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The New Physics and Its Evolution by Lucien Poincare

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The statics of solids under high pressure is as yet, therefore, hardly
drafted, but it seems to promise results which will not be identical
with those obtained for the statics of fluids, though it will present
at least an equal interest.


Sec. 4. THE DEFORMATIONS OF SOLIDS

If the mechanical properties of the bodies intermediate between solids
and liquids have only lately been the object of systematic studies,
admittedly solid substances have been studied for a long time. Yet,
notwithstanding the abundance of researches published on elasticity by
theorists and experimenters, numerous questions with regard to them
still remain in suspense.

We only propose to briefly indicate here a few problems recently
examined, without going into the details of questions which belong
more to the domain of mechanics than to that of pure physics.

The deformations produced in solid bodies by increasing efforts
arrange themselves in two distinct periods. If the efforts are weak,
the deformations produced are also very weak and disappear when the
effort ceases. They are then termed elastic. If the efforts exceed a
certain value, a part only of these deformations disappear, and a part
are permanent.

The purity of the note emitted by a sound has been often invoked as a
proof of the perfect isochronism of the oscillation, and,
consequently, as a demonstration _a posteriori_ of the correctness of
the early law of Hoocke governing elastic deformations. This law has,
however, during some years been frequently disputed. Certain
mechanicians or physicists freely admit it to be incorrect, especially
as regards extremely weak deformations. According to a theory in some
favour, especially in Germany, i.e. the theory of Bach, the law which
connects the elastic deformations with the efforts would be an
exponential one. Recent experiments by Professors Kohlrausch and
Gruncisen, executed under varied and precise conditions on brass, cast
iron, slate, and wrought iron, do not appear to confirm Bach's law.
Nothing, in point of fact, authorises the rejection of the law of
Hoocke, which presents itself as the most natural and most simple
approximation to reality.

The phenomena of permanent deformation are very complex, and it
certainly seems that they cannot be explained by the older theories
which insisted that the molecules only acted along the straight line
which joined their centres. It becomes necessary, then, to construct
more complete hypotheses, as the MM. Cosserat have done in some
excellent memoirs, and we may then succeed in grouping together the
facts resulting from new experiments. Among the experiments of which
every theory must take account may be mentioned those by which Colonel
Hartmann has placed in evidence the importance of the lines which are
produced on the surface of metals when the limit of elasticity is
exceeded.

It is to questions of the same order that the minute and patient
researches of M. Bouasse have been directed. This physicist, as
ingenious as he is profound, has pursued for several years experiments
on the most delicate points relating to the theory of elasticity, and
he has succeeded in defining with a precision not always attained even
in the best esteemed works, the deformations to which a body must be
subjected in order to obtain comparable experiments. With regard to
the slight oscillations of torsion which he has specially studied, M.
Bouasse arrives at the conclusion, in an acute discussion, that we
hardly know anything more than was proclaimed a hundred years ago by
Coulomb. We see, by this example, that admirable as is the progress
accomplished in certain regions of physics, there still exist many
over-neglected regions which remain in painful darkness. The skill
shown by M. Bouasse authorises us to hope that, thanks to his
researches, a strong light will some day illumine these unknown
corners.

A particularly interesting chapter on elasticity is that relating to
the study of crystals; and in the last few years it has been the
object of remarkable researches on the part of M. Voigt. These
researches have permitted a few controversial questions between
theorists and experimenters to be solved: in particular, M. Voigt has
verified the consequences of the calculations, taking care not to
make, like Cauchy and Poisson, the hypothesis of central forces a mere
function of distance, and has recognized a potential which depends on
the relative orientation of the molecules. These considerations also
apply to quasi-isotropic bodies which are, in fact, networks of
crystals.

Certain occasional deformations which are produced and disappear
slowly may be considered as intermediate between elastic and permanent
deformations. Of these, the thermal deformation of glass which
manifests itself by the displacement of the zero of a thermometer is
an example. So also the modifications which the phenomena of magnetic
hysteresis or the variations of resistivity have just demonstrated.

Many theorists have taken in hand these difficult questions. M.
Brillouin endeavours to interpret these various phenomena by the
molecular hypothesis. The attempt may seem bold, since these phenomena
are, for the most part, essentially irreversible, and seem,
consequently, not adaptable to mechanics. But M. Brillouin makes a
point of showing that, under certain conditions, irreversible
phenomena may be created between two material points, the actions of
which depend solely on their distance; and he furnishes striking
instances which appear to prove that a great number of irreversible
physical and chemical phenomena may be ascribed to the existence of
states of unstable equilibria.

M. Duhem has approached the problem from another side, and endeavours
to bring it within the range of thermodynamics. Yet ordinary
thermodynamics could not account for experimentally realizable states
of equilibrium in the phenomena of viscosity and friction, since this
science declares them to be impossible. M. Duhem, however, arrives at
the idea that the establishment of the equations of thermodynamics
presupposes, among other hypotheses, one which is entirely arbitrary,
namely: that when the state of the system is given, external actions
capable of maintaining it in that state are determined without
ambiguity, by equations termed conditions of equilibrium of the
system. If we reject this hypothesis, it will then be allowable to
introduce into thermodynamics laws previously excluded, and it will be
possible to construct, as M. Duhem has done, a much more comprehensive
theory.

The ideas of M. Duhem have been illustrated by remarkable experimental
work. M. Marchis, for example, guided by these ideas, has studied the
permanent modifications produced in glass by an oscillation of
temperature. These modifications, which may be called phenomena of the
hysteresis of dilatation, may be followed in very appreciable fashion
by means of a glass thermometer. The general results are quite in
accord with the previsions of M. Duhem. M. Lenoble in researches on
the traction of metallic wires, and M. Chevalier in experiments on the
permanent variations of the electrical resistance of wires of an alloy
of platinum and silver when submitted to periodical variations of
temperature, have likewise afforded verifications of the theory
propounded by M. Duhem.

In this theory, the representative system is considered dependent on
the temperature of one or several other variables, such as, for
example, a chemical variable. A similar idea has been developed in a
very fine set of memoirs on nickel steel, by M. Ch. Ed. Guillaume. The
eminent physicist, who, by his earlier researches, has greatly
contributed to the light thrown on the analogous question of the
displacement of the zero in thermometers, concludes, from fresh
researches, that the residual phenomena are due to chemical
variations, and that the return to the primary chemical state causes
the variation to disappear. He applies his ideas not only to the
phenomena presented by irreversible steels, but also to very different
facts; for example, to phosphorescence, certain particularities of
which may be interpreted in an analogous manner.

Nickel steels present the most curious properties, and I have already
pointed out the paramount importance of one of them, hardly capable of
perceptible dilatation, for its application to metrology and
chronometry.[13] Others, also discovered by M. Guillaume in the course
of studies conducted with rare success and remarkable ingenuity, may
render great services, because it is possible to regulate, so to
speak, at will their mechanical or magnetic properties.

[Footnote 13: The metal known as "invar."--ED.]

The study of alloys in general is, moreover, one of those in which the
introduction of the methods of physics has produced the greatest
effects. By the microscopic examination of a polished surface or of
one indented by a reagent, by the determination of the electromotive
force of elements of which an alloy forms one of the poles, and by the
measurement of the resistivities, the densities, and the differences
of potential or contact, the most valuable indications as to their
constitution are obtained. M. Le Chatelier, M. Charpy, M. Dumas, M.
Osmond, in France; Sir W. Roberts Austen and Mr. Stansfield, in
England, have given manifold examples of the fertility of these
methods. The question, moreover, has had a new light thrown upon it by
the application of the principles of thermodynamics and of the phase
rule.

Alloys are generally known in the two states of solid and liquid.
Fused alloys consist of one or several solutions of the component
metals and of a certain number of definite combinations. Their
composition may thus be very complex: but Gibbs' rule gives us at once
important information on the point, since it indicates that there
cannot exist, in general, more than two distinct solutions in an alloy
of two metals.

Solid alloys may be classed like liquid ones. Two metals or more
dissolve one into the other, and form a solid solution quite analogous
to the liquid solution. But the study of these solid solutions is
rendered singularly difficult by the fact that the equilibrium so
rapidly reached in the case of liquids in this case takes days and, in
certain cases, perhaps even centuries to become established.




CHAPTER V

SOLUTIONS AND ELECTROLYTIC DISSOCIATION


Sec. 1. SOLUTION

Vaporization and fusion are not the only means by which the physical
state of a body may be changed without modifying its chemical
constitution. From the most remote periods solution has also been
known and studied, but only in the last twenty years have we obtained
other than empirical information regarding this phenomenon.

It is natural to employ here also the methods which have allowed us to
penetrate into the knowledge of other transformations. The problem of
solution may be approached by way of thermodynamics and of the
hypotheses of kinetics.

As long ago as 1858, Kirchhoff, by attributing to saline solutions--
that is to say, to mixtures of water and a non-volatile liquid like
sulphuric acid--the properties of internal energy, discovered a
relation between the quantity of heat given out on the addition of a
certain quantity of water to a solution and the variations to which
condensation and temperature subject the vapour-tension of the
solution. He calculated for this purpose the variations of energy
which are produced when passing from one state to another by two
different series of transformations; and, by comparing the two
expressions thus obtained, he established a relation between the
various elements of the phenomenon. But, for a long time afterwards,
the question made little progress, because there seemed to be hardly
any means of introducing into this study the second principle of
thermodynamics.[14] It was the memoir of Gibbs which at last opened
out this rich domain and enabled it to be rationally exploited. As
early as 1886, M. Duhem showed that the theory of the thermodynamic
potential furnished precise information on solutions or liquid
mixtures. He thus discovered over again the famous law on the lowering
of the congelation temperature of solvents which had just been
established by M. Raoult after a long series of now classic
researches.

[Footnote 14: The "second principle" referred to has been thus
enunciated: "In every engine that produces work there is a fall of
temperature, and the maximum output of a perfect engine--_i.e._ the
ratio between the heat consumed in work and the heat supplied--depends
only on the extreme temperatures between which the fluid is
evolved."--Demanet, _Notes de Physique Experimentale_, Louvain, 1905,
fasc. 2, p. 147. Clausius put it in a negative form, as thus: No
engine can of itself, without the aid of external agency, transfer
heat from a body at low temperature to a body at a high temperature.
Cf. Ganot's _Physics_, 17th English edition, Sec. 508.--ED.]

In the minds of many persons, however, grave doubts persisted.
Solution appeared to be an essentially irreversible phenomenon. It was
therefore, in all strictness, impossible to calculate the entropy of a
solution, and consequently to be certain of the value of the
thermodynamic potential. The objection would be serious even to-day,
and, in calculations, what is called the paradox of Gibbs would be an
obstacle.

We should not hesitate, however, to apply the Phase Law to solutions,
and this law already gives us the key to a certain number of facts. It
puts in evidence, for example, the part played by the eutectic point--
that is to say, the point at which (to keep to the simple case in
which we have to do with two bodies only, the solvent and the solute)
the solution is in equilibrium at once with the two possible solids,
the dissolved body and the solvent solidified. The knowledge of this
point explains the properties of refrigerating mixtures, and it is
also one of the most useful for the theory of alloys. The scruples of
physicists ought to have been removed on the memorable occasion when
Professor Van t'Hoff demonstrated that solution can operate reversibly
by reason of the phenomena of osmosis. But the experiment can only
succeed in very rare cases; and, on the other hand, Professor Van
t'Hoff was naturally led to another very bold conception. He regarded
the molecule of the dissolved body as a gaseous one, and assimilated
solution, not as had hitherto been the rule, to fusion, but to a kind
of vaporization. Naturally his ideas were not immediately accepted by
the scholars most closely identified with the classic tradition. It
may perhaps not be without use to examine here the principles of
Professor Van t'Hoff's theory.


Sec. 2. OSMOSIS

Osmosis, or diffusion through a septum, is a phenomenon which has been
known for some time. The discovery of it is attributed to the Abbe
Nollet, who is supposed to have observed it in 1748, during some
"researches on liquids in ebullition." A classic experiment by
Dutrochet, effected about 1830, makes this phenomenon clear. Into pure
water is plunged the lower part of a vertical tube containing pure
alcohol, open at the top and closed at the bottom by a membrane, such
as a pig's bladder, without any visible perforation. In a very short
time it will be found, by means of an areometer for instance, that the
water outside contains alcohol, while the alcohol of the tube, pure at
first, is now diluted. Two currents have therefore passed through the
membrane, one of water from the outside to the inside, and one of
alcohol in the converse direction. It is also noted that a difference
in the levels has occurred, and that the liquid in the tube now rises
to a considerable height. It must therefore be admitted that the flow
of the water has been more rapid than that of the alcohol. At the
commencement, the water must have penetrated into the tube much more
rapidly than the alcohol left it. Hence the difference in the levels,
and, consequently, a difference of pressure on the two faces of the
membrane. This difference goes on increasing, reaches a maximum, then
diminishes, and vanishes when the diffusion is complete, final
equilibrium being then attained.

The phenomenon is evidently connected with diffusion. If water is very
carefully poured on to alcohol, the two layers, separate at first,
mingle by degrees till a homogeneous substance is obtained. The
bladder seems not to have prevented this diffusion from taking place,
but it seems to have shown itself more permeable to water than to
alcohol. May it not therefore be supposed that there must exist
dividing walls in which this difference of permeability becomes
greater and greater, which would be permeable to the solvent and
absolutely impermeable to the solute? If this be so, the phenomena of
these _semi-permeable_ walls, as they are termed, can be observed in
particularly simple conditions.

The answer to this question has been furnished by biologists, at which
we cannot be surprised. The phenomena of osmosis are naturally of the
first importance in the action of organisms, and for a long time have
attracted the attention of naturalists. De Vries imagined that the
contractions noticed in the protoplasm of cells placed in saline
solutions were due to a phenomenon of osmosis, and, upon examining
more closely certain peculiarities of cell life, various scholars have
demonstrated that living cells are enclosed in membranes permeable to
certain substances and entirely impermeable to others. It was
interesting to try to reproduce artificially semi-permeable walls
analogous to those thus met with in nature;[15] and Traube and Pfeffer
seem to have succeeded in one particular case. Traube has pointed out
that the very delicate membrane of ferrocyanide of potassium which is
obtained with some difficulty by exposing it to the reaction of
sulphate of copper, is permeable to water, but will not permit the
passage of the majority of salts. Pfeffer, by producing these walls in
the interstices of a porous porcelain, has succeeded in giving them
sufficient rigidity to allow measurements to be made. It must be
allowed that, unfortunately, no physicist or chemist has been as lucky
as these two botanists; and the attempts to reproduce semi-permeable
walls completely answering to the definition, have never given but
mediocre results. If, however, the experimental difficulty has not
been overcome in an entirely satisfactory manner, it at least appears
very probable that such walls may nevertheless exist.[16]

[Footnote 15: See next note.--ED.]

[Footnote 16: M. Stephane Leduc, Professor of Biology of Nantes, has
made many experiments in this connection, and the artificial cells
exhibited by him to the Association francaise pour l'avancement des
Sciences, at their meeting at Grenoble in 1904 and reproduced in their
"Actes," are particularly noteworthy.--ED.]

Nevertheless, in the case of gases, there exists an excellent example
of a semi-permeable wall, and a partition of platinum brought to a
higher than red heat is, as shown by M. Villard in some ingenious
experiments, completely impermeable to air, and very permeable, on the
contrary, to hydrogen. It can also be experimentally demonstrated that
on taking two recipients separated by such a partition, and both
containing nitrogen mixed with varying proportions of hydrogen, the
last-named gas will pass through the partition in such a way that the
concentration--that is to say, the mass of gas per unit of volume--
will become the same on both sides. Only then will equilibrium be
established; and, at that moment, an excess of pressure will naturally
be produced in that recipient which, at the commencement, contained
the gas with the smallest quantity of hydrogen.

This experiment enables us to anticipate what will happen in a liquid
medium with semi-permeable partitions. Between two recipients, one
containing pure water, the other, say, water with sugar in solution,
separated by one of these partitions, there will be produced merely a
movement of the pure towards the sugared water, and following this, an
increase of pressure on the side of the last. But this increase will
not be without limits. At a certain moment the pressure will cease to
increase and will remain at a fixed value which now has a given
direction. This is the osmotic pressure.

Pfeffer demonstrated that, for the same substance, the osmotic
pressure is proportional to the concentration, and consequently in
inverse ratio to the volume occupied by a similar mass of the solute.
He gave figures from which it was easy, as Professor Van t'Hoff found,
to draw the conclusion that, in a constant volume, the osmotic
pressure is proportional to the absolute temperature. De Vries,
moreover, by his remarks on living cells, extended the results which
Pfeffer had applied to one case only--that is, to the one that he had
been able to examine experimentally.

Such are the essential facts of osmosis. We may seek to interpret them
and to thoroughly examine the mechanism of the phenomenon; but it must
be acknowledged that as regards this point, physicists are not
entirely in accord. In the opinion of Professor Nernst, the
permeability of semi-permeable membranes is simply due to differences
of solubility in one of the substances of the membrane itself. Other
physicists think it attributable, either to the difference in the
dimensions of the molecules, of which some might pass through the
pores of the membrane and others be stopped by their relative size, or
to these molecules' greater or less mobility. For others, again, it is
the capillary phenomena which here act a preponderating part.

This last idea is already an old one: Jager, More, and Professor
Traube have all endeavoured to show that the direction and speed of
osmosis are determined by differences in the surface-tensions; and
recent experiments, especially those of Batelli, seem to prove that
osmosis establishes itself in the way which best equalizes the
surface-tensions of the liquids on both sides of the partition.
Solutions possessing the same surface-tension, though not in molecular
equilibrium, would thus be always in osmotic equilibrium. We must not
conceal from ourselves that this result would be in contradiction with
the kinetic theory.


Sec. 3. APPLICATION TO THE THEORY OF SOLUTION

If there really exist partitions permeable to one body and impermeable
to another, it may be imagined that the homogeneous mixture of these
two bodies might be effected in the converse way. It can be easily
conceived, in fact, that by the aid of osmotic pressure it would be
possible, for example, to dilute or concentrate a solution by driving
through the partition in one direction or another a certain quantity
of the solvent by means of a pressure kept equal to the osmotic
pressure. This is the important fact which Professor Van t' Hoff
perceived. The existence of such a wall in all possible cases
evidently remains only a very legitimate hypothesis,--a fact which
ought not to be concealed.

Relying solely on this postulate, Professor Van t' Hoff easily
established, by the most correct method, certain properties of the
solutions of gases in a volatile liquid, or of non-volatile bodies in
a volatile liquid. To state precisely the other relations, we must
admit, in addition, the experimental laws discovered by Pfeffer. But
without any hypothesis it becomes possible to demonstrate the laws of
Raoult on the lowering of the vapour-tension and of the freezing point
of solutions, and also the ratio which connects the heat of fusion
with this decrease.

These considerable results can evidently be invoked as _a posteriori_
proofs of the exactitude of the experimental laws of osmosis. They are
not, however, the only ones that Professor Van t' Hoff has obtained by
the same method. This illustrious scholar was thus able to find anew
Guldberg and Waage's law on chemical equilibrium at a constant
temperature, and to show how the position of the equilibrium changes
when the temperature happens to change.

If now we state, in conformity with the laws of Pfeffer, that the
product of the osmotic pressure by the volume of the solution is equal
to the absolute temperature multiplied by a coefficient, and then look
for the numerical figure of this latter in a solution of sugar, for
instance, we find that this value is the same as that of the analogous
coefficient of the characteristic equation of a perfect gas. There is
in this a coincidence which has also been utilized in the preceding
thermodynamic calculations. It may be purely fortuitous, but we can
hardly refrain from finding in it a physical meaning.

Professor Van t'Hoff has considered this coincidence a demonstration
that there exists a strong analogy between a body in solution and a
gas; as a matter of fact, it may seem that, in a solution, the
distance between the molecules becomes comparable to the molecular
distances met with in gases, and that the molecule acquires the same
degree of liberty and the same simplicity in both phenomena. In that
case it seems probable that solutions will be subject to laws
independent of the chemical nature of the dissolved molecule and
comparable to the laws governing gases, while if we adopt the kinetic
image for the gas, we shall be led to represent to ourselves in a
similar way the phenomena which manifest themselves in a solution.
Osmotic pressure will then appear to be due to the shock of the
dissolved molecules against the membrane. It will come from one side
of this partition to superpose itself on the hydrostatic pressure,
which latter must have the same value on both sides.

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