent apparitions and disappearances
correspond to transformations and not to simultaneous creations and
destructions. We thus represent energy to ourselves as taking
different forms--mechanical, electrical, calorific, and chemical--
capable of changing one into the other, but in such a way that the
quantitative value always remains the same. In like manner a bank
draft may be represented by notes, gold, silver, or bullion. The
earliest known form of energy, _i.e._ work, will serve as the standard
as gold serves as the monetary standard, and energy in all its forms
will be estimated by the corresponding work. In each particular case
we can strictly define and measure, by the correct application of the
principle of the conservation of energy, the quantity of energy
evolved under a given form.
We can thus arrange a machine comprising a body capable of evolving
this energy; then we can force all the organs of this machine to
complete an entirely closed cycle, with the exception of the body
itself, which, however, has to return to such a state that all the
variables from which this state depends resume their initial values
except the particular variable to which the evolution of the energy
under consideration is linked. The difference between the work thus
accomplished and that which would have been obtained if this variable
also had returned to its original value, is the measure of the energy
evolved.
In the same way that, in the minds of mechanicians, all forces of
whatever origin, which are capable of compounding with each other and
of balancing each other, belong to the same category of beings, so for
many physicists energy is a sort of entity which we find under various
aspects. There thus exists for them a world, which comes in some way
to superpose itself upon the world of matter--that is to say, the
world of energy, dominated in its turn by a fundamental law similar to
that of Lavoisier.[5] This conception, as we have already seen, passes
the limit of experience; but others go further still. Absorbed in the
contemplation of this new world, they succeed in persuading themselves
that the old world of matter has no real existence and that energy is
sufficient by itself to give us a complete comprehension of the
Universe and of all the phenomena produced in it. They point out that
all our sensations correspond to changes of energy, and that
everything apparent to our senses is, in truth, energy. The famous
experiment of the blows with a stick by which it was demonstrated to a
sceptical philosopher that an outer world existed, only proves, in
reality, the existence of energy, and not that of matter. The stick in
itself is inoffensive, as Professor Ostwald remarks, and it is its
_vis viva_, its kinetic energy, which is painful to us; while if we
possessed a speed equal to its own, moving in the same direction, it
would no longer exist so far as our sense of touch is concerned.
[Footnote 5: "Nothing is created; nothing is lost"--ED.]
On this hypothesis, matter would only be the capacity for kinetic
energy, its pretended impenetrability energy of volume, and its weight
energy of position in the particular form which presents itself in
universal gravitation; nay, space itself would only be known to us by
the expenditure of energy necessary to penetrate it. Thus in all
physical phenomena we should only have to regard the quantities of
energy brought into play, and all the equations which link the
phenomena to one another would have no meaning but when they apply to
exchanges of energy. For energy alone can be common to all phenomena.
This extreme manner of regarding things is seductive by its
originality, but appears somewhat insufficient if, after enunciating
generalities, we look more closely into the question. From the
philosophical point of view it may, moreover, seem difficult not to
conclude, from the qualities which reveal, if you will, the varied
forms of energy, that there exists a substance possessing these
qualities. This energy, which resides in one region, and which
transports itself from one spot to another, forcibly brings to mind,
whatever view we may take of it, the idea of matter.
Helmholtz endeavoured to construct a mechanics based on the idea of
energy and its conservation, but he had to invoke a second law, the
principle of least action. If he thus succeeded in dispensing with the
hypothesis of atoms, and in showing that the new mechanics gave us to
understand the impossibility of certain movements which, according to
the old, ought to have been but never were experimentally produced, he
was only able to do so because the principle of least action necessary
for his theory became evident in the case of those irreversible
phenomena which alone really exist in Nature. The energetists have
thus not succeeded in forming a thoroughly sound system, but their
efforts have at all events been partly successful. Most physicists are
of their opinion, that kinetic energy is only a particular variety of
energy to which we have no right to wish to connect all its other
forms.
If these forms showed themselves to be innumerable throughout the
Universe, the principle of the conservation of energy would, in fact,
lose a great part of its importance. Every time that a certain
quantity of energy seemed to appear or disappear, it would always be
permissible to suppose that an equivalent quantity had appeared or
disappeared somewhere else under a new form; and thus the principle
would in a way vanish. But the known forms of energy are fairly
restricted in number, and the necessity of recognising new ones seldom
makes itself felt. We shall see, however, that to explain, for
instance, the paradoxical properties of radium and to re-establish
concord between these properties and the principle of the conservation
of energy, certain physicists have recourse to the hypothesis that
radium borrows an unknown energy from the medium in which it is
plunged. This hypothesis, however, is in no way necessary; and in a
few other rare cases in which similar hypotheses have had to be set
up, experiment has always in the long run enabled us to discover some
phenomenon which had escaped the first observers and which corresponds
exactly to the variation of energy first made evident.
One difficulty, however, arises from the fact that the principle ought
only to be applied to an isolated system. Whether we imagine actions
at a distance or believe in intermediate media, we must always
recognise that there exist no bodies in the world incapable of acting
on each other, and we can never affirm that some modification in the
energy of a given place may not have its echo in some unknown spot
afar off. This difficulty may sometimes render the value of the
principle rather illusory.
Similarly, it behoves us not to receive without a certain distrust the
extension by certain philosophers to the whole Universe, of a property
demonstrated for those restricted systems which observation can alone
reach. We know nothing of the Universe as a whole, and every
generalization of this kind outruns in a singular fashion the limit of
experiment.
Even reduced to the most modest proportions, the principle of the
conservation of energy retains, nevertheless, a paramount importance;
and it still preserves, if you will, a high philosophical value. M.J.
Perrin justly points out that it gives us a form under which we are
experimentally able to grasp causality, and that it teaches us that a
result has to be purchased at the cost of a determined effort.
We can, in fact, with M. Perrin and M. Langevin, represent this in a
way which puts this characteristic in evidence by enunciating it as
follows: "If at the cost of a change C we can obtain a change K, there
will never be acquired at the same cost, whatever the mechanism
employed, first the change K and in addition some other change, unless
this latter be one that is otherwise known to cost nothing to produce
or to destroy." If, for instance, the fall of a weight can be
accompanied, without anything else being produced, by another
transformation--the melting of a certain mass of ice, for example--it
will be impossible, no matter how you set about it or whatever the
mechanism used, to associate this same transformation with the melting
of another weight of ice.
We can thus, in the transformation in question, obtain an appropriate
number which will sum up that which may be expected from the external
effect, and can give, so to speak, the price at which this
transformation is bought, measure its invariable value by a common
measure (for instance, the melting of the ice), and, without any
ambiguity, define the energy lost during the transformation as
proportional to the mass of ice which can be associated with it. This
measure is, moreover, independent of the particular phenomenon taken
as the common measure.
§ 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.
In whatever way you look at it, the principle of Carnot furnishes, in
the immense majority of cases, a very sure guide in which physicists
continue to have the most entire confidence.
§ 4. THERMODYNAMICS
To apply the two fundamental principles of thermodynamics, various
methods may be employed, equivalent in the main, but presenting as the
cases vary a greater or less convenience.
In recording, with the aid of the two quantities, energy and entropy,
the relations which translate analytically the two principles, we
obtain two relations between the coefficients which occur in a given
phenomenon; but it may be easier and also more suggestive to employ
various functions of these quantities. In a memoir, of which some
extracts appeared as early as 1869, a modest scholar, M. Massieu,
indicated in particular a remarkable function which he termed a
characteristic function, and by the employment of which calculations
are simplified in certain cases.
In the same way J.W. Gibbs, in 1875 and 1878, then Helmholtz in 1882,
and, in France, M. Duhem, from the year 1886 onward, have published
works, at first ill understood, of which the renown was, however,
considerable in the sequel, and in which they made use of analogous
functions under the names of available energy, free energy, or
internal thermodynamic potential. The magnitude thus designated,
attaching, as a consequence of the two principles, to all states of
the system, is perfectly determined when the temperature and other
normal variables are known. It allows us, by calculations often very
easy, to fix the conditions necessary and sufficient for the
maintenance of the system in equilibrium by foreign bodies taken at
the same temperature as itself.
One may hope to constitute in this way, as M. Duhem in a long and
remarkable series of operations has specially endeavoured to do, a
sort of general mechanics which will enable questions of statics to be
treated with accuracy, and all the conditions of equilibrium of the
system, including the calorific properties, to be determined. Thus,
ordinary statics teaches us that a liquid with its vapour on the top
forms a system in equilibrium, if we apply to the two fluids a
pressure depending on temperature alone. Thermodynamics will furnish
us, in addition, with the expression of the heat of vaporization and
of, the specific heats of the two saturated fluids.
This new study has given us also most valuable information on
compressible fluids and on the theory of elastic equilibrium. Added to
certain hypotheses on electric or magnetic phenomena, it gives a
coherent whole from which can be deduced the conditions of electric or
magnetic equilibrium; and it illuminates with a brilliant light the
calorific laws of electrolytic phenomena.
But the most indisputable triumph of this thermodynamic statics is the
discovery of the laws which regulate the changes of physical state or
of chemical constitution. J.W. Gibbs was the author of this immense
progress. His memoir, now celebrated, on "the equilibrium of
heterogeneous substances," concealed in 1876 in a review at that time
of limited circulation, and rather heavy to read, seemed only to
contain algebraic theorems applicable with difficulty to reality. It
is known that Helmholtz independently succeeded, a few years later, in
introducing thermodynamics into the domain of chemistry by his
conception of the division of energy into free and into bound energy:
the first, capable of undergoing all transformations, and particularly
of transforming itself into external action; the second, on the other
hand, bound, and only manifesting itself by giving out heat. When we
measure chemical energy, we ordinarily let it fall wholly into the
calorific form; but, in reality, it itself includes both parts, and it
is the variation of the free energy and not that of the total energy
measured by the integral disengagement of heat, the sign of which
determines the direction in which the reactions are effected.
But if the principle thus enunciated by Helmholtz as a consequence of
the laws of thermodynamics is at bottom identical with that discovered
by Gibbs, it is more difficult of application and is presented under a
more mysterious aspect. It was not until M. Van der Waals exhumed the
memoir of Gibbs, when numerous physicists or chemists, most of them
Dutch--Professor Van t'Hoff, Bakhius Roozeboom, and others--utilized
the rules set forth in this memoir for the discussion of the most
complicated chemical reactions, that the extent of the new laws was
fully understood.
The chief rule of Gibbs is the one so celebrated at the present day
under the name of the Phase Law. We know that by phases are designated
the homogeneous substances into which a system is divided; thus
carbonate of lime, lime, and carbonic acid gas are the three phases of
a system which comprises Iceland spar partially dissociated into lime
and carbonic acid gas. The number of phases added to the number of
independent components--that is to say, bodies whose mass is left
arbitrary by the chemical formulas of the substances entering into the
reaction--fixes the general form of the law of equilibrium of the
system; that is to say, the number of quantities which, by their
variations (temperature and pressure), would be of a nature to modify
its equilibrium by modifying the constitution of the phases.
Several authors, M. Raveau in particular, have indeed given very
simple demonstrations of this law which are not based on
thermodynamics; but thermodynamics, which led to its discovery,
continues to give it its true scope. Moreover, it would not suffice
merely to determine quantitatively those laws of which it makes known
the general form. We must, if we wish to penetrate deeper into
details, particularize the hypothesis, and admit, for instance, with
Gibbs that we are dealing with perfect gases; while, thanks to
thermodynamics, we can constitute a complete theory of dissociation
which leads to formulas in complete accord with the numerical results
of the experiment. We can thus follow closely all questions concerning
the displacements of the equilibrium, and find a relation of the first
importance between the masses of the bodies which react in order to
constitute a system in equilibrium.
The statics thus constructed constitutes at the present day an
important edifice to be henceforth classed amongst historical
monuments. Some theorists even wish to go a step beyond. They have
attempted to begin by the same means a more complete study of those
systems whose state changes from one moment to another. This is,
moreover, a study which is necessary to complete satisfactorily the
study of equilibrium itself; for without it grave doubts would exist
as to the conditions of stability, and it alone can give their true
meaning to questions relating to displacements of equilibrium.
The problems with which we are thus confronted are singularly
difficult. M. Duhem has given us many excellent examples of the
fecundity of the method; but if thermodynamic statics may be
considered definitely founded, it cannot be said that the general
dynamics of systems, considered as the study of thermal movements and
variations, are yet as solidly established.
§ 5. ATOMISM
It may appear singularly paradoxical that, in a chapter devoted to
general views on the principles of physics, a few words should be
introduced on the atomic theories of matter.
Very often, in fact, what is called the physics of principles is set
in opposition to the hypotheses on the constitution of matter,
particularly to atomic theories. I have already said that, abandoning
the investigation of the unfathomable mystery of the constitution of
the Universe, some physicists think they may find, in certain general
principles, sufficient guides to conduct them across the physical
world. But I have also said, in examining the history of those
principles, that if they are to-day considered experimental truths,
independent of all theories relating to matter, they have, in fact,
nearly all been discovered by scholars who relied on molecular
hypotheses: and the question suggests itself whether this is mere
chance, or whether this chance may not be ordained by higher reasons.
In a very profound work which appeared a few years ago, entitled
_Essai critique sur l'hypothese des atomes_, M. Hannequin, a
philosopher who is also an erudite scholar, examined the part taken by
atomism in the history of science. He notes that atomism and science
were born, in Greece, of the same problem, and that in modern times
the revival of the one was closely connected with that of the other.
He shows, too, by very close analysis, that the atomic hypothesis is
essential to the optics of Fresnel and of Cauchy; that it penetrates
into the study of heat; and that, in its general features, it presided
at the birth of modern chemistry and is linked with all its progress.
He concludes that it is, in a manner, the soul of our knowledge of
Nature, and that contemporary theories are on this point in accord
with history: for these theories consecrate the preponderance of this
hypothesis in the domain of science.
If M. Hannequin had not been prematurely cut off in the full expansion
of his vigorous talent, he might have added another chapter to his
excellent book. He would have witnessed a prodigious budding of
atomistic ideas, accompanied, it is true, by wide modifications in the
manner in which the atom is to be regarded, since the most recent
theories make material atoms into centres constituted of atoms of
electricity. On the other hand, he would have found in the bursting
forth of these new doctrines one more proof in support of his idea
that science is indissolubly bound to atomism.
From the philosophical point of view, M. Hannequin, examining the
reasons which may have called these links into being, arrives at the
idea that they necessarily proceed from the constitution of our
knowledge, or, perhaps, from that of Nature itself. Moreover, this
origin, double in appearance, is single at bottom. Our minds could
not, in fact, detach and come out of themselves to grasp reality and
the absolute in Nature. According to the idea of Descartes, it is the
destiny of our minds only to take hold of and to understand that which
proceeds from them.
Thus atomism, which is, perhaps, only an appearance containing even
some contradictions, is yet a well-founded appearance, since it
conforms to the laws of our minds; and this hypothesis is, in a way,
necessary.
We may dispute the conclusions of M. Hannequin, but no one will refuse
to recognise, as he does, that atomic theories occupy a preponderating
part in the doctrines of physics; and the position which they have
thus conquered gives them, in a way, the right of saying that they
rest on a real principle. It is in order to recognise this right that
several physicists--M. Langevin, for example--ask that atoms be
promoted from the rank of hypotheses to that of principles. By this
they mean that the atomistic ideas forced upon us by an almost
obligatory induction based on very exact experiments, enable us to
co-ordinate a considerable amount of facts, to construct a very general
synthesis, and to foresee a great number of phenomena.
It is of moment, moreover, to thoroughly understand that atomism does
not necessarily set up the hypothesis of centres of attraction acting
at a distance, and it must not be confused with molecular physics,
which has, on the other hand, undergone very serious checks. The
molecular physics greatly in favour some fifty years ago leads to such
complex representations and to solutions often so undetermined, that
the most courageous are wearied with upholding it and it has fallen
into some discredit. It rested on the fundamental principles of
mechanics applied to molecular actions; and that was, no doubt, an
extension legitimate enough, since mechanics is itself only an
experimental science, and its principles, established for the
movements of matter taken as a whole, should not be applied outside
the domain which belongs to them. Atomism, in fact, tends more and
more, in modern theories, to imitate the principle of the conservation
of energy or that of entropy, to disengage itself from the artificial
bonds which attached it to mechanics, and to put itself forward as an
independent principle.
Atomistic ideas also have undergone evolution, and this slow evolution
has been considerably quickened under the influence of modern
discoveries. These reach back to the most remote antiquity, and to
follow their development we should have to write the history of human
thought which they have always accompanied since the time of
Leucippus, Democritus, Epicurus, and Lucretius. The first observers
who noticed that the volume of a body could be diminished by
compression or cold, or augmented by heat, and who saw a soluble solid
body mix completely with the water which dissolved it, must have been
compelled to suppose that matter was not dispersed continuously
throughout the space it seemed to occupy. They were thus brought to
consider it discontinuous, and to admit that a substance having the
same composition and the same properties in all its parts--in a word,
perfectly homogeneous--ceases to present this homogeneity when
considered within a sufficiently small volume.
Modern experimenters have succeeded by direct experiments in placing
in evidence this heterogeneous character of matter when taken in small
mass. Thus, for example, the superficial tension, which is constant
for the same liquid at a given temperature, no longer has the same
value when the thickness of the layer of liquid becomes extremely
small. Newton noticed even in his time that a dark zone is seen to
form on a soap bubble at the moment when it becomes so thin that it
must burst. Professor Reinold and Sir Arthur Rücker have shown that
this zone is no longer exactly spherical; and from this we must
conclude that the superficial tension, constant for all thicknesses
above a certain limit, commences to vary when the thickness falls
below a critical value, which these authors estimate, on optical
grounds, at about fifty millionths of a millimetre.
From experiments on capillarity, Prof. Quincke has obtained similar
results with regard to layers of solids. But it is not only capillary
properties which allow this characteristic to be revealed. All the
properties of a body are modified when taken in small mass; M. Meslin
proves this in a very ingenious way as regards optical properties, and
Mr Vincent in respect of electric conductivity. M. Houllevigue, who,
in a chapter of his excellent work, _Du Laboratoire à l'Usine_, has
very clearly set forth the most interesting considerations on atomic
hypotheses, has recently demonstrated that copper and silver cease to
combine with iodine as soon as they are present in a thickness of less
than thirty millionths of a millimetre. It is this same dimension
likewise that is possessed, according to M. Wiener, by the smallest
thicknesses it is possible to deposit on glass. These layers are so
thin that they cannot be perceived, but their presence is revealed by
a change in the properties of the light reflected by them.
Thus, below fifty to thirty millionths of a millimetre the properties
of matter depend on its thickness. There are then, no doubt, only a
few molecules to be met with, and it may be concluded, in consequence,
that the discontinuous elements of bodies--that is, the molecules--
have linear dimensions of the order of magnitude of the millionth of a
millimetre. Considerations regarding more complex phenomena, for
instance the phenomena of electricity by contact, and also the kinetic
theory of gases, bring us to the same conclusion.
The idea of the discontinuity of matter forces itself upon us for many
other reasons. All modern chemistry is founded on this principle; and
laws like the law of multiple proportions, introduce an evident
discontinuity to which we find analogies in the law of electrolysis.
The elements of bodies we are thus brought to regard might, as regards
solids at all events, be considered as immobile; but this immobility
could not explain the phenomena of heat, and, as it is entirely
inadmissible for gases, it seems very improbable it can absolutely
occur in any state. We are thus led to suppose that these elements are
animated by very complicated movements, each one proceeding in closed
trajectories in which the least variations of temperature or pressure
cause modifications.
The atomistic hypothesis shows itself remarkably fecund in the study
of phenomena produced in gases, and here the mutual independence of
the particles renders the question relatively more simple and,
perhaps, allows the principles of mechanics to be more certainly
extended to the movements of molecules.
The kinetic theory of gases can point to unquestioned successes; and
the idea of Daniel Bernouilli, who, as early as 1738, considered a
gaseous mass to be formed of a considerable number of molecules
animated by rapid movements of translation, has been put into a form
precise enough for mathematical analysis, and we have thus found
ourselves in a position to construct a really solid foundation. It
will be at once conceived, on this hypothesis, that pressure is the
resultant of the shocks of the molecules against the walls of the
containing vessel, and we at once come to the demonstration that the
law of Mariotte is a natural consequence of this origin of pressure;
since, if the volume occupied by a certain number of molecules is
doubled, the number of shocks per second on each square centimetre of
the walls becomes half as much. But if we attempt to carry this
further, we find ourselves in presence of a serious difficulty. It is
impossible to mentally follow every one of the many individual
molecules which compose even a very limited mass of gas. The path
followed by this molecule may be every instant modified by the chance
of running against another, or by a shock which may make it rebound in
another direction.
The difficulty would be insoluble if chance had not laws of its own.
It was Maxwell who first thought of introducing into the kinetic
theory the calculation of probabilities. Willard Gibbs and Boltzmann
later on developed this idea, and have founded a statistical method
which does not, perhaps, give absolute certainty, but which is
certainly most interesting and curious. Molecules are grouped in such
a way that those belonging to the same group may be considered as
having the same state of movement; then an examination is made of the
number of molecules in each group, and what are the changes in this
number from one moment to another. It is thus often possible to
determine the part which the different groups have in the total
properties of the system and in the phenomena which may occur.
Such a method, analogous to the one employed by statisticians for
following the social phenomena in a population, is all the more
legitimate the greater the number of individuals counted in the
averages; now, the number of molecules contained in a limited space--
for example, in a centimetre cube taken in normal conditions--is such
that no population could ever attain so high a figure. All
considerations, those we have indicated as well as others which might
be invoked (for example, the recent researches of M. Spring on the
limit of visibility of fluorescence), give this result:--that there
are, in this space, some twenty thousand millions of molecules. Each
of these must receive in the space of a millimetre about ten thousand
shocks, and be ten thousand times thrust out of its course. The free
path of a molecule is then very small, but it can be singularly
augmented by diminishing the number of them. Tait and Dewar have
calculated that, in a good modern vacuum, the length of the free path
of the remaining molecules not taken away by the air-pump easily
reaches a few centimetres.
By developing this theory, we come to consider that, for a given
temperature, every molecule (and even every individual particle, atom,
or ion) which takes part in the movement has, on the average, the same
kinetic energy in every body, and that this energy is proportional to
the absolute temperature; so that it is represented by this
temperature multiplied by a constant quantity which is a universal
constant.
This result is not an hypothesis but a very great probability. This
probability increases when it is noted that the same value for the
constant is met with in the study of very varied phenomena; for
example, in certain theories on radiation. Knowing the mass and energy
of a molecule, it is easy to calculate its speed; and we find that the
average speed is about 400 metres per second for carbonic anhydride,
500 for nitrogen, and 1850 for hydrogen at 0° C. and at ordinary
pressure. I shall have occasion, later on, to speak of much more
considerable speeds than these as animating other particles.
The kinetic theory has permitted the diffusion of gases to be
explained, and the divers circumstances of the phenomenon to be
calculated. It has allowed us to show, as M. Brillouin has done, that
the coefficient of diffusion of two gases does not depend on the
proportion of the gases in the mixture; it gives a very striking image
of the phenomena of viscosity and conductivity; and it leads us to
think that the coefficients of friction and of conductivity are
independent of the density; while all these previsions have been
verified by experiment. It has also invaded optics; and by relying on
the principle of Doppler, Professor Michelson has succeeded in
obtaining from it an explanation of the length presented by the
spectral rays of even the most rarefied gases.
But however interesting are these results, they would not have
sufficed to overcome the repugnance of certain physicists for
speculations which, an imposing mathematical baggage notwithstanding,
seemed to them too hypothetical. The theory, moreover, stopped at the
molecule, and appeared to suggest no idea which could lead to the
discovery of the key to the phenomena where molecules exercise a
mutual influence on each other. The kinetic hypothesis, therefore,
remained in some disfavour with a great number of persons,
particularly in France, until the last few years, when all the recent
discoveries of the conductivity of gases and of the new radiations
came to procure for it a new and luxuriant efflorescence. It may be
said that the atomistic synthesis, but yesterday so decried, is to-day
triumphant.
The elements which enter into the earlier kinetic theory, and which,
to avoid confusion, should be always designated by the name of
molecules, were not, truth to say, in the eyes of the chemists, the
final term of the divisibility of matter. It is well known that, to
them, except in certain particular bodies like the vapour of mercury
and argon, the molecule comprises several atoms, and that, in compound
bodies, the number of these atoms may even be fairly considerable. But
physicists rarely needed to have recourse to the consideration of
these atoms. They spoke of them to explain certain particularities of
the propagation of sound, and to enunciate laws relating to specific
heats; but, in general, they stopped at the consideration of the
molecule.
The present theories carry the division much further. I shall not
dwell now on these theories, since, in order to thoroughly understand
them, many other facts must be examined. But to avoid all confusion,
it remains understood that, contrary, no doubt, to etymology, but in
conformity with present custom, I shall continue in what follows to
call atoms those particles of matter which have till now been spoken
of; these atoms being themselves, according to modern views,
singularly complex edifices formed of elements, of which we shall have
occasion to indicate the nature later.