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‘ATOMS.

BY

JEAN PERRIN

PROFESSEUR DE CHIMIE PHYSIQUE A LA SORBONNE

AUTHORISED TRANSLATION BY

D. Lt. HAMMICK

LONDON
CONSTABLE & COMPANY LTD
ORANGE STREET LEICESTER SQUARE WC
1916

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| TRANSLATOR’S NOTE.

The 4th Revised Edition of Professor Pervin’s

“Les Atomes” has been followed in making the

tvanslation. i ee

D. Lu. H.

GRESHAM’S ScHOOL, Hott,
June 8th, 1916.

Printed in Great Britain.

- PREFACE

Two kinds of intellectual activity, both equally instinc-
tive, have played a prominent part in the progress of physical
science.

One is already developed in a child that, while holding
an object, knows what will happen if he relinquishes
his grasp. He may possibly never have had hold of the
particular object before, but he nevertheless recognises
something in common between the muscular sensations
it calls forth and those which he has already experienced
when grasping other objects that fell to the ground when
_ his grasp was relaxed. Men like Galileo and Carnot, who
possessed this power of perceiving analogies to an extra-
ordinary degree, have by an analogous process built
up the doctrine of energy by successive generalisations,
cautious as well as bold, from experimental relationships
and objective realities.

‘In the first place they observed, or it would perhaps be
better to say that everyone has observed, that not only does
an object fall if it be dropped, but that once it has reached
the ground it will not rise of itself. We have to pay before
a lift can be made to ascend, and the more dearly the
swifter and higher it rises. Of course, the real price is not
a sum of money, but the external compensation given for
. the work done by the lift (the fall of a mass of water, the
combustion of coal, chemical change in a battery): The
money is only the symbol of this compensation.

This once recognised, our attention naturally turns to
the question of how small the payment can be. We know
that by means of a wheel and axle we can raise 1,000 kilo-
grammes through | metre by allowing 100 kilogrammes to
fall 10 metres ; is it possible to devise a more economical
mechanism that will allow 1,000 kilogrammes to be raised

vi PREFACE

1 metre for the same price (100 kilogrammes falling through
10 metres) ?

Galileo held that it is possible to affirm that, under certain
conditions, 200 kilogrammes could be raised 1 metre without
external compensation, ‘“‘for nothing.’ Seeing that we no
longer believe that this is possible, we have to recognise
equivalence between mechanisms that bring about the elevation
of one weight by the lowering of another.

In the same way, if we cool mercury from 100°C. to
0° C. by melting ice, we always find (and the general expres-
sion of this fact is the basis of the whole of calorimetry)
that 42 grammes of ice are melted for every kilogramme of
mercury cooled, whether we work by direct contact, radia-
tion, or any other method (provided always that we end with
melted ice and mercury cooled from 100°C. to 0° C.).
Even more interesting are those experiments in which,
through the intermediary of friction, a heating effect is
produced by the falling of weights (Joule). However widely
we vary the mechanism through which we connect the two
phenomena, we always find one great calory of heat produced
for the fall of 428 kilogrammes through 1 metre.

Step by step, in this way the First Principle of Thermo-
dynamics has been established. It may, in my opinion, be
enunciated as follows :

If by means of a certain mechanism we are able to connect
two phenomena in such a way that each may accurately com-
pensate the other, then it can never happen, however the
mechanism employed is varied, that we could obtain, as the
external effect of one of the phenomena, first the other and then
another phenomenon in addition, which would represent a
gain.*

Without going so fully into detail, we may notice another
similar result, established by Sadi Carnot, who, grasping
the essential characteristic common to all heat engines,
showed that the production of work is always accompanied
“by the passage of caloric from a body at a higher tem-

1 At least, the other phenomenon could only be one of those which we know
can occur without external compensation (such as isothermal change of volume

of a gaseous mass, according to a law discovered by Joule). In that case the
gain may still be looked upon as non-existent.

PREFACE vii

perature to another at a lower temperature.” As we know,
proper analysis of this statement leads to the Second Law of
Thermodynamics.

Each of these principles has been reached by noting
analogies and generalising the results of experience, and our
lines of reasoning and statements of results have related
only to objects that can be observed and to experiments
that can be performed. Ostwald could therefore justly say
that in the doctrine of energy there are no hypotheses.
Certainly when a new machine is invented we at once assert
that it cannot create work ; but we can at once verify our
statement, and we cannot call an assertion a hypothesis if,
as soon as it is made, it can be checked by experiment.

Now, there are cases where hypothesis is, on the contrary,
both necessary and fruitful. In studying a machine, we do
not confine ourselves only to the consideration of its visible
parts, which have objective reality for us only as far as we
can dismount the machine. We certainly observe these
visible pieces as closely as we can, but at the same time we
seek to divine the hidden gears and parts that explain its
apparent motions.

To divine in this way the existence and properties of
objects that still lie outside our ken, to explain the complica-
tions of the visible in terms of invisible simplicity, is the
function of the intuitive intelligence which, thanks to men
such as Dalton and Boltzmann, has given us the doctrine
of Atoms. This book aims at giving an exposition of that
doctrine.

The use of the intuitive method has not, of course, been
used only in the study of atoms, any more than the inductive
method has found its sole application in energetics. A time
may perhaps come when atoms, directly perceptible at last,
will be as easy to observe as are microbes to-day. The true
spirit of the atomists will then be found in those who have
inherited the power to divine another universal structure
lying hidden behind a vaster experimental reality than
ours.

I shall not attempt, as too many have done, to decide
between the merits of the two methods of research. Cer-

Vili PREFACE

tainly during recent years intuition has gone ahead of
induction in rejuvenating the doctrine of energy by the
incorporation of statistical results borrowed from the
atomists. But its greater fruitfulness may well be transient,
and I can see no reason to doubt the possibility of further
' discovery that will dispense with the necessity of employing
any unverifiable hypothesis.

Although perhaps without any logical necessity for so
doing, induction and intuition have both up to the present
made use of two ideas that. were familiar to the Greek
philosophers ; these are the conceptions of fullness (or
continuity) and of emptiness (or discontinuity).

Even more for the _ benefit of the reader who has
just read this book than for him who is about to do
so, I wish to offer a few remarks designed to give
objective justification for certain logical exigencies of the
mathematicians.

It is well known that before giving accurate definitions
we show beginners that they already possess the idea of
continuity. We draw a well-defined curve for them and say
to them, holding a ruler against the curve, “ You see that
there is a tangent at every point.” Or again, in order to
impart the more abstract‘notion of the true velocity of a
moving object at a point in its trajectory, we say, ““ You see,
of course, that the mean velocity between two neighbouring
points on this trajectory does not vary appreciably as these
points approach infinitely near to each other.” And many
minds, perceiving that for certain familiar motions this
appears true enough, do not see that there are considerable
difficulties in this view.

To mathematicians, however, the lack of rigour in these
so-called geometrical considerations is quite apparent, and
they are well aware of the childishness of trying to show,
by drawing curves, for instance, that every continuous
function has a derivative. Though derived functions are
the simplest and the easiest to deal with, they are
nevertheless exceptional; to use geometrical language,
curves that have no tangents are the rule, and regular

PREFACE ix

curves, such as the circle, are interesting though quite special
cases.

At first sight the consideration of such cases seems merely
an intellectual exercise, certainly ingenious but artificial
and sterile in application, the desire for absolute accuracy
carried to a ridiculous pitch. And often those who hear of
curves without tangents, or underived functions, think at
first that Nature presents no such complications, nor even
offers any suggestion of them. |

The contrary, however, is true, and the logic of the mathe-
maticians has kept them nearer to reality than the practical
representations employed by physicists. This may be
illustrated by considering, in the absence of any precon-
ceived opinion, certain entirely experimental data.

The study of colloids provides an abundance of such data.
Consider, for instance, one of the white flakes that are
obtained by salting a soap solution. At a distance its contour
may appear sharply defined, but as soon as we draw nearer
its sharpness disappears. The eye no longer succeeds in
drawing a tangent at any point on it; a line that at first
sight would seem to be satisfactory, appears on closer
scrutiny to be perpendicular or oblique to the contour.
The use of magnifying glass or microscope leaves us just as
uncertain, for every time we increase the magnification we
find fresh irregularities appearing, and we never succeed
in getting a sharp, smooth impression, such as that given,
for example, by a steel ball. So that if we were to take a
steel ball as giving a useful illustration of classical continuity,
our flake could just as logically be used to suggest the more
general notion of a continuous underived function.

We must bear in mind that the uncertainty as to the
position of the tangent plane at a point on the contour is by
no means of the same order as the uncertainty involved,
according to the scale of the map used, in fixing a tangent
at a point on the coast line of Brittany. The tangent would
be different according to the scale, but a tangent could
always be found, for a map is a conventional diagram in
which, by construction, every line has a tangent. An
essential characteristic of our flake (and, indeed, of the coast

x PREFACE

line also when, instead of studying it as a map, we observe
the line itself at various distances from it) is, on the
contrary, that on any scale we suspect, without seeing them
clearly, details that absolutely prohibit the fixing of a
tangent.

We are still in the realm of experimental reality when,
under the microscope, we observe the Brownian movement
agitating each small particle suspended in a fluid. In order
to be able to fix a tangent to the trajectory of such a particle,
we should expect to be able to establish, within at least
approximate limits, the direction of the straight line joining
the positions occupied by a particle at two very close
successive instants. Now, no matter how many experiments
are made, that direction is found to vary absolutely irregu-
larly as the time between the two instants is decreased.
An unprejudiced observer would therefore come to the con-
clusion that he was dealing with an underived function,
instead of a curve to which a tangent could be drawn.

I have spoken first of curves and outlines because curves
are ordinarily used to suggest the notion of continuity and to
represent it. But it is just as logical, and in physics it is more
usual, to inquire into the variation of some property, such
as density or colour, from one point in a given material to
another. And here again complications of the same kind
as those mentioned above will appear.

The classical idea is quite definitely that it is possible to
decompose any material object into practically identical
small parts. In other words, it is assumed that the differen-
tiation of the matter enclosed by a given contour becomes
less and less as the contour contracts more and more.

Now I may almost go so far as to say that, far from being
suggested by experience, this conception but rarely cor-
responds with it. My eye seeks in vain for a small “ practi-
cally homogeneous ”’ region on my hand, on the table at
which I am writing, on the trees or in the soil that I can see
from my window. And if, taking a not too difficult case,
I select a somewhat more homogeneous region, on a tree
trunk for instance, I have only to go close to it to distinguish
details on the rough bark, which until then had only been

PREFACE xi

suspected, and to be led to suspect the existence of others.
Having reached the limits of unaided vision, magnifying
glass and microscope may be used to show each succes-
sive part chosen at a progressively increasing magnifi-
cation. Fresh details will be revealed at each stage, and
when at last the utmost limit of magnifying power has been
reached the impression left on the mind will be very different
from the one originally received. In fact, as is well known,
a living cell is far from homogeneous, and within it we are
able to recognise the existence of a complex organisation of
fine threads and granules immersed in an irregular plasma,
where we can only guess at things that the eye tires itself in
vain in seeking to characterise with precision. Thus the
portion of matter that to begin with we had expected to
find almost homogeneous appears to be indefinitely diverse,
and we have absolutely no right to assume that on going
far enough we should ultimately reach “‘ homogeneity,”
or even matter having properties that vary regularly from point

to point. |

It is not living matter only that shows itself to be indefi-
nitely sponge-like and differentiated. Charcoal obtained by
calcining the bark of the tree mentioned above displays the
same unlimited porosity. The soil and- most rocks do not
appear to be easily decomposable into small homogeneous
parts. Indeed, the only examples of regularly continuous
materials to be found are crystals such as diamonds, liquids °
such as water, and gases. Thus the notion of continuity is
the result of an arbitrary limitation of our attention to a
part only of the data of experience.

It must be borne in mind that although closer observation
of the object we are studying generally leads to the dis-
covery of a highly irregular structure, we can with advantage
often represent its properties approximately by continuous
functions. More simply, although wood may be indefinitely
porous, it is useful to speak of the surface of a beam that we
wish to paint, or of the volume displaced by a float. In
other words, at certain magnifications and for certain
-methods of investigation phenomena may be represented
by regular continuous functions, somewhat in the same

xii PREFACE

way that a sheet of tin-foil may be wrapped round a
sponge without it following accurately the latter’s com-
plicated contour.

If then we refuse to limit our considerations to the part
of the universe we actually see, and if we attribute to matter
the infinitely granular structure that is suggested by the
results obtained by the use of the conception of atoms, our
power to apply rigorously mathematical continuity to
reality will be found. to suffer a very remarkable diminution.

Let us consider, for instance, the way in which we define
the density of a compressible fluid (air, for example) at a
given point and at a given moment. We picture a sphere
of volume v having its centre at that point and including

at the given moment a mass m. The quotient - is the mean

density within the sphere, and by true density we mean the
limiting value of this quotient. This means that at the
~~given moment the mean density within the sphere is
practically constant below a certain value for the volume.
Indeed, this mean density, which may possibly be notably
different for spheres containing 1,000 cubic metres and
1 cubic centimetre respectively, only varies by 1 part in
1,000,000 on passing from 1 cubic centimetre to one-
. thousandth of a cubic millimetre. Nevertheless, even
between these volume limits (the width of which is consider-
ably influenced by the state of agitation of the fluid) varia-
tions of the order of 1 part in 1,000,000,000 occur irregularly.
Suppose the volume to become continually smaller.
Instead of these fluctuations becoming less and less impor-
tant, they come to be more and more considerable and
irregular. For dimensions at which the Brownian move-
ment shows great activity, for one-tenth of a cubic micron,
say, they begin (in air) to attain to 1 part in 1,000, and they
become of the order of 1 part in 5 when the radius of the
hypothetical spherule becomes of the order of a hundredth
a micron. |
One step further and the radius becomes of the same
order as the molecular radius. Then, as a general rule (in a

PREFACE xiii

gas at any rate), our spherule will lie in intermolecular space,
where its mean density will henceforth be nil ; at our given
point the true density will be nil also. But about once in a
thousand times that point will lie within a molecule and
the mean density will then come to be comparable with that
of water, or a thousand times higher than the value we
usually take to be the true density of the gas.

Let our spherule grow steadily smaller. Soon, except
under exceptional circumstances, which have few chances
of occurring, it will become empty and remain so henceforth ~
owing to the emptiness of intra-atomic space; the true
density at any given point will still remain nil. If, however,
as will happen only about once in a million times, the given
point lies within a corpuscle or the central atomic nucleus,
the mean density will rise enormously and will become
several million times greater than that of water.

If the spherule were to become still smaller, it may be
that we should attain a measure of continuity, until we
reached a new order of smallness; but more probably
(especially in the atomic nucleus, which radioactivity shows
to possess an extremely complicated structure) the mean
density would soon fall to nothing and remain there, as will
the true density also, except in certain very rare positions,
where it will reach values enormously greater than any before.

In short, the doctrine of atoms leads to the following :—
density is everywhere nil, except at an infinite number of
isolated points, where it reaches an infinite value.t _

Analogous considerations are applicable to all properties
that, on our scale, appear to be continuous and regular,
such as velocity, pressure or temperature. We find them
growing more and more irregular as we increase the magnifi-
cation of the ever imperfect image of the universe that we
construct for ourselves. Density we have seen to be nil at

1 | have simplified the problem. As a matter of fact, time is a factor, and
mean density, defined in a small volume v surrounding the given point at a

given instant, must be connected with a small lapse of time 7 that includes the
given instant. The mean mass in the volume v during the time 7 would be

of the form ‘fim . dt, and the mean density is a second derivative with respect

to volume and time. Its representation as a function of two variables would
lead to infinitely indented surfaces,

xiv PREFACE

all points, with certain exceptions; more generally, the
function that represents any physical property we consider
(say electric potential) will form in intermaterial space a
continuum that presents an infinite number of singular
points and which we shall be able to study with the aid of the
mathematician.*

An infinitely discontinuous matter, a continuous ether
studded with minute stars, is the picture presented by the
universe, if we remember, with J. H. Rosny, sen., that no
formula, however comprehensive, can embrace Diversity
that has no limits, and that all formule lose thei: significance
when we make any considerable departure from the condi-
tions under which we acquire our knowledge.

The conclusion we have just reached by considering a
continuously diminishing centre can also be arrived at by
imagining a continually enlarging sphere, that successively
embraces planets, solar system, stars, and nebule. Thus
we find ourselves face to face with the now familiar concep-
tion developed by Pascal when he showed that man lies

‘“ suspended between two infinities.”’

Among those whose genius has thus been able to con-
template Nature in her full high majesty, I have chosen one,
to whom I dedicate this work, in homage to a departed friend ;
to him I owe the inspiration that brings to scientific research
a tempered enthusiasm, a tireless energy and a love of
beauty.’

1 Those who are interested in this question will do well to read the works of
M. Emile Borel, particularly the very fine lecture on “ Molecular Theories and
Mathematics ”’ (Inauguration of the University of Houston, and Revue générale
des Sciences, November, 1912), wherein he shows how the physics of discontinuity
may ssibly transform the mathematical analysis created originally to meet
the nc os of the physics of continuity.

2 Prcxessor Perrin’s book is dedicated to the memory of M. Noel Bernard.—
[TR.]

CHAP.

If.

Ii.

IV.

Vil.

VIII.

CONTENTS

PREFACE

CHEMISTRY AND THE ATOMIC THEORY

MOLECULAR AGITATION

THE BROWNIAN MOVEMENT—EMULSIONS .
THE LAWS OF THE BROWNIAN MOVEMENT
FLUCTUATIONS

LIGHT AND QUANTA

THE ATOM OF ELECTRICITY

THE GENESIS AND DESTRUCTION OF ATOMS

INDEX

53

83

109

134

144

164

186

209

°

ATOMS

CHAPTER I
CHEMISTRY AND THE ATOMIC THEORY

MOLECULES.

Some twenty-five centuries ago, before the close of the
lyric period in Greek history, certain philosophers on the
shores of the Mediterranean were already teaching that
changeful matter is made up of indestructible particles in con-
stant motion ; atoms which chance or destiny has grouped
in the course of ages into the forms or substances with which
we are familiar. But we know next to nothing of these
early theories, of the works of Moschus, of Democritus of
Abdera, or of his friend Leucippus. No fragments remain
that might enable us to judge of what in their work was of
scientific value. And in the beautiful poem, of a much later
date, wherein Lucretius expounds the teachings of Epicurus,
we find nothing that enables us to grasp what facts or what
theories guided Greek thought.

1.—PERSISTANCE OF THE COMPONENT SUBSTANCES IN
MixtuREs.—Without raising the question as to whether our
present views actually originated in this way, we may notice
that it is possible to infer a discontinuous structure for
certain substances which, like water, appear perfectly homo-
geneous, merely from a consideration of the familiar pro-
perties of solution. It is universally admitted, for instance,
that when sugar is dissolved in water the sugar and the
water both exist in the solution, although we cannot distin-
guish the different components from each other. Similarly,
if we drop a little bromine into chloroform, the bromine and

the chloroform constituents in the homogeneous liquid thus

A. B

2 ATOMS

obtained will continue to be recognisable by their colour and
) smell.

This would be easily explicable if the different substances
existed in the liquid in the same way that the particles of a
well powdered mixture exist side by side ; though we may
“no longer be able to distinguish the particles from each other
even at close quarters, we can nevertheless detect them (by
their colour or taste, for example, as may readily be verified
by making an intimate mixture of powdered sugar and
flowers of sulphur). Similarly, the persistance of the pro-
perties of bromine and of chloroform in the liquid obtained by
mixing these substances is perhaps due to the existence in
the liquid of small particles, in simple juxtaposition (but
unmodified), which by themselves constitute bromine, and
of other particles which, by themselves, form chloroform.
These elementary particles, or molecules, should be found in
all mixtures in which we recognise bromine or chloroform,
and their extreme minuteness alone prevents us from per-
ceiving them as individuals. Moreover, since bromine (or
chloroform) is a pure substance, in the sense that no single
observation has ever led us to recognise in it the properties of
components of which it could be a mixture, we must suppose
that its molecules are composed of the same substance.

But they may be of various dimensions, like the particles
which make up powdered sugar or flowers of sulphur ; they
may even be extremely minute droplets, capable, under
certain circumstances, of uniting among themselves or of
subdividing without losing their nature. Indefiniteness of
this kind is often met with in physics when we come
to give a precise meaning to a hypothesis put forward
vaguely in the first place. In such circumstances we trace
out as far as possible the consequences of each particular |
precise form of our hypothesis that we can devise. The
necessary condition that they must agree with experiment
or simply their obvious barrenness soon leads us to abandon
most of these tentative forms and they are consequently
omitted from subsequent discussion. \ :

2.—EaAcH CHEMICAL SPECIES IS COMPOSED OF CLEARLY
CHARACTERISED MoLecuLes.—In the present case one only

CHEMISTRY AND THE ATOMIC THEORY 3

of the precise forms which we have been able to devise for
our general hypothesis has proved fruitful. It has been
assumed that the molecules that make. up a pure substance
are exactly identical and remain identical in all mixtures in
which that substance is found. In liquid bromine, in bro-
mine vapour, in a solution of bromine, at all pressures and
temperatures, as long as we can “ recognise bromine ”’ this
material ‘“‘ bromine ”’ is resolvable, at a sufficient magnifica-
tion, into identical molecules. Even in the solid state these
molecules exist, as assemblages of objects each maintaining
their own individuality and separable without rupture, (quite
unlike the way in which bricks are cemented into a wall ;
for when the wall is destroyed, the bricks cannot be recovered
intact, whereas on melting or vapourising a solid the mole-
cules are recoverable with their independence and mobility
unimpaired).

_ If every pure substance is necessarily made up of a parti-
cular kind of molecule, it does not follow that, conversely,
with each kind of molecule we should be able to make up a
pure substance without admixture of other kinds of mole-
cules. We can thus understand the properties of that
singular gas nitrogen peroxide ; it does not obey Boyle’s law,
and its red colour becomes more intense when it is allowed
‘to expand into a larger volume. These abnormalities are
explained in every detail if nitrogen peroxide is really a mix-
ture in variable proportions of two gases, the one red and the
other colourless. It is certainly to be expected that each of
these gases would be composed of a definite kind of molecule ;
but as a matter of fact it is not possible to separate these two
kinds of molecule or, in other words, to prepare in a pure
state the red and the colourless gases. As soon as one brings
about a separation that, for example, increases for a moment
the proportion of red gas, a fresh quantity of colourless gas
is at once re-formed at the expense of the red, until the pro-
portion fixed by the particular pressure and temperature is
reached once more.!

1 A more complete discussion leads us to regard the colourless gas molecule as
formed by the union of two molecules of red gas, the two gases having the
chemical formule (in the sense that will be explained later), N20, and NO».

B 2

4 ATOMS

More generally, it appears that a substance may be easy to
characterise and to recognise as a constituent of various
mixtures, though at the same time we may not know how to
separate it in the pure state from substances that dissolve
it or with which it is in equilibrium. Chemists do not hesi-
tate to speak of sulphurous acid or carbonic acid, although
it is not possible to separate these hydrogen compounds
from their solutions. On our hypothesis a particular kind
of molecule should correspond to each chemical species that
is to be regarded in this manner as existing ; and, conversely,
to each kind of molecule will correspond a chemical entity,
definite though not always capable of being isolated. Of
course, we do not assume that the molecules that make up a
chemical entity are indivisible like “atoms” ; on the contrary,
we are generally led to the view that they are divisible. But
in that case the properties by means of which the chemical
entity is recognised disappear and others make their appear-
ance ; the new properties belong to new chemical entities,
which have for molecules the fragments of the old mole-
cules.+

In short we suppose that any substance whatever, that appears
to be homogeneous to observations on our scale of dimensions,
would be resolved at a sufficient magnification into well defined
molecules of as many different kinds as there are constituents
recognisable from the properties of the given substance.

We shall see that these molecules do not remain at rest.

3.—MoLEcuLAR AGITATION IS MADE MANIFEST BY THE
PHENOMENA OF Dirrustion.—When a layer of alcohol is
superposed upon a layer of water, although the alcohol is on —
top it is well known that the two liquids do not remain
separated, in spite of the fact that the lower layer is the
denser. Reciprocal solution takes place, by the diffusion
of the two substances into each other, and in a few days
renders the liquid uniform throughout. It must therefore
be assumed that the molecules of alcohol and of water are
endowed with movement, at least during the time the act of

solution lasts.

1 A simple example is furnished in sal ammoniac, which has a molecule
capable of splitting up into two portions—.e., an ammonia molecule and a
hydrochloric acid molecule.

CHEMISTRY AND THE ATOMIC THEORY 5

As a matter of fact, if we had superposed water and ether,
a distinct surface of separation would have persisted. But
even in this case of incomplete solubility, water passes into
every layer of the upper liquid and ether penetrates equally
into each layer of the lower liquid. A movement of the
molecules is thus again manifest.

With gaseous layers, diffusion, which is more rapid,
always proceeds until the entire mass becomes uniform. In
Berthollet’s famous experiment a globe containing carbon
dioxide was put in communication by means of a stop-cock
with another globe containing hydrogen at the same pres-
sure, the hydrogen being above the carbon dioxide. In
spite of the great difference in density between the two gases,
_ the composition gradually became uniform in the two
globes and soon each one of them contained as much hydro-
gen as carbon dioxide. The experiment led to the same
result no matter what pairs of gases were used.’

Moreover, the rate of diffusion has no connection with any
difference in properties of the two fluids put in contact. It
may be great or small for very similar bodies as well as for
those that are very dissimilar. We find, for example, that
ethyl alcohol (spirits of wine) and methyl alcohol (wood
spirit), which chemically and physically are very similar, do
not interpenetrate more quickly than ethyl alcohol and
toluene, which differ much more widely from each other.

Now, if diffusion takes place between two layers of fluids
of any kind—between, for instance, ethyl alcohol and water,
ethyl alcohol and methyl! alcohol, ethyl alcohol and propyl
alcohol—may we not assume that diffusion takes place in
just the same way between ethyl alcohol and ethyl] alcohol ?
In the light of the preceding considerations, it seems difficult
to avoid the conclusion that diffusion probably does take
place but that we are no longer able to perceive it on account
of the identical nature of the two interpenetrating bodies.

We are thus forced to imagine a continual diffusion taking
place between any two contiguous sections of the same fluid.
If molecules do exist, it comes to the same thing if we say
that every surface traced in a fluid is traversed incessantly
by molecules passing from one side to the other, and hence

6 ATOMS

that the molecules of any fluid whatever are in constant
motion.

If these conclusions are well founded, our ideas on fluids
“in equilibrium ’’ must undergo a very profound readjust-
ment. Like homogeneity, equilibrium is only apparent and
disappears when we change the “ magnification’ under
which we observe matter. More exactly, such equilibrium
represents a particular permanent condition of. uncoordinated
agitation. From our observations on the ordinary dimen-
sional scale we can get no inkling of the internal agitation of
fluids, because each small element of volume at each instant
gains as many molecules as it loses and preserves the same
mean condition of uncoordinated movement. As we pro-
ceed we shall find that these ideas will become more precise,
and we shall come to understand better the important
position the theories of statistics and probability must occupy
in physics.

4.—MOLECULAR AGITATION EXPLAINS THE EXPANSIBILITY
or FLuips.—Having once admitted the existence of mole-
cular agitation, we can readily understand the expansibility
of fluids, or, which comes to the same thing, why they always
exert a pressure on the walls of the vessels that contain them.
This pressure is due, not to a mutual repulsion between
diverse portions of the fluid, but to the incessant impact of
the molecules of the fluid against the walls.

This somewhat vague hypothesis was given a precise form
and developed towards the middle of the eighteenth century
for the case of a fluid rarefied sufficiently to possess the pro-
perties characteristic of the gaseous state. It is assumed
that under such conditions the molecules roughly correspond
to elastic spheres having a total volume very small compared
with the space they traverse, and which are on the average
so far from each other that each moves in a straight ine for
the greater part of its path, until impact with another mole-
cule abruptly changes its direction. We shall see later how
this hypothesis explained all the known properties of gases
and how by means of it other properties then unrecognised
were predicted.

Suppose that a gaseous mass is heated at constant volume ;

CHEMISTRY AND THE ATOMIC THEORY 7

we know that its pressure then rises. If this pressure is due
to the impacts of the molecules upon the containing walls, we
must suppose that the molecules are now moving with
speeds that, on the average, have increased, so that each
square centimetre of the containing surface is subjected to
impacts that are more violent and more numerous. JMole-
cular agitation must therefore increase with rise in temperature.
If, on the other hand, the temperature falls, the molecular
agitation will slacken and should tend towards zero along
with the pressure of the gas. At the “absolute zero” of
temperature the molecules should be absolutely motionless.

In this connection we may remember that in all cases,
without exception, rate of diffusion becomes slower the
lower the temperature. Thus molecular agitation and tem-
perature always vary in the same sense and appear to be
fundamentally connected with each other.

ATOMS.

_ §—SrmpeLe Susstances.—Amid the vast aggregate of

known substances (which are in general mixtures in varying
proportions) the various chemical species serve as co-
ordinating centres in the same way that the four apices of a
tetrahedron act as points of reference for all points inside it.
But even then their number is enormous. As we know,
since Lavoisier’s time the study and classification of all such
species has been simplified by the discovery of ‘ simple
substances,’ indestructible substances obtained by pushing
as far as possible the “‘ decomposition” of the different
available materials.

The meaning of this word ‘‘ decomposition’ will be made
clear by the discussion of some particular case. It is pos-
sible, for instance, by merely heating, to transform sal
ammoniac, a well-defined pure solid substance, into a mix-
ture of gases that can be separated by a suitable fractiona-
tion (diffusion or effusion) into ammonia gas and hydrochloric
acid gas. Ammonia gasis in its turn transformable (by means
of a stream of sparks) into a gaseous mixture of nitrogen
and hydrogen, which in their turn are easily separable.
Then, having dissolved the hydrochloric acid gas in a little

8 ATOMS

water, it is possible (by electrolysis) to recover, firstly, the |
added water, and, secondly, chlorine and hydrogen (which
separate at the electrodes) from the hydrochloric acid gas.
From 100 grammes of the salt we can produce 26°16 grammes
of nitrogen’, 7°50 grammes of hydrogen, and 66°34 grammes
of chlorine, these masses being equal to that of the salt that
has disappeared. |

All other ways of decomposing sal ammoniac, pushed to
their utmost limit, are always found to end with the pro-
duction of these three elementary. bodies in exactly the same
proportions. Speaking more generally, an enormous number
of decompositions has led to the recognition of about 100
simple substances (nitrogen, chlorine, hydrogen, carbon, etc.,
etc.) possessing the following property :—

Any material system whatsoever can be decomposed into
masses each composed of one of these simple substances ; these
masses are absolutely independent, in quantity and in their
nature, of the operations that the given system has been made to
undergo.

Thus, if we start with fixed masses of these different simple
substances, we can, after making them react with each other
in every conceivable way, always recover the mass of each
simple substance originally taken. If the element oxygen
is represented to begin with by 16 grammes, it is not within
our power to bring about an operation at the end of which
we do not regain 16 grammes of oxygen, neither more nor
less, on decomposing the system obtained.!

It is therefore hard to avoid the conclusion that the oxygen
has actually persisted throughout the series of compounds
produced, disguised but certainly present ; one and the same
“elementary substance’’ must exist in all substances con-
taining oxygen, such as water, oxygen, ozone, carbon
dioxide, or sugar.”

1 It is of course understood that oxygen and ozone, which are transformable
in their entirety into each other, are regarded as equivalent. Similarly with all
simple substances capable of existing in various allotropic modifications.

2 Incidentally, it is not quite correct to speak of this particular elementary
substance as “ oxygen.’”’ Clearly, we might just as well call it “ ozone,” since
oxygen and ozone can be completely transformed into each other. One and the
same substance, to which a distinct name should be given, appears to us,
according te circumstances, sometimes in the form “‘ oxygen ”’ and sometimes
in the form “ozone.” We shall perceive the significance of this later on (par. 7).

CHEMISTRY AND THE ATOMIC THEORY 9

But if sugar, for instance, is made up of identical mole-
cules, oxygen, with its usual properties masked, must have a
place in the structure of each, and similarly with carbon and
hydrogen, which are the other elements in sugar. We shall
endeavour to make out in what form the elementary sub-
stances exist in molecules.

6.—THe Law or CuemicaL Discontinurry.—Certain
fundamental chemical laws will help us in this task. We
have first that the proportion of an element that enters into
a molecule cannot have all possible values. When carbon
burns in oxygen, it produces a pure substance (carbon
dioxide), containing 3 grammes of carbon to every 8 grammes
of oxygen. It would not be irrational to expect (and indeed
eminent chemists have in the past regarded it as possible)
that, by changing the conditions under which combination
takes place (by working, for example, under high pressures
or by substituting slow for rapid combustion), we might be
able to change slightly the proportions of combined carbon
and oxygen. ‘Thus we might not unreasonably expect to be
able to obtain a pure substance possessing properties approxi-
mating to those of carbon dioxide and containing, for
example, for 3 grammes of carbon, 8 grammes plus | deci-
gramme of oxygen. No such substance is produced, and the
fact that the absence of such substances is general gives us
the “ Law of Definite Proportions’ (mainly due to Proust’s
work) :—.

The proportions in which two elements combine cannot vary
continuously.

This is not meant to imply that carbon and oxygen can
unite in one single proportion only ; it is not difficult (as in
the preparation of carbon monoxide) to combine 3 grammes
of carbon, not with 8, but with 4 grammes of oxygen. Only
the variation, as we see, is in this case very large ; it is, in
fact, a discontinuous leap. At the same time the properties
of the compound thus obtained have become very different
from those of carbon dioxide. The two compounds are
marked off from each other, as it were, by an insuperable
gap.

The above example immediately suggests another law,

10 ATOMS

discovered by Dalton. It might be merely fortuitous that
3 grammes of carbon should unite with either 4 grammes of
oxygen or with exactly double that amount. But we find
simple ratios figuring in so large a number of cases that we
cannot regard them as so many accidental coincidences.
And this leads us to the Law of Multiple Proportions, which
we can enunciate as follows :—

If two definite compounds are taken-at random from among

the multitude of those containing the simple substances A and

B, and if the masses of the element B that are found to be com-
bined with the same mass of the element A are compared, it is
found that those masses are usually in a very simple ratio to
each other. In certain cases they may be, and in fact frequently
are, exactly equal.

Thus the ratio of chlorine to silver in silver chloride and
in silver chlorate is found to be the same, or at least the error
does not exceed that conditioned by the degree of accuracy
reached in the operations of analytical chemistry. Now
analytical accuracy has been increasing continuously, and
in this particular case (of Stas’ measurements) exceeds 1 part
in 10,000, so that we cannot possibly doubt that rigorous
equality holds.

7.—Tue Atomic HypotHesis.—We owe to Dalton the
happy inspiration that, embracing in the simplest manner
both Proust’s law and the law he had discovered himself, »
finally gave capital importance to molecular theories in the
coordination and prediction of chemical phenomena (1808).

Dalton supposed that each of the elementary substances of
which all the various kinds of materials are composed is made
up of a fixed species of particles, all absolutely identical ; !
these particles pass, without ever becoming subdivided,
through the various chemical and physical transformations
that we are able to bring about, and, being indivisible by
means of such changes, they can therefore be called atoms,
in the etymological sense.

1 ‘Identical when once isolated, even if they are not absolutely interchangeable
at any given moment. ‘Two springs when compressed to different extents may
be regarded as identical if, when released, they become identical. There will
thus be no difference between an iron atom extracted from ferrous chloride and
one obtained from ferric chloride.

CHEMISTRY AND THE ATOMIC THEORY 11

Any single molecule necessarily contains a whole number
of atoms of each elementary substance present. Its com-
position therefore cannot vary continuously (which is Proust’s
law), but only by discontinuous leaps, corresponding to the
gain or loss of at least one atom (which leads us to Dalton’s
law of multiple proportions).

It is clear, moreover, that if a molecule were a highly com-
plex body, containing several thousand atoms, niakvtical
chemistry would be found to be too inexact to give us any
information as to the entry or exit of a few atoms more or
less. That the laws of discontinuity were discoverable when
chemical analysis was not always reliable to within aig
than 10 per cent.’ is clearly due to the fact that the mole-
cules. studied by chemists contained but few atoms.

A molecule may be monatomic (composed of single atoms) ;
more usually it will contain several atoms. A particularly
interesting case is that in which the atoms combined in the
same molecule are of the same kind. We are then dealing
with a simple substance which can nevertheless actually be
regarded as a compound of a particular elementary sub-
stance with itself. We shall see that this is of frequent
occurrence and that it explains certain cases of allotropy
(we have already pointed out the case of oxygen and ozone).

In short, the whole material universe, in all its extra-
ordinary complexity, may have been built up by the coming
together of elementary units fashioned after a small number
of types, elements of the same type being absolutely identical,
It is easy to see how greatly the atomic hypothesis, if it is
' substantiated, will enable us to simplify our study of matter.

8.—THE RELATIVE WEIGHTS OF THE ATOMS WOULD BE
KNOWN, IF IT WERE KNOWN HOW MANY OF EACH SORT THERE
ARE IN THE MoLecuLe.—Once the existence of atoms is

1 We would expect, moreover, that very complicated molecules would be more
fragile than molecules composed of few atoms and that they would therefore
have fewer chances of coming under observation. We should also expect that
if a molecule were very large (albumins ?) the entry or exit of a few atoms would
not greatly affect its properties and, moreover, that the separation of a pure
substance corresponding to such molecules would present no little difficulty,
even if its isolation did not become impossible. And this would still further
increase the probability that a pure substance easy to prepare would be com-
posed of molecules containing few atoms.

12 ATOMS

assumed, the question arises as to how many atoms of each
kind are to be found in the molecules of the better known
substances. The solution to this problem will give us the
relative weights of the molecules and the atoms.
If, for example, we find that the water molecule contains
p atoms of hydrogen and g atoms of oxygen, we can easily
O
h
the mass hf of the hydrogen atom. Each molecule of water
will in fact contain a mass p X h of hydrogen and a mass
q X o of oxygen ; now, since all water molecules are identical
each will contain hydrogen and oxygen in the same propor-
tion as any mass of water whatsoever—that is to say (accord-
ing to the well known analytical result), 1 part .of hydrogen
to 8 parts of oxygen. The mass qg x o should therefore
weigh 8 times as much as the mass p x h, from which we get

for the quotient ; the value 8 x : , which will be known

obtain the quotient 7 for the mass o of the oxygen atom by

when p and q are known.

At the same time we shall know the relation between the
masses (or weights) of the water molecule and its constituent
atoms. Since by weight p atoms of hydrogen make one-
ninth and gatoms of oxygen make eight-ninths of 1 moleculem

9
of water, the two ratios 7 and < are necessarily 9 p and 3d:

If now we know the atomic composition of some other
hydrogen compound, say of methane (which contains
3 grammes of carbon to 1 gramme of hydrogen) we may

c
obtain by quite similar reasoning the ratio h of the mass of

the carbon atom to that of the hydrogen atom ; then the

/ /

ratios =; 5 of the mass m’ of 1 methane molecule to the

/

masses of its constituent atoms. Knowing h and 7 we

_ m
then arrive, by simple division, at the ratio = of the masses

of the water and methane molecules.
It is thus obvious that it is sufficient to know the atomic

CHEMISTRY AND THE ATOMIC THEORY § 13

composition of a small number of molecules to obtain, as we
have shown, the relative weights of the different atoms (and
of the molecules under consideration).

$.—PROPORTIONAL NUMBERS AND CHEMICAL FORMULA.—
Unfortunately gravimetric analysis, which, by demonstrat-
ing chemical discontinuity, has led us to formulate the
atomic hypothesis, provides no means for solving the prob-
lem that has just been propounded. To make this quite
clear we may state that the laws of discontinuity are all sum-
marised in the following (law of ‘‘ proportional numbers’’) :—

Corresponding to the various simple substances :

Hydrogen, Oxygen, Carbon .
we are able to find numbers (called proportional numbers) :
| NOME & Jae, Omori

such that the masses of the simple substances found in compounds
are to one another as :

WEE? GUN es

p,q,7 ... . being whole numbers that often are quite simple.

We can therefore express all that analysis can tell us about
the substance under investigation by representing it by the
chemical formula :—

TR Reg ee Se

' It would not meet the case to state that these numbers are whole numbers.
Let 7 and y be the masses of hydrogen and carbon combined together in an
analysed specimen of a hydrocarbon. These masses are only known to a certain
degree of approximation, which depends on the accuracy of the analysis.
However exact the analysis may be, and even if H and C were numbers chosen
quite at random, there would always be whole numbers p and r which, within the
limits of experimental error, satisfy the equation :—

1p H
+ ro ;
But the smaller values possible for p and q must increase as the analytical

2
accuracy increases. If for an accuracy of within 1 per cent. we had found 3 asa

2027
possible value for f then the simplest possible value should become, say, 3041

when the degree of accuracy gets within one in a hundred thousand. Such,
however, is not the case, and the value : still holds with the latter degree of

accuracy. The law is that the ratios of the whole numbers p, q, 7... - have
fixed values which appear to be simpler, and the more surprisingly so the higher
the degree of accuracy attained.

14 ATOMS

Let us now replace any one of these terms in the series of
proportional numbers, say C, by a term C’, obtained by

2
? 3?
and let the other terms remain unchanged. The new series:

Be 0

is still a series of proportional numbers. For the compound
that contains, for instance, pH grammes of hydrogen to gO
grammes of oxygen and rC grammes of carbon also contains
2 pH grammes of hydrogen to 2 gO grammes of oxygen and
3 rC’ grammes of carbon. Its formula, which was H,,, O,, C,,
may now be written :

multiplying C by a simple arbitrary fraction, for instance

’
H,,,, O, C 3r?

209
and if p, g, r are whole numbers, 2p, 2q, and 3r will be whole
numbers also.

Moreover, the new formula may possibly be simpler than
the old.. A compound having originally the empirical
formula H, C, gets with the new proportional numbers the
formula H, C,, or, in other words, the formula HC.

Each of the two formule completely expresses all the
information given by analytical chemistry. We therefore
ean obtain from analysis no evidence that will decide whether
the atoms of carbon and hydrogen are in the ratio of C to H
or of C’ to H, nor any means of estimating how many atoms
of each kind a given molecule contains.

in other words :

There is a wide choice of distinct series of proportional
numbers—distinct in the sense that they do not give the
same formula to the same compound. We pass from any
one series to another by multiplying one or more terms by
simple fractions. Neither analytical chemistry, however,
nor the laws of discontinuity furnish the slightest clue to the
recognition, among the possible series, of one in which the
terms are in the same ratio as the masses of the atoms
(supposing the latter to exist).

10.—SitmiLtar Compounps.—Fortunately there are other

1 Two series are not distinct if one is got by nine yee all the terms in the
other by the same number.

CHEMISTRY AND THE ATOMIC THEORY | 15

considerations that can help us in our choice, which, from
the point of view of analysis alone, must remain indeter-
minate. And in fact we have never seriously hesitated in
our choice between more than a few lists of proportional
numbers.

For from the first it has been held that analogous formule
must be used to represent compounds that resemble each
other. We find a case of this resemblance between the
chlorides, bromides, and iodides of any given metal. These
three salts are tsomorphous, that is to say, they have the
same crystalline shape} and may be made (by the evapora-
tion of a mixed solution) to yield mixed crystals which still
retain that shape (these mixed crystals are homogeneous solid
mixtures of arbitrary composition). In addition to this
physical resemblance, which in itself is sufficiently remark-
able, the three salts resemble each other in their several
chemical reactivities. The atoms of chlorine, bromine, and
iodine therefore probably play very similar rdles, and their
masses are probably in the same ratios as the masses of the
three elements that combine with a given mass of the same
metal. This at once enables us to eliminate from the series
of proportional numbers which a priori might provide atomic
ratios, all these in which the proportional numbers Cl, Br, [,
corresponding to chlorine, bromine and iodine, are not in
the ratios of 71 to 160 to 254.

The probable values for the atomic ratios of various alkali
metals are readily obtainable, and this will still further reduce
the number of possible series. But we shall not (or, at any
rate, we have not up to the present) obtain by this means the
ratio between the atomic masses of chlorine and potassium,
since these two elements do not play analogous rdles in any
class of compound. Nor, up to the present, have we been
able to pass, by any definite isomorphic relation or chemical
analogy, from one of the alkali metals to one of the other
metals.

In short, the original lack of definiteness, so discouraging
at the outset, is thus very considerably diminished. It has

' In the crystallographic sense ; it is possible to orientate two such crystals
so that each facet of one is parallel to a facet on the other.

16 ATOMS

not been done away with altogether. And, to quote an
example which has provided material for much lively con-
troversy in the past, the study of isomorphism has furnished
no adequate reason for giving to water the formula H,O rather
than HO; that is to say, for assigning the value 16 to the

Oe:
ratio H instead of 8.

11.—EQuIvALENTS.—We must remember that, for many
chemists, who felt that little importance could be attached
to the atomic theory, the question had no great interest.
It appeared to them more dangerous than useful to employ
a hypothesis deemed incapable of verification in the exposi-
tion of well-ascertained laws. They also held that there
was nothing to guide their choice among the possible series
of proportional numbers except the one condition that the
facts should be expressed in language as clear as_ possible.
The use of the hypothesis was of advantage in that the
memorisation and prediction of reactions was facilitated,
and the representation of similar compounds by analogous
formule was made possible ; but, apart from this, it only
remained to assign the simplest formule to compounds
deemed the most important. For instance, it seemed
reasonable to write HO for the formula of water, thus
arbitrarily choosing the number 8 from the possible values

for the ratio =

In this way scientists hostile or indifferent to the atomic
theory agreed in using a particular series of proportional
numbers under the name of “ equivalents.’ This equivalent
notation, adopted by the most influential chemists and pre-
scribed in France in the curriculum used in elementary
schools,! hindered the development of chemistry for more
than fifty years. In fact, putting all question of theory on
one side, it has shown itself very much less successful in
representing and suggesting phenomena than the atomic
notation proposed by Gerhardt about 1840. In this notation
those proportional numbers are used that Gerhardt and his
successors, for reasons which will appear presently, regarded

1 Until about 1895.

CHEMISTRY AND THE ATOMIC THEORY § 17

as giving the ratios of atomic weights that isomorphism
and chemical analogy had not been able to determine.

AvoGADRO’s HyPorTHEsIs.

_12.—Laws.or Gaseous EXPANSION AND CoMBINATION.—
The considerations that have made these most important
numbers known to us depend upon the now familiar gas laws.

To begin with, it has been known since the time of Boyle
(1660) and Marriotte (1675) that at a fixed temperature the
density of a gas (mass contained in unit volume) is pro-
portional to the pressure.! Let, therefore, n and n’ be the
numbers of molecules present per cubic centimetre of two
different gases at the same temperature and pressure. If
we multiply the pressure common to the two gases by the
same number, say 3, the masses contained per cubic centi-
metre are multiplied by 3, and, consequently, the numbers

: ae ae
n and n’ also; for a given temperature the ratio a of the

numbers of molecules present per cubic centimetre in the
two gases at the same pressure is independent of that
pressure.

Further, Gay-Lussac showed (about 1810) that, at fixed
pressure, the density of a gas varies with the temperature in
a manner independent of the particular nature of the gas ?
(thus oxygen and hydrogen expand equally as their tem-
perature is raised). Consequently in this case also, since
the numbers 7 and n’ change in the same way, their ratio
does not alter.

In short, within the limits of applicability of the gas laws,
whether hot or cold, under high or low pressure, the numbers
of molecules present in two equal globes of oxygen and
hydrogen remain in a constant ratio, provided the tempera-
ture and pressure are the same in the two globes. And
similarly for all gases.

As a matter of fact, we are dealing with a law valid only within limits Itis
fairly well satisfied (to within about 1 per cent.) for the various gases when their
pressure is less than ten atmospheres, much better at still lower pressures, and
apparently becomes rigorously exact as the density tends to zero.

* Here again the law is limited in its application, being better satisfied the

smaller the density.
A. Cc

18 ATOMS

These various fixed ratios must be simple. This appears
from other experiments carried out about the same time
(1810) by means of which Gay-Lussac showed that :—

The volumes of gas that appear or disappear in any reaction
are in simple ratios to each other .+

An example will make this clear. Gay-Lussac found that
when hydrogen and oxygen combine together to form water,
the masses of hydrogen, oxygen, and water vapour that are
concerned, when reduced. to the same conditions of tempera-
ture and pressure, occupy volumes that are to one another
exactly as 2:1:2. Let » be the number of oxygen mole-
cules per cubic centimetre and n’ the number of water vapour
molecules. . The oxygen molecule contains a whole number,
which is probably small, say p, of oxygen atoms. The
water molecule similarly contains p’ atoms of oxygen. If
no oxygen is lost, the number mp of the atoms making up the
oxygen that disappears must equal the number 2n’p’ present

in the water that makes its appearance. The ratio 7’ 8

therefore equal to oP and is moreover a simple fraction,

since p and p’ are small whole numbers.

But as yet we have had no indication that the matter is
even simpler than might be supposed ; in other words, that
the numbers n and 7’ must invariably be equal.

13.—Avocapro’s Hyporuesis.—The famous hypothesis
of Avogadro (1811) asserts this equality. Having made the
preceding observations on the subject of Gay-Lussac’s laws,
this chemist laid it down that equal volumes of different gases,
under the same conditions of temperature and pressure, contain
equal numbers of molecules. The hypothesis may also be
enunciated with advantage as follows :—

When in the gaseous condition, equal numbers of molecules
of any kind whatever, enclosed in equal volumes at the same
temperature, exert the same pressure.”

1 As a matter of fact, Gay-Lussac did not draw from this statement the
proposition in molecular theory that is indicated here.
2 Tt is obvious that the hypothesis, supposing that it holds good, will be the
more rigorously applicable the more accurately the laws of “ perfect ” gases are
obeyed ; that is to say, the smaller the gas density.

CHEMISTRY AND THE ATOMIC THEORY 19

This proposition, which was at once defended by Ampeére,
-provides, if it be true, as we shall see later and as Ampére
pointed out, ‘‘a method for determining the relative masses of
the atoms and the proportions according to which they enter
into combination.” But Avogadro’s theory, put forward
as it was accompanied by other inexact considerations, and
being as yet without sufficient experimental foundation,
was received by chemists with great suspicion. We owe
the recognition of its supreme importance to Gerhardt, who,
not content with vague suggestions that had convinced
nobody, proved in detail! the superiority of the notation
which he deduced from the theory and which from his time
forward has gained so many adherents that it is now accepted
everywhere without opposition. An account of these past
controversies would have no interest for us at this point, and
we are solely concerned with understanding how Avogadro’s
hypothesis is able to give us the ratios of the atomic weights.

14.—AtTomic COoOEFFICIENTS.—Let us imagine certain
absolutely identical vessels, of volume V, filled with the
various pure substances known in the gaseous state, at the
same temperature and pressure. If Avogadro’s hypothesis
is correct, the gaseous masses thus obtained will contain the
same number of molecules, say N, which is proportional
to V.

Let us consider the hydrogen compounds in particular.
In every case the molecule contains the mass h grammes of
the hydrogen atom a whole number of times, say p; the
corresponding containing vessel therefore contains Nph
grammes of hydrogen, that is, p times H grammes, where
H. is the product Nh, which is independent of the given
substance since N is the same for all. Hqual volumes of
different hydrogen compounds therefore all contain a simple
multiple of a fixed mass of hydrogen.?

1“ Précis de chimie organique.”

2 But the reverse is not necessarily the case ; thus, suppose that Avogadro’s
hypothesis is incorrect, so that N, N’, NN”... . are the numbers of molecules
in volume V of various hydrogen compounds ; let p, p’, p” be the whole
numbers of hydrogen atoms present in each respective gaseous molecule. To
say that the masses of hydrogen Nph, N’p’h, N’p’h . . . . contained in equal

volumes V are simple multiples of a fixed mass Ht only implies that Np, N’p’
N’p”, . . . . and consequently N, N’, N” . .. . are to one another in penis

c2

20 ATOMS

Similarly, for the oxygen compounds, each of our contain-
ing vessels must contain a whole number of times, say q, the
mass O grammes of oxygen (which is equal to No, where o is
the mass of an oxygen atom); for the carbon compounds,
each vessel must contain 7 times (7 being a whole number)
the mass C grammes of carbon (equal to Ne, ¢ being the mass
of the carbon atom); and so on. Since finally the numbers
H, O,C ... . are proportional to N, we could, if desired,
choose the volume V in such a way that one of these numbers,
say H, has any desired value, unity, for instance. All the
others will then be fixed. |

These consequences of Avogadro’s hypothesis have been
fully confirmed by chemical analysis and the measurement of
densities in the gaseous state, for thousands of substances, no
single exception! having been discovered. At the same
time, the numbers H, O, C . . . . corresponding to every
value of the volume V are available.

In other words, temperature and pressure being fixed, a
volume V can be found (about 22 litres under normal con-
ditions *), such that those of our containing vessels that
contain hydrogen will contain exactly, or almost exactly,
1 gramme (hydrochloric acid, chloroform), or almost exactly
2 grammes (water, acetylene, hydrogen), or almost exactly
3 grammes (ammonia), or almost exactly 4 grammes
(methane, ethylene), or almost exactly 5 grammes (pyridine),
or almost exactly 6 grammes (benzene), but never inter-
mediate qualities, such as 1-1 or 3-4 grammes.

For the same volume each vessel will contain, if the simple
substance oxygen enters into the composition of the com-
pound enclosed therein, either exactly 16 grammes of oxygen

ratios. This is the important proposition deduced in para. 12 from the law of
gaseous combination (which, incidentally, is thus established on a far broader
experimental basis than was available to Gay-Lussac); it is not, however,
Avogadro’s more exact theorem.

1 We might regard bodies such as nitrogen peroxide (see para. 2), which do not —
obey the laws of Boyle and Gay-Lussac, and which consequently do not come
within the scope of the present discussion, as constituting exceptions. But we
have pointed out that nitrogen peroxide does not obey the gas laws, because it
is not a single gas but a mixture in varying proportions of two gases. Analogous
remarks apply to certain anomalous cases that at first sight seem important
(e.g., sal ammoniae vapour).

2 At the temperature of melting ice and under atmospheric pressure (76 cms.
of the barometric mercury column, at Paris).

CHEMISTRY AND THE ATOMIC THEORY 21

(water, carbon monoxide), or exactly twice 16 grammes
(carbon dioxide, oxygen), or exactly 3 times 16 grammes
(sulphuric anhydride, ozone), etc. ... . but never inter-
mediate quantities, such as 5-19 or 3-7 grammes.

Still at the same volume, our containing vessels will
contain either no carbon at all or exactly 12 grammes of
the substances (methane, carbon monoxide), or exactly
twice 12 grammes (acetylene), or exactly 3 times 12 grammes
(acetone), etc., always without intermediate quantities.

Similarly the vessels will contain, if chlorine, bromine, or
iodine exist therein, a whole number of times 35-5 grammes
of chlorine, 80 grammes of bromine, and 127 grammes of
iodine, so that (according to Avogadro’s hypothesis) the
masses of the three atoms corresponding should be to one
another as 35-5: 80:127. It is very remarkable that we
should obtain in this way numbers in the very same ratio
that was suggested by the isomorphism and chemical analogies
between chlorides, bromides, and iodides (para. 10). This
agreement obviously supports Avogadro’s hypothesis.

Thus step by step it has been possible to obtain experi-
mentally, from the densities of gases, a series of remarkable
proportional numbers—

Pet 0 = 16. C= 19.2 Ol = 35S) SS.

which are in the same ratios as the atomic weights, if
Avogadro’s hypothesis is correct, and which are, at any rate
for those among them that can be subjected to the test,
quite in accordance with the ratios fixed already by the facts
_ of isomorphism and chemical analogy.

For the sake of brevity, it has become customary to call
these numbers atomic weights. It is more correct (since they
are numbers and not weights or masses) to call them atomic
coefficients. Moreover, it is customary to speak of the mass
of a simple substance that, in grammes, is measured by its
atomic coefficient as a gramme atom of that body: 12
grammes of carbon or 16 grammes of oxygen are the gramme
atoms of carbon and oxygen.

15.—DvuLoneG AND Petit’s Law.—We shall now, in order
to deal with the isomorphism and analogies between simple

22 ATOMS

substances that form no volatile compounds, examine more
closely the atomic ratios of all the simple substances. Where
some uncertainty still exists in regard to a small number of
metals that show no obvious analogies to substances having
atomic weights that are already known, we can remove it by
_ the application of a rule discovered by Dulong and Petit.

According to this rule, when the specific heat of a simple
substance in the solid state is multiplied by its atomic weight,
very nearly the same number is obtained in all cases; this
number is about 6. We may express this result more clearly
as follows :—

In the solid state nearly the same quantity of heat is required,
namely, about 6 calories, to raise the temperature of any gramme
atom through 1° C.

If, therefore, there is any doubt as to the value to be
assigned to an atomic coefficient—for example, to that of
gold—we need only observe that the specific heat of gold is
‘03 to conclude that its atomic coefficient must be in the
neighbourhood of 200. It can then be accurately fixed by
the chemical analysis of gold compounds; gold chloride,
for instance, contains 653 grammes of gold to 35-5 grammes
of chlorine, so that the ‘‘ atomic weight ”’ of gold must be a
simple multiple or sub-multiple of 65-3. Seeing that it must
be in the neighbourhood of 200, it is therefore most probably
equal to 197, which is 3 times 65-7.

It goes without saying that a determination of this kind,
depending as it does upon an empirical rule, cannot be held
to have the same value as those based upon isomorphism and
Avogadro’s hypothesis. Such a reservation is all the more
necessary since certain elements (boron, carbon, silicon) do
not obey Dulong and Petit’s rule with certainty, at any rate
at ordinary temperatures.1 The number of such exceptions,
and the seriousness of the discrepancies they show, increases
moreover as the temperature falls and the specific heat
ultimately tends towards zero ? for all elements (Nernst), so

1 The specific heat of the gramme atom is at ordinary temperatures, instead
of 6, 4-5 for silicon, 3 for boron, 2 for carbon.

2 Cf. the recent work of Dewar (Proc. Roy. Soc., 1913, A 89); he has shown
that the specific heat of the elements at very low temperatures is a “ periodic ”’
function of their atomic weights ['TR.].

CHEMISTRY AND THE ATOMIC THEORY 23

that the rule becomes entirely false at low temperatures
(for instance, the atomic heat for diamond below — 240° C.
is less than -01).

We cannot, however, regard the very numerous instances
of agreement pointed out by Dulong and Petit (and after-
wards by Regnault) as entirely fortuitous, and we need only
modify their statement, giving it the following form, which
includes all recent results :—

The quantity of heat required to Said at constant volume,
the temperature of a solid mass through 1° C. is practically
nothing at very low temperatures, but increases as the tem-
perature rises, finally becoming very nearly constant.” It
is then about 6 calories per gramme atom, independent of
the nature of the atoms composing the solid mass.

This limit is reached the more rapidly with the elements
of higher atomic weight ; thus it is practically reached in
the case of lead (Pb = 207) at about—200° C., and in the case
of carbon not until above 900° C.

It is important to remember that compound substances
obey the law. ‘This is the case at ordinary temperatures for
the fluorides, chlorides, bromides, and iodides of various
metals, but not ‘for oxygen compounds. A piece of quartz
weighing 60 grammes, made up of 1 gramme atom of silicon
and 2 of oxygen, absorbs only 10 calories per degree. But
above 400° C.? it absorbs uniformly 18 calories per degree,
which is exactly 6 for each gramme atom.

We are led to suspect that some important law lies behind
the above facts ; the atomic notation has brought it to our
notice, but the kinetic theory alone is able to furnish an
approximate explanation of it (para. 91).

16.—A CorRrECcTION.—We have seen that one of the atomic

1 The heat used up in the form of work done against the forces of cohesion can
easily be calculated if the compressibility is known, and must be deducted,
according to Nernst, from the gross value obtained in the usual determination
of specific heat. Ultimately (cf. Pierre Weiss’ work on ferromagnetic bodies) it
would be necessary to subtract the heat required to destroy the natural
-magnetisation of the body. In order to obtain ‘“ absolute” results, only that
portion of the heat absorbed must be taken into account that appears to be
concerned in increasing the potential and kinetic energy of the various atoms,
which are maintained at a constant mean distance from each other.

$ a course, if the body melts or volatilises, the above proposition no longer
applies.

* According to Piouchon’s measurements, carried out up to 1,200° C.

De
>
.

24 ATOMS:

coefficients is arbitrarily fixed, and we have agreed that the
smallest among them, that of hydrogen, should be taken as |.
This, indeed, was the convention first adopted, and it gives,
as we have seen, 16 and 12 for the atomic coefficients of
oxygen and carbon. But more accurate measurements sub-
sequently showed that these values are somewhat too high’
by about 1 per cent. It then seemed desirable to alter the
original convention and to agree to give to oxygen (which
takes part more often than hydrogen in well-defined quanti-
tative changes) the atomic coefficient 16 exactly. Hydrogen
then becomes, to within 1 part in about 2,000, 1-0076 (as
the mean of concordant values obtained by very different
methods). Carbon remains 12-00 to within less than 1 part
in 1,000. |

Beyond this the preceding considerations require no further
qualification, except that the volume V of our identical recep-
tacles (filled with various gaseous substances at a fixed
temperature and pressure) are to be regarded as chosen so —
that those which contain oxygen will contain exactly 16
grammes or some multiple of 16 grammes.

17..—Prout’s HypotuEsis : MENDELEJEFF’S RuLE.—We
have seen, in the preceding paragraph, that the difference
between the atomic coefficients of carbon and oxygen is
exactly 4, which is very nearly 4 times the coefficient of
hydrogen.! To account for this and other similar cases
Prout supposed that the different atoms are built up by the
union, without loss of weight (into extremely stable com-
plexes, which cannot be decomposed), of a necessarily whole
number of proto-atoms, all of the same kind, which are the
universal constituents of all matter and which are possibly
identical with our hydrogen atoms or perhaps weigh 2 or 4
times less. :

Exact determinations which have accumulated since show
that Prout’s hypothesis, in this its simplest form, is untenable.
The proto-atom, if it exists, weighs much less. That the
hypothesis has nevertheless some claim to be retained
becomes obvious on reading through the list of atomic
coefficients, of which the first twenty-five are printed below,

1 And equal to that of helium (see para. 108).

CHEMISTRY AND THE ATOMIC THEORY 25

in order of increasing magnitude (with the exception of one

juxtaposition, of little importance, in the case of argon).
Hydrogen, H = 1-0076.

Helium, He = 4-0; lithium, Li = 7-00; glucinum, Gl = 9-1;

boron, B = 11-0; carbon, C = 12-00; nitrogen, N = 14-01;

oxygen, O = 16; fluorine, F = 19-0.

Neon, Ne = 20:0; sodium, Na = 23:00; magnesium,
Mg = 24:3; aluminium, Al = 27-1; silicon, Si = 28-3;
phosphorus, P = 31:0; Sulphur, S = 32-0; chlorine,
Cl = 35-47.

Argon, A = 39:9; potassium, K = 39-1; calcium,
Ca = 40-1; scandium, Sc = 44; titanium, Ti = 48-1;
vanadium, V = 51-2; chromium, Cr = 52-1; manganese,
Mn = 55-0.

If the values of the atomic coefficients were distributed at
random, we should expect five out of these twenty-five
elements to have a whole number for coefficient to within
-1,1 whereas, excluding oxygen (for which a whole number
coefficient has been assumed), twenty elements are found to
do so. We should expect that one element would have a
whole number coefficient to within -02, whereas this is
actually the case with nine of them. A cause at present
unknown therefore maintains the majority of the differences
between atomic weights in the neighbourhood of whole numbers.
We shall see later that this cause may be connected with a
spontaneous transmutation of the elements.

It may be objected that we are limiting our considera-
tions to the smaller atomic coefficients; as a matter of |
fact, we might extend our list without finding cause to
alter our conclusions. But an uncertainty of the order
of -25 is frequent among the higher atomic weights, which
therefore cannot be taken into account in the present
discussion.

Another very surprising regularity, pointed out by
Mendélejeff, is brought out in the preceding list of coefficients,
in which helium, neon, and argon (of zero valency) ; lithium,

! For the fifth part of a large number of points marked at random along a scale

graduated in centimetres subdivided into millimetres fall into the sections, each
_2 millimetres wide, that contain the centimetre divisions.

26 ATOMS

sodium, potassium (univalent alkali metals); glucinum,
magnesium, calcium (divalent alkaline earth metals), and
so on, are found in corresponding positions. We have here
an indication of the following law, which, however, cannot
now be discussed at any length:

When atoms are arranged in the order of their ascending
atomic masses, we find, at least approximately, that analogous
atoms appear periodically, starting from any particular atom.

It may be of interest to recall that after Mendélejeft
had pointed out two probable gaps in the atomic series,
between zinc (Zn = 65-5) and arsenic (As = 75), these
gaps were soon filled in by the discovery of two elements,
namely, gallium (Ga = 70) and germanium (Ge = 72),
having respectively the properties predicted for them. by
Mendelejeff.

No theory has up to the present given any explanation of
the Law of Periodicity.

18:.—GRAMME MOLECULES AND AVoGADRO’S NUMBER.—
In order to arrive at the atomic coefficients we have con-
sidered certain identical receptacles, full of various sub-
stances in the gaseous state, at the same temperature and
under a pressure such that there are exactly 16 grammes of
oxygen, or some multiple of 16 grammes, in those receptacles
containing oxygen compounds. The masses of pure sub-
stances that fill our receptacles under these conditions are
often called gramme molecules.

The gramme molecules of various substances are those masses
which in the rarefied gaseous state (at the same temperature and
pressure) all occupy equal volumes, the common value for these
volumes being fixed by the condition that, among those which
contain oxygen, the ones containing the least oxygen shall contain
exactly 16 grammes of it.

More briefly, but without bringing out the theoretical
significance of the theorem, we can say :—

The gramme molecule of a body is the mass of it in the
gaseous state that occupies the same volume as 32 grammes
of oxygen at the same lone and pressure (?.€., very
nearly 22,400 c.c. under ‘“‘ normal ”’ conditions).

_ According to Avogadro’s hypothesis, every gramme mole-_

CHEMISTRY AND THE ATOMIC THEORY 27

cule should be made up of the same number of molecules.
This number N is what is called Avogadro’s Constant or
Avogadro's Number.

Suppose that 1 gramme molecule contains | gramme atom
of a certain element ; in other words, suppose that each of
the N molecules of the gramme molecule is composed of
1 atom of that element, so that its gramme atom is made up
of N atoms. The mass of each of these atoms is then
obtainable by dividing the corresponding gramme atom by
Avogadro’s number, just as that of a molecule is obtained
by dividing the corresponding gramme molecule by this

16
number N. The mass o of the oxygen atom is NX the

1:0076
mass h of the hydrogen atom is > the mass co, of the

+4 .
carbon dioxide molecule is N? and-so on. Having found

Avogadro’s number, we should be able to find the masses
of all molecules and atoms.

19.—MotecuLtarR Formuta.—A gramme molecule con-
taining N molecules composed of p hydrogen, ¢ oxygen, and
ry carbon atoms also contains pH grammes of hydrogen,
gO grammes of oxygen, and rC grammes of carbon. The
formula H,0,C,, which clearly expresses the number of
atoms of each kind in the gramme molecule, is called a
molecular formula.

The examples given above (para. 14) in demonstrating how
gas densities indicate the atomic ratios show that the mole-
cular formula of water is H,O (and not HO), that of methane
being CH, and that of acetylene C,H,. It is also evident,
and moreover a point of considerable interest, that the
formula H, must be assigned to hydrogen (which thus
appears to be a diatomic compound), O, representing
oxygen and QO, ozone.1 There are monatomic molecules
also—such as those that make up the vapours of mercury,
zinc, and cadmium.

1 It is clearly no more logical to speak of an “ atom of oxygen ”’ than of an
‘atom of ozone.’ ‘To each variety of atom should correspond a name distinct

from the names of the various bodies that can be formed by the combination of
such atoms among themselves.

28 ATOMS

MoLECULAR STRUCTURE.

20.—SUBSTITUTION.—_The importance of the chemical
notation imposed by Avogadro’s hypothesis is particularly
well illustrated in the power it gives us of representing and
predicting chemical reactions. In particular, the idea of
chemical substitution, which is so important in organic
chemistry, is directly suggested by this notation.

Suppose that we mix chlorine with some methane, which
has the mloecular formula CH,, and that we expose the
mixture to the action (indirect) of light. The mixture
undergoes a change, and soon, besides hydrochloric acid, it
will be found to contain as components! four substances,
having the following molecular formule :—CH, (methane),
CH,Cl (methane monochloride or methyl chloride), CH,Cl,
(methylene dichloride,) CHCl, (chloroform), and CCl,
(carbon tetrachloride).

We pass from each formula to the next by writing Cl for
an H, and the question inevitably arises whether the corre-
sponding chemical reaction does not consist merely in the
substitution of 1 atom of chlorine for 1 atom of hydrogen
without further disturbance and without modification of
the molecular structure. However natural such a hypothesis
may seem, it is nevertheless still a hypothesis, for some
alteration might certainly be expected to result in the
situation and nature of the atomic unions when the grouping
loses 1 atom of hydrogen and gains 1 atom of chlorine.

21.—Awn ATTEMPT TO DETERMINE ATOMIC WEIGHTS FROM
PURELY CHEMICAL CONSIDERATIONS.—Some chemists have
thought to find in substitution an accurate means for
arriving at the ratios of the atomic weights, thus dispens-
ing with the necessity of appealing to gas densities and
Avogadro’s hypothesis. It seems desirable to give some
account of their line of reasoning, which, though instructive,
is certainly not sufficiently rigorous.

Thus if we could, when in complete ignorance of mole-
cular formule (which is the whole point), consider ourselves

1 Which could be separated by fractionation or simply identified in the
mixture, if it is assumed that we know how to prepare by other means these
same bodies in the pure state.

CHEMISTRY AND THE ATOMIC THEORY 29

justified in regarding it as probable that the hydrogen in
methane can be “replaced” in four stages, we could not
avoid the conclusion that the methane molecule probably
contains 4 atoms of hydrogen. Now this molecule, like any
other mass of methane, weighs (according to gravimetric
analysis) 4 times as much as the hydrogen it contains ; the
methane molecule therefore weighs 16 times as much as the
hydrogen atom. We should find, with a like degree of
probability and by similar processes, that the benzene
molecule contains 6 atoms of hydrogen and weighs 78 times
as much as one hydrogen atom. The molecular masses of
methane and benzene are thus in the ratio of 16 to 78.
Further, the carbon in the methane molecule (as in any mass
of methane) weighs 3 times more than the hydrogen it con-
tains, and hence is 12 times as heavy as the hydrogen atom ;
and this carbon probably constitutes a single atom, for no
substance studied in this way, by substitution methods,
ever gives a smaller ratio between the carbon contained in
its molecule and the hydrogen atom. The carbon in the
benzene molecule, which weighs 12 times as much as the
6 hydrogen atoms in it, that is to say 72 times as much as
the hydrogen atom, is therefore made up of 6 carbon atoms.

We “bie thus obtain, from a purely chemical standpoint,
the ratio = for the atomic mass of hydrogen to that of carbon,
with the molecular formule CH, and C,H, for methane and
benzene and the ratio < between their molecular masses.

Two masses of these substances in the ratio of 16 to 78
will therefore each contain as many molecules as the other.
Now density measurements show that the masses of methane
and benzene which, in the gaseous state, occupy the same
volume at the same temperature and under the same
pressure are to each other as 16 is to 78 exactly, and should
in consequence contain the same number of molecules.
This result, established on a general basis, would give us
Avogadro’s hypothesis, but this time as a law and not as a
hypothesis.

There will be no difficulty in filling in the details of this

30 ATOMS —

seductive theory, which has recently assumed great import-
ance in French education,! mainly owing to the efforts of
Lespieau and L.-J. Simon. Its yalue is undoubted in the
sense that it is only through the consideration of the
phenomena of substitution that we are able to obtain certain
molecular formule (such as that of acetic acid, for example).
I nevertheless am strongly of G. Urbain’s opinion that it is
not capable of providing practically and in a logical fashion
the ratios of the weights of all atoms.

In the first place, I know of no case where it has actually
been of use in obtaining atomic weights, all of them having
been fixed already by the means summarised above. -More-
over, whilst admitting that the theory might have developed
independently, I very much doubt whether it would have
proved convincing. Certainly, if we incautiously grant that
it is proved by experiment that the hydrogen in methane can
be replaced in four stages, then the rest follows. But would
the word “‘ replaced,” which the molecular formule, if we
suppose that they are known, at once suggest, have been
suggested by chemical reactions alone and by the exami-
nation, without preconceived ideas, of the products of
reaction ?

It is, of course, true that the products of the progressive
action of chlorine on methane resemble each other as closely
as do the various alums, for example, or the chlorides,
bromides, and iodides of the same metal. In the latter case
the analogy is so striking that the idea of substitution is
forced upon one (although as a matter of fact the word has
never been used in connection with such cases), and, indeed,
it has proved, as we have seen, a most valuable guide in our
choice of atomic weights, at a time when no other guidance
was available. .

On the other hand, it is doubtful whether chemists really
ignorant of the formula of methane would have been able
to recognise analogies between methane and methyl! chloride
complete enough to establish identity of molecular structure.
They might equally well have assumed (taking one only of

1 It has been prescribed (to the exclusion of any other) in the French scheme
for girls’ secondary education.

CHEMISTRY AND THE ATOMIC THEORY § 31

the possible hypotheses) that, taking the atomic weights of
earbon and hydrogen as 6 and 1, the formule of the two
substances in question are CH, and CH,.CH,Cl, thus
making methyl chloride an additive compound. And need
it be pointed out that, for half a century, the majority of
chemists, although perfectly well aware that potassium
displaces a portion only of the hydrogen in the water it
attacks, actually gave to water the formula HO and the
formula KO .HO to potassium hydroxide, thus regarding
the latter substances as an additive compound, whereas we
now look upon it as a substitution product of formula KOH,
because we have assigned to water the formula HOH 2

In short, a purely chemical theory that is able to yield us
atomic coefficients and molecular formule has not yet been
put forward, and it seems doubtful whether, starting with
the facts actually known, it is possible to formulate one that
does not tacitly assume a previous knowledge of the
coefficients and of certain fundamental molecular formule,
such as, for instance, that of water.

22.—MInIMUM INTERNAL DISLOCATION OF THE MOLECULE
DURING REACTION: VALENCY.—As we have seen above,
the possibilities of substitution suggested by the examination
of the molecular formulz enable us to predict and interpret
an immense number of reactions and in this way provide
a striking confirmation of Avogadro’s hypothesis. Fresh
hypotheses are necessary, however, to define and expand
the conception of substitution.

When we say that methane CH, and methyl chloride
CH,Cl have the same molecular structure, we imply that the
group CH; has not been modified by the chlorination and
that it is connected with the Cl atom in the same way that
it was with the H atom. This is a postulate constantly
used in chemistry ; we argue continually (without always
saying so clearly enough) as though the reacting molecule
always undergoes the smallest possible internal disturbance
compatible with reaction. It is assumed, for example, that
the group CH, in methyl chloride exists in the molecule
CH,0O of methyl alcohol (which is consequently written
CH,OH) because the action of hydrochloric acid HCl on this

32 | ATOMS

alcohol gives (together with water HOH) the methyl chloride
CH,Cl with which we are already familiar.

Thus, when a structure made up of parts held together
with screws and bolts is taken to pieces, it may be possible
to remove and keep intact the whole of one important part
and ultimately to incorporate it, making use of the same bolts
or fastenings, into a second structure. This rough image
makes it sufficiently clear how it is possible to have sub-
stitution, not only of one atom by another, but also of one
group of atoms by another group; and even the nature of
the union devised to maintain our imaginary: structure is
found to correspond well enough with our ideas on chemical
combination.

We have not yet put forward any suggestion as to the
nature of the forces that keep the atoms grouped together
within the molecule. It may be that each atom in the mole-
cule is joined to each of the others by an attraction that
varies according to their nature and decreases rapidly with
the distance between them. But such a hypothesis leads
to no verifiable conclusions and presents considerable
difficulties. If all hydrogen atoms are attracted by all other .
hydrogen atoms, why is it that the only molecule built up
of hydrogen atoms is H,, the capacity of the hydrogen atom
for combining with itself being exhausted directly two atoms
become united? It appears as though each atom of
hydrogen stretches out a single hand only. Directly this
hand succeeds in gripping another hand, the capacity for
combination of the atom is exhausted ; the hydrogen atom
is therefore said to be monovalent (or better, univalent).

Speaking more generally, we regard the atoms in a mole-
cule as being held together by hooks or “ hands ”’ of some
kind, each bond uniting two atoms only, without any dis-
turbing effect whatever on the other atoms present. Of course,
no one imagines that there actually are little hooks or hands
on the atoms, though the absolutely unknown forces that
unite them would seem to be equivalent to bonds of some
such kind, which are called valencies to avoid the use of:
expressions that are too anthropomorphic.

- If all atoms were monovalent a single molecule could never

CHEMISTRY AND THE ATOMIC THEORY ~ 33

contain more than two atoms; there must therefore be
polyvalent atoms. Since there is no limit to the number of
atoms in a molecule except that set by the latter’s fragility,
which becomes progressively greater the more atoms there
are in the molecule (whereas the number of children that
can form a circle by taking hold of hands is not limited).
Oxygen, for instance, is at least bivalent, since its atoms can
form the ozone molecule O, as well as the molecule O,.

The image we have used above serves to suggest: that
the number of valencies assumed by an atom may vary
from one compound to another. Ifa man with his two hands
is taken to represent a bivalent atom, it is obviously possible
for him to put one hand in his pocket and thus to represent
a monovalent atom ; finally, bringing into play a valency of a
different kind, he might seize an object with his teeth and thus
represent a trivalent atom, irrespective of the fact that in
more ordinary circumstances the possibility of his so doing
might be neglected.

Similarly every atom usually retains the same number of
valencies in the various compounds into which it enters.
We have never had any reason to suppose that hydrogen
is polyvalent ; chlorine, bromine, and iodine, which can
replace hydrogen atom for atom, are univalent also. Oxygen
is usually bivalent, as in water HOH, nitrogen being triva-
lent, as in ammonia NH, or pentavalent, as in ammonium
chloride NH,Cl, and carbon quadrivalent, as in methane
CH,. But the indisputable existence of molecules of nitric
oxide NO serves to remind us that oxygen and nitrogen
_ are not always bivalent and trivalent respectively ; again,
carbon and oxygen cannot both retain their usual valencies
in carbon monoxide CO. Obviously, if such anomalies were
of frequent occurrence, the notion of valency, though well
founded, would lose much of its usefulness.

23.—ConsTITUTIONAL ForMULa&.—When the conditions
under which a compound is formed are known, it is often
possible, by assuming that a minimum of internal dislocation
occurs, to make a complete determination of the manner in
which the atoms are united in the molecule of the com-
pound and of the number of valencies by which they are held

A, D

34 ATOMS

together. This is what is known as establishing the con-
stitution of the compound. ‘The result is open to doubt so
long as the constitution is fixed by a single series of reactions.
But the doubt is considerably lessened if several series of
different reactions point to the same constitution. Repre-
- senting each saturated valency by a line, we can then repre-
sent the compound by a constitutional formula, which will
possess a wide power of representation with respect to the
possible reactions of the compound. For example, we are
led by various paths to the opinion that the bonds in the
acetic acid molecule are expressed in the formula :

H O

| |
H—C—C—O—H -

tes,
H
which suggests at once the different rdles played by the
hydrogen atoms (three being replaceable by chlorine and the
fourth by a metal), by the oxygen atoms (the group OH
being removed in the formation of acetyl chloride CH,COCl),
and by the carbon atoms themselves (the action of a base
KOH on an acetate CH,COOK splits the molecule up into
methane and carbonate).

Constitutional formule have taken a position of capital
importance in the chemistry of carbon. I shall draw atten-
tion to the readiness with which they explain the difference
in properties between isomeric substances (molecules made
up of the same atoms united in different ways ') and enable
us to predict the number of possible isomers. But I cannot
dwell at greater length on the services they have rendered
to chemistry, and must content myself with the observation
that the 200,000 constitutional formule ? with which organic
chemistry is concerned provide just so many arguments

1d
Tae,
1 For example, ethanolal HO—-C—C—H, which possesses both an aldehydic
H

and an alcoholic function, is an isomer of acetic acid,
2 See Beilstein’s Dictionary.

CHEMISTRY AND THE ATOMIC THEORY — 35

in support of the atomic notation and the theory of
valency.!

24.—_STEREOCHEMISTRY.—Once the constitution of the
molecule is known, with regard to the modes of union of its
component atoms, we may ask ourselves, reasoning as though
the molecule were an almost rigid edifice of definite shape,
what may be the configuration in space of its various atoms.
We require to construct in some way a model in three dimen-
sions that will indicate the respective positions of the atoms
in space. This new problem, which at first sight would seem
to have no meaning (for the valencies might be expected to
behave like flexible bonds fixed to a mobile point on the
atom, and therefore permitting no definite configuration),
has advanced a step towards solution as a result of the
splendid work of Pasteur, Le Bel, and van’t Hoff, to which I
‘wish to make some reference.

Let us replace successively the four hydrogen atoms in a
methane molecule CH, by four monovalent groups R,, Ro,
Rg, and R,, which all differ from each other. If these four
groups could occupy any position whatever about the
carbon, one single substitution product only could then be
obtained. Now two are found, actually very analogous and
identical even in certain particulars (they have the same
melting points, the same solubility, the same vapour pressure,
etc.), but differing sharply in other respects. Their crystals,
for instance, which at first sight seem identical, differ in the
same way that right-hand gloves differ from left-hand ones,
the two kinds being, of course, not mutually replaceable.

Such isomerism is comprehensible if we suppose that the
four carbon valencies are attached to the four corners of a
practically indeformable tetrahedron. Now there are two
non-superimposable ways in which four different objects can

1 It is moreover possible, and even probable, that, independently of the
valencies proper, bonds of a different nature and not so powerful, though equally
limited in saturation capacity, may exist between atoms or molecules, giving
rise to “ molecular compounds ”’ such as double salts or complex salts, which are
met with more especially in the solid state. Merely to illustrate the possibility
of different kinds of union, we may imagine that the ordinary valencies are due
to electrostatic attraction and that, in addition, 2 molecules (or even 2 atoms)
may. attract one another like magnets, which can form astatic systems having

no external magnetic action (cf. polymerisation by doubling of the molecule,
which is frequently observed).

D2

36 ATOMS

be distributed at the corners of such a tetrahedron, ‘and the
two arrangements are symmetrical with respect to a mirror
as a right-hand glove is to a left-hand one. If, moreover,
the tetrahedron is not regular, more than two arrangements
producing different solids would be possible (and it should
therefore be possible to obtain several di-substitution
derivatives having the same formula CH,R’R”, which is
contrary to experience).

It therefore seems probable that the molecular edifices are
to be regarded, at least approximately, as solid structures,
the configuration of which stereochemistry (from drepeos =
solid) aims at determining. Rigidity of the bonds between
atoms will appear even more probable when the specific
heats of gases (para. 42) have been discussed.

SoLUTION.

25.—Raovutt’s Laws.— The physical and chemical
methods that have been described above are not always |
sufficient to fix the constitution or even the molecular
formula of certain substances. Fortunately a valuable
auxiliary is found in the experimental study of dilute
solutions.

‘The formule of certain non-volatile substances are
difficult to determine. This is the case with numerous
‘‘ carbohydrates ’’ which by analysis can only be proved to
have the formula C,,H,,0,, their chemical properties not
always being sufficient to determine n.

Now for a long time it has been definitely known that
when a non-volatile substance is soluble in a liquid, for
example in water, the solidifying temperature is lower, the
vapour pressure less, and the boiling point higher, than is
the case with the pure solvent. Thus sea water solidifies
at — 2° C. and boils (under normal conditions) at
100°6° C.

But from the restricted study of aqueous solutions of salts
it has not been found possible to give precision to these
qualitative rules. From his experiments on sclutions that,
in contradistinction to saline solutions, are not noticeably

CHEMISTRY AND THE ATOMIC THEORY © 37

conductors of electricity and hence are not “ electrolytes,”
Raoult established the following laws (1884) :—

(1) The influence of each dissolved substance is proportional
_ to its concentration.1 The lowering of the freezing point is
5 times greater for a sugar solution that contains 100 grammes
of sugar per litre than for one containing only 20 grammes.

(2) Any two substances exert the same influence when their
molecular concentrations are equal. More strictly, two
solutions (in the same solvent) that in equal volumes contain
the same number of gramme molecules have the same
freezing point, the same vapour pressure, and the same
boiling point. :

For the present it is sufficient to recognise the facts
embodied in these rules; it may, however, be added that
Raoult’s more complete statements express the influence due
to a given molecular concentration. If n gramme molecules
(of any kind) are dissolved in # gramme molecules of a sol-
vent, which exerts a vapour pressure p, a solution being thus
obtained of vapour pressure p’, the relative lowering of

2p

vapour pressure, that is - , 1s sensibly equal to & “py
n

dissolving 1 gramme molecule of any substance in 100
grammes of solvent the vapour pressure is lowered by one-
hundredth of its value.”

With regard to all these laws, it is, of course, understood
that the solution must be dilute ; that is to say, the molecular
concentrations must be comparable with those at which gases
obey Boyle’s Law (more or less of the order of 1 gramme
molecule per litre).

It may reasonably be supposed that the above laws are
applicable to bodies having molecular formulz which we do
not yet know, as well as to those of known formule. If,
therefore, a mass m of a substance of unknown formula
produces in the boiling point of, for instance, an alcohol
solution a variation 3 times smaller than that produced by

' This law was stated before Raoult by Wiillner and Blagden, but with refer-
ence to electrolytes, the very substances for which it is inaccurate.

* The exact expressions relating to the variation in boiling point and freezing
point follow thermodynamically from the expression giving the variation in
vapour pressure ; further reference is not necessary here.

38 ATOMS

any of the known gramme molecules when dissolved in the
same volume, then the unknown gramme molecule is equal
to 3m. In this way our ability to determine molecular
coefficients is enormously increased.

26.—ANALOGY BETWEEN GASES AND DILUTE SOLUTIONS :
Osmotic PRreEssuRE.—Raoult’s laws, though clear and
precise, were nevertheless merely empirical rules. Van’t:
Hoff gave a deeper significance to them when he connected
them with the laws characteristic of the gaseous state, which
he was able to show apply also to dilute solutions.

The idea of certain laws being common to all attenuated
forms of matter, whether gaseous or in solution, was suggested
to him by various botanical researches on osmosis. All
living cells are enclosed by a membrane that allows water to
pass through but stops the diffusion of certain dissolved sub-
stances, the cell gaining or losing water according to the
concentration in the aqueous medium in which it is placed
(de Vries), which causes the pressure in the interior of the cell
to increase or diminsh (it is well known that flowers revive
when their stems are placed in pure water but “ fade ”’ if
the water contains salt or sugar).

Pfeffer succeeded in making indeformable artificial cells
which were enclosed by a copper ferrocyanide membrane
and which showed the properties described above.t_ When
one of these cells, fitted with a manometer and filled with
sugar solution, is placed in pure water, the internal pressure
steadily rises owing to the entry of water. Qn the other
hand, it is easy to show that no sugar leaves the cell. The
ferrocyanide membrane is said to be semi-permeable. The
excess of internal pressure over the external, moreover, tends
to a limit, proportional to the concentration for each tempera-
ture; this limit rises when the temperature is raised and
returns to its former value (the cell losing water) when
the original temperature is reached again. ‘This limiting

1 Battery jars, of porous porcelain, impregnated with a precipitated membrane
of copper ferrocyanide. The cell, previously soaked in water, is filled with a
solution of copper sulphate and placed in a solution of potassium ferrocyanide.
The precipitated membrane is formed in the pores of the porcelain, from which .
it cannot escape. ‘The cell is washed, filled with a sugar solution, and sealed
with a firm cement.

CHEMISTRY AND THE ATOMIC THEORY _ 39

difference, which is reached when equilibrium is attained, is
the osmotic pressure of the solution."

If then at the bottom of a cylinder we have a sugar solution,
~above which is pure water, separated from it by a semi-
permeable piston, we can concentrate or dilute the sugar
solution, according as we press on the piston with a
force greater or less than the force just required to balance
the osmotic pressure. Moreover, since this pressure, being
proportional to the concentration, is inversely proportional
to the volume occupied by the sugar, it would not be apparent,
considering only the work required for compression, whether
it was being applied to a gas or a dissolved substance.

Van’t Hoff, who regarded Pfeffer’s experiments from this
point of view, was led to the conclusion (van’t Hoff’s law)
that :— 3

All dissolved substances exert, on a partition that stops them
but which allows the solvent to pass, an osmotic pressure equal
to the pressure that would be developed in the same volume by a
gaseous substance containing the same number of pome
molecules.

Assuming Avogadro’ s hypothesis, this is the same as :—

Either as a gas or in solution, the same numbers of any kind
of molecules whatever, enclosed in the same volume at the same
temperature, exert the same pressure on the walls that confine
them.

Van’t Hoff’s theorem, when applied to sugar (which has a
gramme molecule of 342 grammes), gives to within 1 per cent.
the osmotic pressures measured by Pfeffer. This agreement,
though striking, might be accidental. But van’t Hoff
removed all doubts by showing that his theorem follows
necessarily from certain known laws. Thus, if Raoult’s
laws are exact, van’t Hoff’s law must necessarily be so also
(and vice versd).”

! The order of magnitude is: 4 atmospheres at ordinary temperatures for a

6 per cent. sugar solution.
2 This readily follows from the proof given below (Arrhenius). Let there be,
in a region where the gravitational intensity is g and in a vessel free from air, a
vertical column of solution in communication with pure solvent through a semi-
permeable plug. Let the solution contain » gramme molecules of the . dissolved
substance (non-volatile) in #2 gramme molecules of solvent. Let equilibrium be
reached when the difference in level between the two surfaces is h ; d is the

40 ATOMS

27.—lons-- ARRHENIUS’S HypotTuesis.—As yet we do not
know why it is that a conducting solution, such, for instance,
as salt water, does not obey Raoult’s laws (and consequently
van’t Hoff’s).

Let us first make clear the nature of this discrepancy ; a
mass of salt water containing 1 gramme atom of sodium
(Na = 23) and 1 gramme atom of chlorine (Cl = 35°5), which
is 1 gramme molecule 58°5 grammes of sodium chloride,
freezes at a lower temperature than the same volume of
solution containing 1 gramme molecule of a non-conducting
substance, such as sugar. As the dilution increases, the
ratio between the lowerings of freezing point produced by
1 gramme molecule of salt and 1 gramme molecule of sugar
increases and tends towards 2, so that, in very dilute solu-
tions, 1 gramme molecule of salt exerts exactly the same
influence as 2 gramme molecules of sugar.

This is just what might be expected to happen if, in
solution, the salt were partially dissociated into two com-
ponents that separately obey Raoult’s laws, and if, when the
dilution is very great, the dissociation were to become com-
(mean) density of the vapour, D the very much greater density of the solvent
(it is very nearly equal to the density of the solution). Let p’ and p be the
vapour pressures at the surfaces of the solu-
tion and solvent respectively. Then, from
the definition of the osmotic pressure P,
the pressure at the bottom of the solution is

(p + P). The fundamental theorem of
hydrostatics, applied to the solution and

ment tf its vapour, then gives :—
EE ery p— p = ghd
A |=-*=| Solution and p+P=p'+ghD
: whence, eliminating gh, we get approxi-
mately
,D_P—P Pp
that is to say, in accordance with Raoult’s
eee —3 law stated above,
Fi. 1. j p=.” p.
Ad:

Let v be the volume, in the gaseous state at pressure p, of 1 gramme molecule

Mot the s6l int 4 Pr)
of the solvent (so that d M

volume V that is occupied by a gramme molecule of the dissolved substance
when in solution, we then have :

knowing also that i is the.

| PV = pr,
which is van’t Hoff’s law.

CHEMISTRY AND THE ATOMIC THEORY 41

plete. We must therefore conclude that the molecules
NaCl split up into atoms Na of sodium and Cl of chlorine, and
that a very dilute salt solution does not really contain salt,
but sodium and chlorine in the form of free atoms. This is
the hypothesis that was put forward with such boldness and
supported with such brilliance by Arrhenius in 1887, he
being then a young man of twenty-five.

His conception appeared irrational to many chemists,
and this is all the more curious because, as Ostwald at once
pointed out, it was really quite in accordance with well-
known facts and also with the binary nomenclature used to
represent salts. Thus all the chlorides in solution have
certain reactions in common, whatever the metal associated
with the chlorine may be, which is readily explained if the
same kind of molecule (which can only be the Cl atom) is to
be found in all such solutions; with the chlorates, which
have a different set of reactions in common, the common
molecule would not be Cl but the group ClO,, and so on.

Disregarding this argument, the opponents of Arrhenius
held it to be absurd to assume the existence of free atoms
of sodium in water. ‘It is well known,” they said, “ that
when sodium is placed in contact with water the latter is
immediately decomposed with liberation of hydrogen. And
further, if chlorine and sodium do co-exist in a solution of
salt, simply mixed together like two gases occupying the same
vessel, should not means analogous to those applicable to
the gaseous state be also available for separating the two
elements from each other ; by superimposing, for example,
above the solution a layer of pure water into which the
constituents Na and Cl would certainly diffuse at unequal
rates? But attempts to separate them by such means fail,
not only in the case of ordinary salt (in which case the rates
of diffusion, as an exceptional case, might happen to be
equal), but for all electrolytes.”’

Arrhenius met these objections by insisting upon the fact
that the abnormal solutions conduct electricity. This con-
ductivity is explicable if the atoms Na and Cl, which one salt
molecule gives on dissociation, are charged with opposite
kinds of electricity (in the same way that discs of copper

42 ATOMS

and zinc become charged when separated after previous
contact). Speaking more generally, every molecule of an
electrolyte can dissociate in the same way into atoms (or
groups of atoms), electrically charged, called ions. It is
assumed that each ion of the same kind, each of the Na ions,
_ for instance, in a solution of NaCl, carries exactly the same
charge (necessarily equal therefore to the charge of opposite
sign carried by the Cl ion, since otherwise the salt solution
would not be electrically neutral, as is actually the case).
The N atoms which, when neutral, make up 1 gramme atom,
constitute, when in the ionic condition, what may be called
] gramme ion.

When placed in an electric field (such a field is produced
when positive and negative electrodes are placed in the salt
solution) the positive ions will be attracted towards the
negative electrode or kathode, and the negative ions will
move in a like manner towards the positive electrode or
anode. A double stream of matter in two opposite directions
will thus accompany the passage of electricity. On coming
in contact with the electrodes, the ions will lose their charges
and acquire other chemical properties at the same instant.

For an ion, which differs by reason of its charge from the
corresponding atom (or group of atoms), cannot possess at
all the same chemical properties as the latter. As a further
result of the charges, diffusion will not be sufficient to effect
a separation of the oppositely charged ions. It might well
happen, and this is in general the case, that certain ions,
say the positive, tend to move faster than the others. They
therefore charge positively the region in the liquid where
they are in excess ; but at the same time this charge attracts
the negative ions, which accelerates their rate of progression
and retards at the same time the positive ions. A dog may
be more active than a man, but if the dog is held on a leash
neither can get along faster than the other. .

28.—DEGREE OF DISSOCIATION OF AN ELECTROLYTE.—
Finally, the degree of an electrolyte’s dissociation into ions
can easily be calculated, for each temperature and dilution,
if it is assumed, with Arrhenius, that ions obey Raoult’s
laws as if they were neutral molecules. If a solution con-

CHEMISTRY AND THE ATOMIC THEORY 43

taining 1 gramme molecule of salt for each volume V has
the same vapour pressure that it would have if it contained
3 gramme molecules of sugar, then it is assumed that it
actually contains 3 gramme molecules, necessarily composed
of (1 — 2) undissociated gramme molecules of salt and 2 x 3
gramme ions, positive and negative. The degree of dissocia-
tion 2 is thus found by the application of Raoult’s laws.
On the other hand, let us consider a cylindrical column of
solution, having a cross-section such that the volume of a
section 1 cm. long is the volume we should expect to contain
1 gramme molecule, if we were not aware of its dissociation.
As a matter of fact it contains 2 of the ions it would contain
if the dissociation were complete. For the same electro-
motive force the quantity of electricity transmitted per
second will therefore be 3 of what would be transmitted at
extremely high dilution. More briefly, the conductivity of
our cylinder should be, for each centimetre of its length,
only 2 of a limiting conductivity that is reached at infinite
dilution. Now this is precisely what is found by experiment.
We find the same thing with other salts and at other
dilutions ; and the degree of dissociation calculated by the
application of Raoult’s laws is equal to that deduced from
the electrical conductivity (Arrhenius’s law). Such remark-
able concordance, which proves a fundamental connection
between properties at first sight as widely different as
freezing point and electrical conductivity (a connection so
intimate that the one can be predicted when the other is
known), clearly lends great support to Arrhenius’s theory.
29.—Tue First IpzEA or A Minimum ELEMENTARY
CHARGE.—We have just decided that all the Cl ions in a solu-
tion of salt bear the same charge, and we have attributed the
difference in chemical properties between the atom and the
ion to the existence of this charge. Instead of a solution
of sodium chloride, let us now consider one of potassium
chloride. Its chemical properties due to the presence of
chlorine ions (precipitation with silver nitrate, etc.) are the
same as with sodium chloride. The chlorine ions in potassium
chloride are therefore probably identical with those of
sodium chloride and consequently bear the same charge.

44 ATOMS

Since the solutions are electrically neutral, the sodium and
potassium ions must have the same charge also, but with
opposite sign. We are thus led step by step to the con-
clusion that all monovalent atoms or groups of atoms
(Cl, Br, 1, ClO; NOy.: andi Na; NE eee
when they become free in the form of ions, bear the same ~
elementary charge e, positive or negative.

Chlorine ions also possess the same properties, and there-
fore the same charge, in a solution of barium chloride
BaCl,. But in this case a single Ba ion only is formed
along with two Cl ions; the charge carried by the Ba ion,
which is derived from a bivalent atom, is therefore equiva-
lent, and opposite in sign, to twice the charge on the Cl ion.
Similarly the Cu ion, derived from copper chloride CuCl,,
carries two elementary charges ; so also does the sulphate
ion SO,, but the charges bear the same ‘sign as the Cl ion.
The trivalent lanthanum atom in the same way is found to
carry three elementary charges when separated from the
three chlorine atoms of the chloride LaCl, ; and so on.

An important relationship is thus brought out between
valency and ionic charge ; each valency bond ruptured in
an electrolyte corresponds with the appearance of a charge,
which is always the same, on the atoms held together by
that bond. Moreover, the total charge on an ion must
always be an exact multiple of this constant elementary
charge, which is, indeed, an actual atom of electricity.

The above view is completely in accordance with the
knowledge we have gained from the careful study of electro-
lysis. I feel that some account of this is very desirable,
since, in my opinion, the usual methods of presenting it are
not at all satisfactory.

30.—THE CHARGE CARRIED BY A GRAMME [ON—ELEC-
TRICAL VALENCY.—When two electrodes are placed in an
electrolyte, changes are at once observed to take place in
their immediate neighbourhood. Bubbles of gas, solid
particles, or drops of liquid make their appearance on the
electrode surfaces, rising or sinking according to their density
and so tending to contaminate regions which the passage
of the current alone would perhaps have left unaltered.

CHEMISTRY AND THE ATOMIC THEORY 45

Complications of this kind can be avoided by making the
current follow a curvilinear course; for example, in the
way indicated in the diagrammatic sketch shown below.
The electrolyte is divided into two parts, contained in two
beakers, an electrode being fixed in each. The two beakers
are connected by means of a siphon containing a column of
liquid, through which the current must pass but which
cannot be entered by substances rising or falling from the
electrodes. Precautions are taken, moreover, to prevent
the loss of these substances. |

It is then easy to show conclusively that the mere passage
of the current does not affect the electrolyte ; we have only

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to remove the siphon tube after the passage of a certain
quantity Q of electricity (which can easily be measured by
a galvanometer) and to analyse the liquid contained in it,
when the solution will be found to have undergone no
change.

At the same time the substances produced from the rest
of the materials concerned in the experiment have been
separated into two compartments. It will be possible to
analyse the contents of each compartment (including the
products formed on the electrodes) and to determine the
number of gramme atoms of each kind found therein.

Let us suppose that a solution of salt has been electro-
lysed. In the kathode compartment we shall find, first,

46 ATOMS

part of the kathodic material (supposing the kathode to
have been attacked); then (in terms of hydrogen,
oxygen, chlorine, and sodium) the materials that constitute
a salt solution ; and, finally, an excess of sodium, so that
the total composition of the compartment can be expressed
by a formula such as :—

(kathode) + a(2H + O) + 6(Na + Cl) + aNa.

It must be clearly understood that we are here dealing with
a formula expressing the gross composition of the contents of
the compartment, independently of any hypothesis as to the
particular compounds that may be present. It is immaterial
that the 2 a gramme atoms of estimated hydrogen are partly
in the form of gaseous hydrogen and partly combined in
water or sodium hydroxide molecules; we are concerned
with their total number only. pees

At the same time, since no matter has been lost, the total
formula for the anode compartment must be :—

(original anode) + a’(2H + O) + b’(Na + Cl) + 2Cl,
the number of gramme atoms 2 of chlorine present in excess
in this compartment being equal to the number of gramme
atoms of sodium present in excess in the kathode com-
partment.

Thus, by causing the quantity of electricity Q to pass
through the solution, x gramme molecules of salt have been
decomposed into sodium and chlorine, which have been
obtained separate, one component in each of the two
compartments.

No matter how the experimental conditions are varied
(dilution, temperature, nature of the electrodes, current density,
etc.) the passage of the same quantity of electricity always
decomposes the same number of gramme molecules. Thus,
when twice, three times or four times more electricity is
passed through, twice, three times or four times more electro-
lyte is Cecomposed. The quantity of electricity F (equal
to 96,500 coulombs) which is accompanied in its passage
through the solution by the decomposition of 1 gramme
molecule of salt, is often called a faraday, after Farecsy:
who first observed this exact proportionality.

CHEMISTRY AND THE ATOMIC THEORY 47

It is easy to see that the charge carried by a gramme ion
must be 1 faraday exactly. If not, let F’ be its charge,
supposed to be different from F. Let 1 faraday be caused to
pass through the electrolyte ; if m gramme atoms of sodium
pass across a section midway between the electrodes, in
the direction of the kathode (carrying with them mF’
positive faradays), (1—m) gramme atoms of chlorine
must cross the section in the opposite direction (carrying with
them (1 — m)F’ negative faradays). The faraday passed
through is therefore equal to (m + 1—m)F’ ; thatis, to the
charge F’ carried by 1 gramme ion of sodium or of chlorine.

If, instead of sodium chloride, we electrolyse a solution
of potassium chloride KCl, we find, by exactly similar
experiments, that the passage of 1 faraday again decomposes
1 gramme molecule. As we should expect from chemical
reasons, the gramme ion of chlorine bears the same charge
in potassium chloride as in sodium chloride. And in the
same way it may be shown that every monovalent ion
carries with it 1 faraday, positive or negative. This is so,
for instance, for the hydrogen ion H*, which is characteristic
of acids, and for the hydroxyl ion OH, characteristic of bases.

It is also found, as might be expected, that 2 faradays
must pass in order to bring about’ the decomposition of
1 gramme molecule of barium chloride, BaCl,, the ion Ba * *
thus bearing two elementary charges. And we find that
the passage of 2 faradays decomposes 1 gramme molecule
of copper sulphate CuSO, (producing in the anode compart-
ment an excess of 1 gramme atom of the group SO,), so that
the ions SO,” ~ and Cu* * each carry twice the charge borne
by the ions Cl- and Na*; and so on.

In short, all monovalent ions carry the same elementary
charge e, either positive or negative in sign, e being equal to

the quotient : obtained by dividing the faraday by Avo-
gadro’s number, in accordance with the equation
eo Ne:

and all polyvalent ions carry as many of these charges as
they have valencies.

48 ATOMS

It does not appear to be possible to obtain a sub-multiple
of this elementary charge, which thus possesses the essential
characteristic of an atom, as Helmholtz first pointed out in
1880. It is indeed an atom of electricity. Its absolute value
will be known when we succeed in obtaining N.

Some indication of the vastness of the charges transported
by the ions may be given with advantage. It can be shown,
by the application of Coulomb’s law, that if it were possible
to obtain two spheres each containing 1 milligramme atom
of monovalent ions, placed 1 centimetre apart, they would
repel or attract each other (according to the signs of the
two lots of ions), with a force equal to the weight of 100
trillion tons. This is sufficient explanation of the fact that a
separation of the Na and Cl ions present in a solution of
salt to any great extent, such as was demanded of Arrhenius,
cannot be effected, either by spontaneous diffusion or in any
- other way.

An Upprr Limtr to MoLEcULAR SIZE.

31.—Drvistpitiry or Marrer.—Up to the present it has
been my endeavour to collect together the arguments that
led to a belief in an atomic structure for matter and elec-
tricity and yielded us the ratios between the weights of the
atoms, supposing that they exist, before any idea as to
the absolute values of these magnitudes had been formed.

I need scarcely point out that these magnitudes elude
direct observation. As far as the subdivision of matter has
been pushed up to the present, there has been no indication
that any limit has been approached and that a granular
structure lies beyond the limits of direct perception. A few
examples will be useful in reminding us of this extreme
divisibility.

Gold workers prepare gold-leaf having a thickness of only
one-thousandth of a millimetre or, more shortly, one-tenth of a
micron. These leaves, which are familiar to all of us and
which are transparent and transmit green light, appear
nevertheless to be continuous in structure ; we cannot push
the subdivision any further, not because the gold ceases to
be homogeneous, but because it becomes more and more

CHEMISTRY AND THE ATOMIC THEORY 49

difficult to manipulate the thin leaves without tearing them.
If gold atoms do exist, their diameter is therefore less than
one-tenth of a micron (-lu or 10~° cms.), and their mass must
be less than the mass of gold that fills a cube of that dia-
meter ; that is, it must be less than the hundred-thousandth
of a milligramme (1074 grs.). The mass of the hydrogen
atom, which is, as we have seen, about 200 times lighter,
is thus so minute that certainly more than 20 million atoms
are needed to make up 1 milligramme ; in other words, its
mass is less than $ x 10~1® grammes.

Microscopical examination of various bodies enables us
to go much further, particularly in the case of strongly
fluorescent substances. Indeed, I have satisfied myself that
a solution of fluorescein containing one part in a thousand,
illuminated at right angles to the microscope by a parallel
beam of very intense light (the ultra-microscopic arrange-
ment), still shows a uniform green fluorescence in volumes
of the order of a cubic micron. The mass of the bulky
fluorescein molecule, which we know (from its chemical
properties and Raoult’s laws) to be 350 times heavier than
the hydrogen atom, is therefore certainly less than the one-
thousandth part of the mass of a cubic micron of water.

This means that the hydrogen atom certainly weighs less
' than the one-thousandth part of one-thousandth part of a
milligramme. Briefly, the hydrogen atom has a mass less
than 10°74. Avogradro’s number N is therefore greater
than 10-21, so that there are more than 1,000 milliards of
milliards of molecules in a gramme molecule.

Since the hydrogen atom weighs less than 10-1 grammes,
the water molecule, which is 18 times as heavy, must weigh
less than 2 x 10°? grammes. Its volume is therefore less
than 2 x 10 -*° cubic centimetres (since 1 cubic centimetre
of water contains 1 gramme).-and its diameter is less than the
cube root of 2 x 10°; less, that is to say, than four
hundred thousandths of a millimetre (+ x 10~® cms.).

32.—TuIn Fitms.—The study of “thin films” leads us
still further. During the blowing of a soap bubble we often
notice, in addition to the familiar brilliant colours, small
round black spots with well-defined edges. These spots

A. E

50 ATOMS

-might be taken for holes, and their appearance is almost
immediately followed by the rupture of the bubble. They
may be readily observed while washing the hands, on a film
of soapy water stretched over the space between the thumb.
and forefinger. When this film is held vertically the water
in it gradually drains to the bottom, while the upper part
becomes thinner and thinner, which process can be followed
by the colour changes that occur. After it has become
purple and then pale yellow, the appearance of the black
spots will soon be noticed ; they run together, forming a
black space, which may fill a quarter of the height of the
film before it breaks. If this rough experiment is repeated -
with certain precautions, the thin films being produced on a
fine framework inside a case to protect them from evapora-
tion, it is possible to maintain these black surfaces in
equilibrium for several days and even weeks and so to
observe theni at leisure.

In the first place, these black spots are not holes, for it is
easy to show, as was done by Newton, who first studied
them, that, though black by contrast, they nevertheless
reflect light, and also that new round, sharp-edged spots
ultimately appear within the original spots ; the new spots
are still darker and hence thinner, and also reflect feeble
images of bright objects, such as the sun.

It is possible to measure the thickness of the black spots
by several concordant methods,! and it has been found that
Newton’s, that is, the blackest and thinnest, have a thick-
ness of about 6 thousandths of a micron or 6 millimicrons
(about 6 X 10-7 cms.). The primary series of black spots
have approximately double this thickness, which is some-
what remarkable.

The films produced by the spreading of oil drops on a
water surface may become even thinner than the black
spots on soap bubbles, as Lord Rayleigh has shown. It is
known (and the fact can easily be verified) that small
pieces of camphor thrown upon quite pure water commence |

1 Either by measuring their electrical resistance (Reinold and Riicker) or by
arranging a hundred of them one behind the other and measuring the thickness
-of water found to be equivalent to the series of black films with respect to the
absorption experienced by a ray of light traversing them (Johonnott).

CHEMISTRY AND THE ATOMIC THEORY 51

to dart about in all directions on the surface of the water
(for the solution of the camphor is accompanied by a con-
siderable lowering of the surface tension, with the result
that each piece is continually being urged into regions
where solution is less active). This phenomenon is not
observed if the water surface is greasy (and has in conse-
quence a surface tension much lower than that of pure
water). Lord Rayleigh has attempted to determine the
weight of the smallest drop of oil that, when placed on the
surface of a large basin full of water, is found to be just
sufficient to prevent the movement of the camphor at all
points on the surface. This weight was so small that the
thickness of oil thus spread over the surface of the water
could not have reached two-thousandths of a micron.

Devaux has made a comprehensive study of these thin
films of oil, which he very happily compares with the black
spots on soap bubbles. Thus, when a drop of oil spreads
upon water, an iridescent film is seen to form, in which
black circular, sharp-edged spots soon appear. Within
them the liquid surface is still covered with oil, since it still
possesses the properties described by Lord Rayleigh. But
this oil has not yet reached its maximum extension; by
allowing a drop of a dilute standard solution of oil in benzene
(which evaporates rapidly) to fall on a large water surface,
Devaux has obtained an oil film free from thick spots and
with sharply defined edges. He demonstrated the presence |
of the oil film, not with camphor (which moves on the film
as if it were pure water), but with powdered talc. When °
sprinkled onto pure water with a s‘eve, this powder is
easily shifted by blowing horizontally on the liquid, and
collects on the opposite side of the basin, where the surface
is dimmed. But its motion is stopped by the edges of the
oil film and marks their limits. In this way it is possible
to measure the surface of the film with an accuracy bordering
on the one-hundreth of a millimicron. The corresponding
thickness is very little more than a millimicron (1-:10up or
atx, 10-7 cms.).

It must be borne in mind that in these measurements it is
assumed that the material of the film is uniform in thickness ;

E 2

52 ATOMS

and that, after all, it is not certain, considering only the facts
at present known, that the films have not a reticular or
fine-meshed structure, like a spider’s web, wh‘ch at a distance
may appear homogeneous.

It seems more probable, however, that the thin films are
nowhere thicker than the mean measured thickness, and
that the maximum diameter possible for the oil molecule —
is in consequence of the order of a millimicron. It will be
considerably less for the constituent atoms ; the maximum
mass possible for a molecule of oil (glycerine tri-oleate,
C,,H4940,) would be of the order of one thousand-millionth
of one thousand-millionth of a milligramme, and the mass of
a hydrogen atom, which is nearly a thousand times less,
would be of the order of a trillionth of a trillionth of a gramme
(10~*4 ors.) |

We may summarise this discussion by stating that the
different atoms are certainly less than a hundred-thousandth
(perhaps a millionth) of a millimetre in diameter, and that
the masses even of the heaviest (such as the gold atom)
are certainly ‘ess than a hundred-thousandth (perhaps a
hundred-m‘lionth) of a trillionth of a gramme.

However small these superior limits, which mark the
actual boundaries of our direct perception, may appear,
they may nevertheless be vastly greater than the actual
values.. Certainly when we review, as has been done above,
all that chemistry owes to the conceptions of atom and
molecule, it is hard to doubt at all seriously the existence
- of such elements in matter. But at present we are not in a
position to decide whether they lie just on the threshold of
the directly perceptible magnitudes or whether they are so
inconceivably small that we must regard them as infinitely
removed from our sphere of cognisance.

This is a problem which, once stated, should prove a
powerful incentive to research. The same ardent and dis-
interested curiosity that has led us to weigh the stars and
map out their courses urges us towards the infinitely small as
strongly as towards the infinitely large. Striking advances
already made give us the right to hope that our knowledge
of both atoms and of stars may become equally complete.

CHAPTER II
MOLECULAR AGITATION

THE transference of matter that occurs during solution or
diffusion has led us to suppose that the molecules in a fluid
are in incessant motion. By developing this idea in con-
formity with the laws of mechanics, which are assumed to be
applicable to molecules, an important collection of proposi-
‘tions has been brought together under the name of the
kinetic theory. This theory has shown great fertility in the
explanation and prediction of phenomena, and was the first
to yield a definite indication of the absolute values of the
molecular magnitudes.

MoLECULAR SPEEDS.

33.—MOLECULAR AGITATION A PERMANENT CoNDITION.—
As long as the properties of a fluid appear to us invariable,
we must suppose that molecular agitation in that fluid
neither increases nor decreases.

Let us endeavour to define this rather vague proposition.
In the first place (as is shown by experiment), equal volumes
contain equal masses, that is to say, equal numbers of mole-
cules. More accurately, if n, denotes the number of mole-
cules that should be found in a certain volume if their dis-
tribution were absolutely uniform, then, if n is the number
actually found at a given moment, the fluctuation n —n,,
which varies from instant to instant with the random motion
of the molecular agitation, will be of less importance the
greater the volume considered. In practice, it is quite
negligible for the smallest volumes observable.

Similarly there is practical equality, in any arbitrary
portion of the fluid, between the number of molecules
moving with a certain velocity in one direction and the
number moving with the same velocity in the opposite

54 ATOMS

direction. More generally, if we consider a large number
of molecules, taken at random at a given moment, then the
projection of all the molecular speeds onto any arbitrary
axis (in other words, their resultant along that axis) will
have a mean value of zero; no particular direction will be
- privileged.

Similarly, the aggregate energy of motion or kinetic
energy associated with a given portion of matter will ex-
perience none but the most insignificant fluctuations for
those portions that can be observed. More generally, if we
consider at any given moment two groups of equal (and
sufficiently large) numbers of molecules, which have been
separately chosen at random, then the sum of the kinetic
energies of the molecules is practically the same for the two
groups. This comes to the same thing as saying that the
molecular energy has a fixed mean value W, which is always
found to be the same on taking the mean, at any -given
moment, of the molecular energies of molecules chosen at
random in any number, so long as it is large.

The same value W would be obtained if we took the mean
of the energies possessed by the same molecule at different
instants (a large number must be considered) distributed at
random over a considerable period of time.

The above remarks hold for every kind of energy that can
be attributed to the molecule. They apply particularly to
the kinetic energy of translation 4mV?, m being the mass
and V the velocity of the centre of gravity of the molecule.
The mass being constant, if there is a definite mean value w
for this energy of translation, there will be a definite mean
value U? for the square of the molecular velocity. en

Similar remarks apply to all definable properties of the
molecules in a fluid. There is, for example, a definite value
G for the mean molecular velocity. This value is not U, as

a+b.

will be obvious when it is recalled that the mean Sao

1 Clearly this mean value W’ is the same for any two molecules (which do not
differ in their capacity for acquiring energy); thus, let the energies of a
large number of molecules p be estimated at ¢ successive instants (q being very
great). The sum of the energies thus measured may be written either g times
pW or p times gW’, which shows that W is equal to W’.

MOLECULAR AGITATION 55

two different numbers a and b is always less than the square
2 2
root of the mean of the squares of these numbers, ened

U is sometimes called the mean quadratic velocity.

Maxwell showed that when the mean square U? is known,
the mean speed G follows from the probability law that fixes
the proportion of molecules that have a certain velocity at
each instant.

He arrived at these results, which are of great importance
in our study of the permanent condition of molecular agita-
tion, by assuming that the proportion of the molecules
having a definite velocity component in a given direction
is the same both for all the molecules together and for that
proportion of their number known independently to possess
another definite component in a perpendicular direction.
(More briefly, if we consider two walls at right angles, and if
we suppose that at a given moment a molecule is moving
with a velocity of 100 metres per second towards the first
wall, then, according to Maxwell, we can gain no information
from this fact as to the probable value of its velocity
towards the second wall.) This hypothesis as to the dis-
tribution of velocities, which is probable though by no means
certain, is justified by its results.

By a calculation involving no other hypothesis, and
which may therefore be omitted in detail without affecting
our attitude towards the phenomena under discussion, it is
possible to determine completely the velocity distribution,
which is the same for all fluids in which the mean square of
the molecular velocity has the same value U?. In this way
it is possible to calculate the mean velocity G, which is

found to be less than U and to be approximately equal to
12 , |

1 To be precise, out of #2 molecules, the number dn of them that have a com-
ponent along Ox lying between 2 and x + dz is given by the equation :—

‘25 5 PY i as
Re ee ae Cone
fa BY Sct a
and, moreover, we have :— is ra
“ory 8

56 ATOMS

34.—CALCULATION OF THE MoLEecuLAR VELocITIES.—If
the fluid is gaseous, a simple theory gives, with considerable
accuracy, the value of the mean square U? of the molecular
velocity, from which the mean velocity and velocity distri-
bution follow.

We have already decided that the pressure exerted by a
gas is due to the continual impact of the molecules against
the walls of the containing vessel. In developing this idea
we will assume that the molecules are perfectly elastic.
Then, in order to find their velocity, it is merely necessary
to calculate the constant pressure supported by unit surface
of a rigid wall uniformly bombarded by a regular stream of
projectiles, which move with equal and parallel velocities
and which rebound from the wall without gain or loss of
energy. This is a mechanical problem, into which no
physical difficulties enter ; I shall therefore omit the calcula-
tion (which is, moreover, simple) and give only its solution,
namely, that the pressure is equal to twice the product of
the velocity component perpendicular to the wall (which
component changes its sign during the impact) into the
total mass of the projectiles striking unit surface in unit
time.

Under equilibrium conditions, the assemblage of mole-
cules near a partition may be regarded as a large number of
streams of this kind, moving in all directions and without
the least influence on each other if the molecules occupy but
little of the space they move in (this is the case when the
fluid is gaseous). Let x be the velocity perpendicular to the
partition for one of these streams and g the number of mole-
cular projectiles per cubic centimetre ; then ga projectiles
per second, of total mass gam, will strike each square centi-
metre of the partition, which will in consequence be acted
- on by a partial pressure 2gma?. The sum of the pressures due
to all the streams will be 2 -mX?, where X? is the mean
square of the component 2, and n is the total number of
molecules per cubic centimetre (of which only a fraction are
moving towards the partition). Hence, since the mass
m X nof each unit volume is the density (absolute) of the gas,

MOLECULAR AGITATION 57

we see that the pressure p is equal to the product X2d of the
density by the mean square of the velocity parallel to an
arbitrary direction Incidentally, we find at the same time
that the mass of gas which strikes a square centimetre of the
partition per second is equal to X’d, where X’ is the mean
value of those of the components x that are directed towards
the partition ; since X’ (which becomes doubled or tripled
when the velocities are doubled or tripled) is proportional to
the mean speed G, this mass is proportional to Gd (a result
we shall use later on).

The square of a velocity, that is to say, the square on the
diagonal of a parallelepiped constructed from three rectan-
gular components, is equal to the sum of the squares on the
three components, and hence the mean square U? is equal to
3 X? (the three rectangular projections having by symmetry
the same mean square). The pressure p, equal to Xd, is
therefore also equal to : Ud or 3 2 . U?, where M is the
mass of gas occupying volume v.

We have thus established the equation

3pv = MU?,
which may be written
| Fog
a0 ge

and may be stated as follows :—

For any given mass of gas, the product of the volume by the
pressure is equal to two-thirds of the energy of translation
associated with the molecules in the mass.

We know, moreover (Boyle’s Law), that at constant
temperature the product pv is constant. The molecular
kinetic energy is therefore, at constant temperature, inde-
pendent of the rarefaction of the gas.

It is now easy to calculate this energy, as well as the mole-
cular velocities, for any gas, at any temperature. The mass
M may be taken equal to the gramme-molecule. Since all
gramme molecules occupy the same volume under the same
pressure (para. 18), which/means that the product pv is the
same for all, we see that,;in the gaseous condition :—

58 ATOMS

The sum of the energies of translation of the molecules
contained in a gramme-molecule is the same for all gases at
the same temperature.

At the temperature of melting ice this total energy is
34,000,000,000 ergs. Expressed in other terms, the work
done by the stoppage, at this temperature, of all the mole-
cules contained in 32 grammes of oxygen or 2 grammes of
hydrogen would be sufficient to raise 350 kilogrammes
through 1 metre ; this shows what a reserve of energy lies in

molecular motion.
2

MU
Knowing the energy <a of a known mass M, we can at

once obtain U and in consequence the mean velocity G.-

Again, at the temperature of melting ice, the kinetic energy
for oxygen (M = 32) is the same as if, supposing that all the
molecules were stopped, the mass considered had as a whole
the velocity U of 460 metres per second. The mean velocity

G, which is slightly less, is 425 metres per second. This is ©

not much less than the speed of a rifle bullet. Im hydrogen
(M = 2) the mean velocity r.ses to 1,700 metres ; it falls to
170 metres for mercury (M = 200).

35.—ABSOLUTE TEMPERATURE (PROPORTIONAL TO THE
Mo.LeEcuLAR ENERGY).—The product pv of the volume by
the pressure, which is constant for a given mass of gas at a
fixed temperature (Boyle), changes in the same way for all
gases as the temperature is raised (Gay-Lussac). In point of

1
fact, it increases by 273 of its value on passing from the

temperature of melting ice to that of boiling water. As we
know, this enables us to define (by means of the gas ther-

mometer) a degree of temperature as being the increment

of temperature that raises the product pv (or simply the

; 1
pressure if we work at constant volume) for any gas by 273

of the value it has at the temperature of melting ice (so that

1 For each gramme molecule occupies 22,400 cubic centimetres when the
pressure corresponds to 76 centimetres of mercury, which gives for the product

ao the value 34 x 10° in C.G.S. units.

.
———_ | -

MOLECULAR AGITATION 59

there are 100 such degrees between the temperature of
melting ice and that of boiling water).

Now we have seen above that the molecular energy is
proportional to the product pv. Thus for a long time we
have unwittingly been accustomed to mark equal steps on
the temperature scale by equal increments of molecular

1
energy, the increment of energy per degree being 273 of the

molecular energy at the temperature of melting ice. As we
have already shown (para. 4), heat and molecular agita-
tion are in reality the same thing viewed under different
magnifications.

Since the energy due to molecular agitation cannot become
negative, the absolute zero of temperature, corresponding to
molecular immobility, will be reached 273 degrees below
the temperature of melting ice. Absolute temperature,
which is proportional to the molecular energy, is reckoned
from this zero ; the absolute temperature of boiling water,
for example, is 373 degrees absolute.

It appears that for any gaseous material the product pv
is proportional to the absolute temperature T; this gives
us the equation for a perfect gas :—

py = rT.

Let R be the particular value,’ independent of the nature
of the gas, that r takes when the quantity of gas chosen is a
gramme molecule. If the quantity considered contains n
gramme molecules, the preceding equation can be written

pv = mRT.
Finally, since the molecular kinetic energy is, as we have

3
seen, equal to 9 PY, we can write, for a gramme molecule M :—
| MU? _ 3
: ag eM
36.—PRooF oF AvoGapRo’s Hypotuesis.—It appears
that any two gramme molecules, considered in the gaseous

RT.

1 From thie fact that 1 gramme molecule occupies 22,400 cubic centimetres at
atmospheric pressure at the temperature of melting ice (T = 273° A), we get
that R is equal to 83:2 « 10° C.G.S. units.

60 ATOMS

state at the same temperature, each conta’n the same

3
amount of molecular energy of translation (5 RT), Now,

according to Avogadro’s hypothesis, two such quantities of
gas each contain the same number of molecules N. At the
same temperature, the molecules of the respective gases
therefore possess the same mean energy of translation w

3 R
(equal to 2°N- T). The hydrogen molecule is 16 times

lighter than the oxygen molecule, but ib moves on the
average 4 times more quickly.

In a gaseous mixture each molecule, of whatever kind,
has this same mean energy. For we know (from the law of
gaseous mixtures) that each of the gaseous masses mixed in
a receptacle exerts on its walls the same pressure that it
would exert if present alone. From the expression giving
the partial pressure of each gas (which we may treat exactly
as in the case of a single gas), it follows that the molecular
energies must be the same before and after mixing. What-
ever the nature of the constituents of a gaseous mixture, any
two molecules chosen at random will possess the same mean
energy.

This equipartition of energy between the various mole-
cules of a gaseous mass, worked out above as a consequence
of Avogadro’s hypothesis, can be demonstrated without.
reference to that hypothesis, if it is assumed, as has already
been done, that the molecules are perfectly elastic.

The demonstration is due to Boltzmann! He considered
a gaseous mixture containing molecules of two kinds, having
masses m and m’. If we are given the velocities (and hence
the energies) of the two molecules m and m’ before an impact
and the direction of the line joining their centres after
impact, the laws of mechanics enable us to calculate what
their velocities will be after impact. The gas is, moreover,
in a state of internal equilibrium ; the disturbing effect on
the distribution of velocities caused by one kind of impact
must therefore be compensated continuously by impacts of
the “ opposite ”’ kind (the quantity of motion of the colliding

! “ Theorie cinétique,” chap. 1.

MOLECULAR AGITATION 61

molecules being just the same as before, but of opposite
sign). Boltzmann then succeeded in showing, without any
further hypothesis, that this continuous compensat-on implies
equality between the mean energies of the molecules m and
m'. Thus the law of gaseous mixture requires that these
mean energies should remain the same for the gases when
separate (which is the result arrived at above).

Since, moreover, we have shown that the total molecular
energy is the same for masses of different gases occupying
the same volume under the same conditions of temperature
and pressure, it follows that such masses must contain the
same number of molecules, which is Avogadro’s hypothesis.
Justified by its results but introduced nevertheless in a
somewhat arbitrary manner (para. 13), the hypothesis now
finds its logical basis in Boltzmann’s theory.

37.—EFFUSION THROUGH SMALL OrIFICcES.—The values
derived above for the molecular velocities cannot as yet be
verified directly. But the values obtained for gas pressure
also give us a quantitative expression of two quite different
phenomena, which provides us with a valuable check upon
the theory. ;

One of these phenomena is the effusion, or progressive
passage, of a gas through a very small opening pierced in a
very thin partition enclosing the gas. To understand the
mechanism of this effusion we must bear in mind that the
mass of gas striking per second against a given element of
the partition is proportional to the product of the mean
molecular velocity G into the density of the gas. Now
suppose that this element of the partition is suddenly
removed ; the molecules which were about to strike against
it will now disappear through the opening. The initial loss
will be proportional to Gd; it will remain constant if the
opening is so small that the balance between the various
molecular motions is not disturbed to any great extent. _

The mass thus effused being proportional to Gd, its
volume under the pressure in the enclosure must’ be pro-
portional to the molecular velocity G, or, which is the same

. ; ; 13
thing, to the mean quadratic velocity U (equal to 12 G).

62 ATOMS

Since, in short, at constant temperature the product MU2
is independent of the nature of the gas, it follows that :—

The volume effused in a given time must be inversely pro-
portional to the square root of the molecular weight of the gas.

The various common gases have been found to obey ! this
law. Hydrogen, for instance, oruses 4 times more rapidly
than oxygen.

38.—WIpTH OF SPECTRAL LingEs.—The phenomenon of
effusion provides us with a means for checking the ratios
of the molecular speeds of the various gases but leaves
indeterminate the absolute values of those speeds, which,
according to what has been said above, should reach several
hundred metres per second.

Now attention has recently been directed towards a
phenomenon that has no apparent connection with the
pressure exerted by gases, but which again enables us to
calculate the speeds of molecules, supposing that they exist,
and which yields exactly the same values.

It is well known that the electric discharge causes rarefied
gases to glow. When examined with the spectroscope, the
light emitted from “ Geissler tubes ”’ in action is seen to be
resolved into fine “lines,’”’ each corresponding to a single
homogeneous beam of light, which may be compared with
sound of a definite pitch. Nevertheless, if the contrivance
for splitting up the light is made sufficiently powerful (by
the use of diffraction gratings and, best of all, of interfero-
meters), the finest lines are found ultimately to have an
appreciable thickness.

That this should be so was predicted by Lord Rayleigh,
from the following very ingenious considerations. He
supposed that the light emitted by each vibrating centre
(atom or molecule) is in reality homogeneous; but such
centres being always in motion, the light perceived has a
longer or shorter period, according as the vibrating centre is
approaching or receding.

1 Once established, this law enables us to determine unknown molecular
weights ; if it takes, for instance, 2-65 times as long to empty by effusion the _

' same enclosed space to the same extent when it contains radium emanation as —

when it contains oxygen, the molecular weight of the emanation can be found
by multiplying the molecular weight of oxygen, 32, by (2-65), or about 7.

MOLECULAR AGITATION 63

In the case of sound we are familiar with a phenomenon
of this kind. It is known that the sound of a motor horn,
emitted at a pitch that is obviously fixed, appears to alter
when the motor is in motion. It becomes sharper as the
motor approaches (for then more vibrations are perceived
per second than are emitted in the same time), and’ suddenly
becomes deeper as soon as it has passed (for then fewer
vibrations are received). A simple calculation shows that if
v is the velocity of the source of sound and V that of sound
itself, the pitch of the sound perceived may be obtained by

multiplying or dividing the real pitch by (1 _ 4 , according

as the source is approaching or receding. (This will cause a
sudden variation, of the order of a third, when the source
passes us.)

The same considerations apply to light, in which connec-
tion they are known as the Doppler-Fizeau principle. In
the first place this principle explains why, with ordinarily
good spectroscopes, the lines characteristic of the metals
found in different stars are sometimes all seen to be displaced
slightly towards the red (receding stars) and sometimes
towards the violet (stars that are approaching us). The
velocities of the stars measured in this way are on the
average of the order of 50 kilometres per second.

But with better instruments velocities of several hundred
metres per second can be detected. If the bright capillary
section of a Geissler tube containing mercury vapour and
immersed in melting ice is observed at right angles to the
electric force, the nature of the light perceived proves the
existence of an enormous number of atoms moving in all
directions with velocities of the order of 200 metres per
second ; rigorously homogeneous light can no longer be per-
ceived, and an apparatus of sufficiently high dispersive
power will reveal a diffuse band instead of an indefinitely
thin line. The mathematical theory enables us to calculate
the mean molecular velocity corresponding to the broadening

1 Neglecting the increase in velocity in its own direction that this force can
communicate to the luminous centre, if the latter is charged (Stark has estab-
lished the Doppler effect in the positive “canal” rays in Crookes’ tubes).

64 ATOMS

observed. It then only remains to be seen whether this
velocity agrees with that derived, according to the theory
discussed above, from a knowledge of the gramme molecule
and the temperature.

Experimental work has been carried out by Nicholson,
and also by Fabry and Buisson, whose experiments were
more accurate and in some cases more numerous. Their
results leave no room for doubt ; the velocities calculated by
the two methods agree to within nearly 1 per cent. (Quali-
tatively, a line is broader the smaller the molecular mass of
the glowing gas and the higher its temperature.)

Having once established this remarkable agreement for
certain gases and for particular lines, it will be legitimate to
regard it as still holding in cases where either the molecular
mass or the temperature is unknown and to use it for deter-
mining the latter magnitudes. In this way Buisson and
Fabry showed that in a Geissler tube in action containing
hydrogen the luminous centre is the hydrogen atom and not
the molecule.t

Mo.LEecuLAR ROTATIONS AND VIBRATIONS.

39.—THE SpreciFIC Heat oF GasEs.—Up to the present
we have confined our attention to the translatory move-
ments of the molecules. But the molecules. probably spin
round while they move, and if they are not rigid, other
more complicated motions may occur.

Consequently, when the temperature is raised, the energy
absorbed during the heating of 1 gramme molecule of a gas
must be greater than the increase in molecular energy of

3
translation, which we know to be equal to 9 BT. For each

rise of 1° C., at constant volume (in which case all the energy
acquired by the gas is communicated by heating alone and

1 Following up this brilliant piece of research, Buisson and Fabry aim at
determining the temperature of the nebulz from the observed broadening of the
lines corresponding to known atoms (hydrogen or helium); having done that
they will be able to determine the atomic weight of the body (nebulium), which,
in the same nebule, produces certain lines belonging to no known terrestrial
element. In this way the atom of a simple substance will have been discovered
and weighed in regions so far distant that light from them takes centuries to
reach us !

MOLECULAR AGITATION 65

none by work done in compression), the quantity of heat
absorbed per gramme molecule of the gas (specific heat at
constant volume) will therefore be greater than or equal to

3 Soar
38 C.G.S. units of energy (ergs); that is, to 2-98,1 or

approximately 3 calories.

This is a very remarkable limitation. A single well-
established case where the heat lost by 3 grammes of water
in cooling 1 degree raises the temperature of 1 gramme
molecule of a gaseous substance by more than 1 degree
would be sufficient to jeopardise the kinetic theory. But no
such case has ever been recorded.

40.—Monatomic GasEes.—The question arises whether the
molecular specific heat at constant volume (which we shall
call c) can actually fall to the above inferior limit of 3
calories. In any case where this occurred, the inference
would be that not only does the internal energy of the
molecule remain unchanged as the temperature rises but
that its rotational energy also remains constantly at zero,
-so that two molecules striking against each other must
behave like two perfectly smooth spheres, there being no
frictional effect at the moment of impact. |

If any mole les should happen to possess this property,
they might | expected to be molecules consisting of single
atoms. Su : molecules are found in mercury vapour, and
consequently the determination of c for that substance is of
particular interest. Experiments carried out by Kundt and
Warburg gave the value 3 exactly. (The same result has
been obtained for the monatomic vapour of zinc.)
~ Furthermore, Rayleigh and Ramsay have discovered
certain gaseous, chemically inactive simple substances
(helium, neon, argon, krypton, xenon), which, owing to
their inactivity, had remained hitherto unknown to chemists.
These bodies, which cannot be made to combine with any -
other substances, are probably composed of atoms of zero
valency which are no more able to combine among them-

1 For sR is equivalent to 12-5 x 10! ergs, or (since a calorie is equivalent to

4°18 107 ergs) 2-98 calories.

A F

66 ATOMS

selves than with other atoms ; the molecules of these gases
are thus probably monatomic. And as a matter of fact the
specific heat c for each of these gases is found to be exactly
equal to 3, at all temperatures (experiments were carried
out up to 2,500° C. with argon).

In short, when molecules are monatomic, they ant not be
caused to spin when they strike each other even at speeds
of the order of a kilometre per second. In this respect the
atoms behave as though they were perfectly rigid, smooth
spheres (Boltzmann). But this is only one possibility, and
ali that is suggested by absence of rotation is that two atoms
approaching one another are repelled by a force directed
towards the centre of gravity of each of the atoms, which
therefore cannot be caused to spin. In the same way (with
the difference that attractive forces are operative) a co net
that is strongly deviated by its passage close to the sun does
not communicate any rotation to the latter.

In other words, at the instant when two atoms rushing
towards each other undergo the sudden change in velocity
that constitutes an impact, they affect each other as though
they were two point centres of repulsion, of dimensions
infinitely small by comparison with their distance
apart.

In fact, we shall ultimately (para. 94) come to the con-
clusion that the material part of the atom is probably
enclosed within a sphere, of extremely small diameter,
which repels with great violence all other atoms that
approach within a certain limiting distance, so that the
‘minimum distance between the centres of two atoms
moving towards each other with velocities of the order of a
kilometre per second lies well above the real atomic dia-
meter. In the same way, the range of the guns on a battle-
ship very greatly exceeds the circumference of the ship itself.
This minimum distance ‘s the radius of a sphere of protection
which is concentric to the atom and vastly greater than it.
We shall find that an altogether new phenomenon is pro-
duced when we succeed in increasing greatly the speeds that
precede impact, and that. then the atoms pierce the pro-
tecting spheres ins‘ead of rebounding from them (para. 113).

MOLECULAR AGITATION _ 67

41.—A Serious Dirricutty.—Even if the material part
of the atom is concentrated within a sphere very small
relative to the distance from it at which impact takes place,
it appears impossible to suppose that its symmetry can both
be and remain such that, at the moment of impact, the
repelling forces should always be accurately directed along
might be expected from a superficial inquiry, this is a case
where a very close approximation is not sufficient and we
have therein an exception to the general rule of very great
interest. However few in number the atoms may be that
deviate from the standard condition of symmetry, they will
end by gaining rotational energy equal to their energy of
translation. And it is easy to see that the more difficult it
is to impart rotation by impact, the more difficult will it be
to check rotation already acquired, so that only the time
taken to reach statistical equilibrium between the two
energies will be affected, and not the ratio between them
once equilibrium has been reached. Boltzmann has laid
stress upon this point and has raised the question whether
the time taken to reach equilibrium might not be con-
siderable in comparison with the duration of our measure-
ments. ?

But this is quite inadmissible, for whether conducted very
rapidly (during an explosion) or of long duration, such
measurements always give the same values for the specific
heat of argon, for example. We are thus faced with a
fundamental difficulty. It can be removed, but only by
postulating a new and rather peculiar property of matter.

42—RoTaTionAL ENERGY OF THE PoLyatTomic MOLE-
CULES.—It is now natural to inquire what the value of the
specific heat c will be when the molecules can cause each
other to spin on striking.

Boltzmann has succeeded, without making any fresh
hypothesis, in generalising the results of the statistical
calculations by which he established the equality between
the mean translational energies of the molecules. He has
_ thus been able to calculate what, under standard conditions
of molecular agitation, the ratio between the mean trans-

F2

68 . ATOMS

lational and rotational energies should be, for a given mole-
cule, if that molecule may be regarded as a solid body.

In the general case where this solid body has no axis of
revolution, a very simple result is obtained and it is found
that the two kinds of energy are equal. Increase in rotation
- will therefore absorb 3 calories per degree, the same as the
translational increase, which will make 6 calories in all (or,
more exactly, 5-96) for the molecular heat c.?

But if the molecule is composed, dumb-bell like, of two
atoms only, each separately comparable to a perfect'y smooth
sphere (or better, as we have seen, to mutually repulsive
centres of force), no kind of impact can impart rotation to
the atoms about the axis of revolution joining the centres
of the spheres, and Boltzmann’s statistical calculation
shows that in that case the mean energy of rotation of the
molecule will be 2 only of the mean translational energy.
Rotational energy will then absorb 2 calories per degree,
the translational energy absorbing 3, making 5 in all (or
more accurately 4-97) for the heat c.

If finally the molecule is not solid, any deformation or
internal vibration caused by impact will absorb still more
energy, and the specific heat will rise above 5 calories if the
molecule is diatomic and < bove 6 if it is polyatomic. On
the whole, these results agree with the experimental data.

To begin with, for a large number of diatomic gases the
specific heat c has sensibly the same value, equal to 5
calories, as is demanded for molecules that may be regarded
as smooth, rigid dumb-bells. This is the case (the measure-
ments having been made at about the ordinary tempera-
tures) for oxygen O,, nitrogen N,, hydrogen H,, hydro-

1 It may be remembered that in stereo-chemistry (para. 24) at least approxi-
mate rigidity is attributed to the molecule

2 In other terms (which are often used) :—-The condition of a molecule is
defined, from the point of view of energy, by the three components along three
fixed axes of the speed of translation «..d by the three components of the speed
of rotation. These six components, which can be independently chosen, there-
fore represent six degrees of freedom. For every rise in temperature of 1°, and
considering 1 gramme molecule, the energy corresponding to each component
will take up 1 calorie: energy is equally divided between the degrees of freedom.
(For a rigid, smooth, spinning diatomic molecule, two only of the components

of the rotational energy are independent, and the number of degrees of freedom
falls to five.)

MOLECULAR AGITATION 69

chloric acid HCl, carbon monoxide CO, nitric oxide, NO,
etc.

For other diatomic gases (iodine I,, bromine Br,, chlorine
Cl,, iodine monochloride ICl) the heat c is from: 6 to 6-5
calories. Now these gases happen to be those that split up
into monatomic molecules at temperatures that we are able
to reach (in the case of iodine dissociation is already complete
at about 1,500°). Weare perhaps justified in supposing that
this dissociation is preceded by some internal modification
of the molecule and that the union between the atoms is
slackened, energy being absorbed, before the final rupture
occurs.

Finally, for polyatomic gases, we should expect, with
Boltzmann, that the heat c would be equal or greater
than 6 calories. And such indeed is found to be the
case with the values obtained for water vapour and methane.
More often the number found is considerably greater (8 for
acetylene, 10 for carbon bisulphide, 15 for chloroform, 30 for
ether). Since the probability of interna! modification as a
result of impact might be expected to be higher the more
complex the molecule, these high values are not to be
wondered at.

43.—TuHr INTERNAL ENERGY OF THE MOLECULE OAN
VARY ONLY IN DisconTINuous Sreps. — The various
monatomic gases (such as mercury or argon) have shown
us that the internal energy of the atoms does not depend
upon the temperature. We may reasonably suppose, there-
fore, that the internal energy absorbed by a polyatomic
molecule reappears entirely in the form of oscillations of
the unchanged atoms that make up the molecule about
their positions of equilibrium, which means that at any
instant the moving atoms possess both kinetic an
potential energy, due to their oscillation. ;

It is very remarkable that we cannot take the energy of
this oscillation as having a continuous value, capable of
variation by insensible degrees. If we could do so, Boltz-
mann’s statistical argument could be extended to the case
of vibrating atoms, and, considering only diatomic "mole-
cules, the increment of heat absorbed as kinetic energy of

70 ATOMS

weed R
oscillation would be gq? oF 1 calorie, for each rise of 1°,

besides the heat absorbed as mean potential energy of
oscillation .1 ,

The specific heat c of a diatomic gas, probably equal to 7,
could not therefore in any case lie between 5 and 6 and,
more simply still, would never be lower than 6, for oscilla-
tion of continuously varying amplitude would begin to
appear only above a certain temperature.

Now we have seen that this is not the case. The specific
heat of the diatomic gases is generally about 5; it increases
slowly with rise in temperature. Thus its value (Nernst)
for oxygen is 5:17 at 300°, 5:35 at 500°, and 6 at 2,000°, at
which temperature oxygen behaves like chlorine or iodine
in the neighbourhood of the ordinary temperatures.

The above values for the specific heat, which are in all
cases lower than those demanded by the very natural
hypothesis of a continuously variable internal energy oi
oscillation, are explicable if certain molecules, increasing
progressively in number, become modified in a discontinuous
fashion as the temperature rises.

Since these low values are always met with as the molecule
approaches the point of dissociation into atoms (first iodine,
bromine and chlorine ; then oxygen, nitrogen and hydrogen),
it is reasonable to suppose that these discontinuities accom-
pany the sudden loosening of the valencies that bind the
atoms together, each diminution in solidity absorbing a
definite quantum of energy. Similarly, when we wind up a
clock, we can feel through the fingers the energy stored in
the spring increasing by indivisible quanta.

We are therefore left with the probable conclusion that
the energy in each quantum is stored within the molecule in
the form of oscillatory energy ; but we must suppose, con-
trary to our experience of vibrating systems on the usual

1 This second increment is also found to be 1 calorie, if, as in the pendulum,
the force urging each atom towards its position of equilibrium is proportional to
its elongation (or distance from the equilibrium position), in which case, as with
the pendulum, the mean potential and kinetic energies of oscillation would be
equal (this is an extension of the theorem indicated in the note to para. 42, on the
equipartition of energy).

MOLECULAR AGITATION 71

dimensional scale, that the internal oscillatory energy of a
molecule can vary only by discontinuous steps. Though at
first sight discontinuity of this kind may seem strange, we
are as a matter of fact prepared to support the assumption
in view of Einstein’s brilliant extension of the hypothesis
that enabled Planck to explain the mechanism of isothermal
radiation, as we shall see later (para. 88).

According to this hypothesis, the energy of each oscillator
varies by equal quanta. Each of these quanta, each of
these specks of energy, is moreover the product hv of the
frequency v (the number of vibrations per second) peculiar
to the oscillator, by a universal constant h, independent of
the nature of the oscillator.

Having once granted this, it is possible, as Einstein
showed, by making certain simple hypotheses as to the
probable distribution of energy between the oscillators, to
calculate the specific heat at any temperature as a function
of the frequency v. When the frequency is sufficiently
small or the temperature sufficiently high, we find, as on
Boltzmann’s theory, that the energy is equally divided
between the degrees of freedom corresponding to translation,
rotation, and oscillation.

44.—_MoLECULES IN A STATE OF CONSTANT Impact:
Sprecrric Heat or Soxtip Bopires.—Up to the present I have
not considered the potential energy developed at the actual
moment of impact—when, for instance, two molecules
approach each other with equal velocities and come to rest
the one against the other before rebounding with their
velocities reversed. For each molecule, the potential energy
of impact is in the mean zero in a gas where the duration
of impact is very small by comparison with the time that
elapses between two impacts; in other words, at any
instant chosen at random, the potential energy of impact
of a molecule is in general non-existent and its mean value
is nothing. This commonsense argument, which I owe to
M. Bauer, is sufficient to show, without calculation, that the
principle of equipartition of energy cannot be extended to
the case of energy of impact.

But if the gas is progressively compressed, impacts will

72 ATOMS

become more and more numerous and the fraction of the
total energy present at each instant in the form of potential
energy of impact will continuously ncrease. After a certain
compression has been reached, the condition of the gas will
be such that practically no single molecule can be regarded
as free. |

Though direct evidence is lacking, it is possible that the
molecule may then be much less rigid than a gaseous mole-
cule composed of two or three atoms, because each atom
will be attracted towards neighbouring atoms outside the
molecule by cohesive forces comparable in magnitude with
those which urge them towards the other atoms in the
molecule. This brings us to the conclusion that each atom
is readily displaceable in all directions about a certain mean
equilibrium position. ;

The laws of elasticity for solids (reaction proportional to
deformation) leads to the supposition that the force urging
the atom back ‘ owards its position of equilibrium is propor-
tional to its displacement, which means that the atomic
vibrations are harmonic and that in the mean the potential
energy is equal to the energy of motion.

Finally, assuming, as was done in Boltzmann’s statistical
calculations, that molecular agitation is a permanent condi-
tion, and considering a solid in thermal equilibr-um with a
gas, we shall find that the mean kinetic energy has the same
value for each atom of the solid and each molecule of the gas.
On raising the temperature by 1°, each gramme atom of the
solid body absorbs 3 calories due to increase in energy of
motion of the component atoms, and, according to what has
been said as to the equality between the kinetic and potential
energies, it also absorbs 3 calories due to increase in the
potential energies-of these atoms. This makes 6 calories in
all, and we obtain Dulong and Petit’s Law (para. 15).

But this gives us no explanation of why the specific heat
of solids tends to zero at very low temperatures, Dulong
and Petit’s law becoming quite inaccurate. As we shall see
later (para. 90), Einstein has succeeded in explaining this
variation of specific heat with temperature, but only by
assuming (as he had done for the internal oscillations of

MOLECULAR AGITATION 73

gaseous molecules) that the actual energy of oscillation of
each atom varies by indivisible quanta, of the form hv,
greater or less according as the frequency v of the oscillation
possible for the atom is high or low.

45.—GASES AT VERY LOW TEMPERATURES: EVEN
ROTATIONAL ENERGY VARIES DISCONTINUOUSLY.—At very
low temperatures peculiarities, at first sight hard to explain,
are observed with gases as well as with solids.

Even at the temperature of melting ice (273° absolute)
the specific heat of hydrogen is only 4:75, and is thus dis-
tinctly lower than the theoretical value 4-97. The discre-
pancy is not great, but, as Nernst has justly pointed out, it
lies in the direction absolutely irreconcilable with Boltz-
mann’s results on rotational energy. Under his direction
investigations have been carried out by Eucken at a very
low temperature, and have led to the surprising result that,
below 50° absolute, the specific heat of hydrogen becomes 38,
as with the monatomic gases! For other gases the specific
heat at low temperatures also falls below the theoretical
value (though at much lower temperatures than hydrogen),
and in fact it seems probable that at sufficiently low
temperatures all gases have the same specific heat as the
monatomic gases, namely 3; that is to say, the molecules,
although not spherical, no longer by their impacts impart
to each other rotational energy comparable with their
energy of translation.

This is incomprehensible, after what has been said above,
if the rotational energy can vary by insensible degrees. And
we are therefore forced to conclude, with Nernst, that this
rotational energy does indeed vary by indivisible quanta
like the atomic oscillation within the molecule. We may
express this result by stating that the angular velocity of
rotation varies in a discontinuous manner. This is indeed
strange, but if we bear in mind that we are dealing, as we
shall see later, with rotations so rapid that each molecule
revolves more than a million times in one thousandth of a
second,” we need not be surprised at the possibility of other

1 Liquefaction can always be avoided by working under reduced pressure.
? Which means that the acceleration must have a colossal value

74 ATOMS

properties of matter becoming manifest, which are quite
imperceptible when looked for in rotating systems of the
kind to which we are accustomed.

Coming back, therefore, to the case of monatomic mole-
cules, we begin to suspect the solution to the problem that
at first sight seemed so perplexing. If two of these atoms
are not caused to spin when they strike each other, although
they are not mutually repelled by forces acting exactly
between their centres, the cause is certainly to be sought
‘in a very marked discontinuity in the energy of rotation.
Obliged to spin rapidly or not at all, they would in general
be able to acquire the high minimum rotational energy by
impact only at very high temperatures, and it has not been
possible up to the present to measure specific heats at such
temperatures. This idea will be developed later (para. 94),
where it will be shown that the atom in reality occupies but
little of the space at the centre of its sphere of protection.

MoLEcULAR FREE PATHS.

46.—TueE Viscosity oF GAses.—Although molecules move
with velocities of several hundreds of metres per second,
even gases mix but slowly by diffusion. This can be ex-
plained if we remember that each molecule, being continually
driven in all directions by the impacts it receives, may take
a considerable time to move from its original position.

Thus bearing in mind the way in which the movements
of a molecule are obstructed by neighbouring molecules, we
are led to the conception of the mean free path of a molecule,
which is the mean value of the path traversed in a straight
line by a molecule between two successive impacts. It has
been found possible to calculate this mean free path (and we
shall find a knowledge of it of service later on in calculating
the size of molecules) by establishing its connection with
the viscosity of gases.

We are certainly not accustomed in practice to regard
gases as viscous substances. As a matter of fact, they are
very much less viscous than liquids, but their viscosity is
measureable nevertheless, Thus, suppose we have a well-

MOLECULAR AGITATION 75

polished horizontal disc, placed in a gas, and revolving with
a uniform motion about a vertical axis passing through its
centre. It will not merely slip round in the layer of gas in
immediate contact with it but will carry the layer round too.
The first layer will then carry round with it, owing to its
friction, an adjacent layer, and so on, until gradually the
movement will be transmitted throughout the gas by
“internal friction,’ just as in a liquid; consequently a
second disc parallel to the first and suspended above it by
a torsion thread will ultimately be carried round by the
tangential forces thus transmitted, until the torsion balances
them (which makes it possible to measure them).

The phenomenon is easily explained by the molecular
agitation hypothesis. To make this clear, let us first imagine
two train loads of travellers moving in the same direction
along parallel sets of rails and at nearly equal speeds. We may
imagine these travellers amusing themselves by constantly
leaping from one train to the other, alighting with a slight
impact at each leap. As a result of these impacts the travel-
lers alighting on the slower train would slowly increase its
speed, and on the other hand would diminish the speed of
the faster train when they leaped upon it. The two. speeds
would thus ultimately become equal, just as if they had been
equalised by direct friction ; indeed, the process is actually
a frictional effect, with a mechanism that we are able to
perceive.

The same effect will be produced when two gaseous layers
slide the one upon the other. We may express this condition
by supposing that the molecules in, say, the lower layer have,
on the average, a certain excess of velocity, in a fixed horizon-
tal direction over the molecules in the upper layer. But the
molecules are moving in all directions, and in consequence
they will continually be projected from the lower into the
upper layer. They will carry with them their excess speed,
which will soon be distributed among the molecules in the
upper layer, thus increasing slightly its velocity in the given
direction ; at the same time, as a result of the action of
molecules projected from the upper layer, the speed of the
lower layer will diminish slightly. Equalisation of the two

‘Bee ATOMS

speeds will therefore ensue, unless, of course, their constant
difference is artificially maintained by some external
means.

The effect of a molecular projectile on a layer will be the
greater the farther off the layer is from which it comes, for
then it must necessarily bring with it a larger excess of
speed ; this will occur the more often the greater the mean
free path. Furthermore, the effect of the bombardment,
for the same free path, must be proportional to the number
oi projectiles that a layer receives from others in its vicinity.
We are therefore prepared to accept the results of the more
detailed 1 mathematical analysis by which Maxwell showed
that the coeffic'ent of viscosity ¢ (or tangential force per
square centimetre for a velocity gradient equal to 1) should
be very nearly equal to one-third of the product of the
following three quantities: d the gas density, G the mean
molecular velocity, and L the mean free path :—

l |

It is fairly obvious that for a density, say, 3 times less
the free path will be 3 times greater. If, therefore, L varies
inversely with d, the product G.L.d is constant; the
viscosity 1s independent of the pressure (at a given tempera-
ture). This law appeared very remarkable when it was
first announced, and its verification (Maxwell, 1866) con-
stituted one of the first important successes of the kinetic
theory.

Since the viscosity is measurable ? (a method for so doing
has been indicated), it appears that all the quantities in»
Maxwell’s equation are known except the free path L,
which can therefore be calculated. For oxygen or nitrogen
(under normal conditions) the mean free path is wey nearly

1 ‘The reasoning is very similar to that which gives gas pressure in terms of the
molecular velocity.

2 Under very low pressures care must be taken that the dimensions of the
measuring apparatus (such as the distance between the plates that are caused to
rotate by the internal friction of the gas) are sufficiently large by comparison
with the free path ; otherwise the theory i is inapplicable.

8 Order of magnitude : ‘00018 dyne for oxygen (under normal conditions).
Water at 20° C. is about 50 times more viscous.

MOLECULAR AGITATION 77

one ten-thousandth of a millimetre (-1u). - For hydrogen it is
very nearly double that value. Under the very low pressures
reached in Crookes’ tubes, it often happens that a molecule.
may move several centimetres in a straight line without
_ meeting another molecule.

During one second a molecule describes as many free paths
as it receives impacts, and its total path traversed in the
same time should be the mean speed G ; the number of im-
pacts per second is therefore the quotient of that speed by the
mean free path. This gives a total of very nearly five
thousand million for air molecules under normal conditions.

47.—TuHE MoLEecuLAR DIAMETER, AS DEFINED BY IMPACT.
—The mean free path has been calculated from a_knowledge
of its relation to the viscosity of gases. It can also be deduced
from the simple postulate that the free paths must be the
greater the smaller the molecules (they would never strike
each other at all, if they were points without magnitude).

Clausius was of the opinion that molecules might be
regarded, without great error, as spherical balls having a
diameter equal to the distance between the centres of two
molecules at the moment of impact. This condition of
sphericity might be expected to hold approximately for
monatomic molecules. It must be borne in mind, as has
been pointed out above, that the distance between the
centres at the moment of impact (probably slightly variable
according to the violence of the impact) is equal to the
radius of a sphere of protection maintained by the intense
forces of repulsion, and is not necessarily equal to the dia-
meter of the material portion of the molecule. Several diffi-
culties in connection with the kinetic theory arise solely
from the fact that the same expression ‘“‘ molecular diameter ”
is used to denote magnitudes that may be widely different=4
To avoid all confusion we shall call the quantity that
Clausius calls the molecular diameter the diameter of impact
or radius of protection. When two molecules strike against
each other their spheres of impact are tangential.

' Diameter of the actual molecular mass, diameter of impact, diameter as
defined by the state of the molecules when brought close together in the solid

state and when cold, the diameter of the conducting sphere having the same
effect as the molecule, etc

78 ATOMS

With these reservations, let the volume occupied by a
gramme molecule of a gas be v, so that there are . molecules.

in unit volume, moving with mean velocity G. Suppose that
at a given moment all the molecules become fixed in their
positions, with the exception of one that retains the velocity
G and rebounds from molecule to molecule with a mean free
path L’ (which differs, as we shall see, from the free path L
that obtains when all the molecules are in motion). Con-
sider the series of cylinders of revolution having as axes the
successive directions of the moving molecule and a circle of
radius D as base, D being the distance we have just defined
as the diameter of impact; the mean volume of these
cylinders is 7D?L’. After a large number of impacts, say
p, the total volume of the whole series of cylinders, which is
equal to p 7D?L’, will include just as many of the fixed
molecules as there are separate cylinders. Since unit volume -

are. }
contains — molecules, we have :—
v

is BAP vere Pa
Prd L’ = p, or Nr D?: = L”

Clausius was satisfied with this equation, in which he
inadvertently assumed equality between L and L’. Maxwell
pointed out that the chances of impact are greater for a
molecule moving with a mean speed G when the other mole-
cules are in motion also ; for then the speed of two molecules
with respect to each other ! takes the higher mean value of
G Pf 2. From this it follows that L’ must be equal to if aS a

In short, Clausius’ calculation, corrected by Maxwell, gives

the total surface of the spheres of impact of the N molecules in |
a gramme molecule, according to the equation

v
2— -—.
where L stands for the free path when the volume of the

1 Let R be a relative velocity, the resultant of the velocities w and u’, 6 being
the angle between their directions; we then have for R? the value (u? + w? —
2u . u’ cos 6) or, in the mean, the value 2U?.

MOLECULAR AGITATION 79

gaseous gramme molecule is v ; this free path can be deduced
from the viscosity of the gas.

Applying this equation in the case of oxygen (v equal to
22.400 cubic centimetres and L equal to -1), we find that the
spheres of impact of the molecules in 1 gramme molecule
(32 grammes) have a total surface of 16 hectares ; placed side
by side in the same plane, they would cover an enormous
surface ; slightly more, in fact, than 5 hectares.

_ A further relation between Avogadro’s number N and the
diameter D of the sphere of impact would give us these two
magnitudes.

In the first place, it may be pointed out that the diameter
D, determined when molecular impact is violent, is probably
a little less than the distance to within which the centres of
molecules approach when the body under consideration is
liquid (or vitreous) and as cold as possible. Moreover, in a
liquid, the molecules cannot be more closely packed together
than are the shot in a pile of shot. The total volume of the

3
spheres of impact (the volume N . ee of the spheres of pro-

tection) is consequently less than ? of the limiting volume
reached by the gramme molecule when liquefied or solidified
at very low temperatures, and this limiting volume is known.
The rough relationship thus obtained, combined with the
exact expression for the surface (N 7 D?), leads to values too
high for the diameter D and too low for Avogadro’s number N.

The calculation for mercury (which is monatomic) gives
the diameter of impact for mercury atoms as less than one-
millionth of a millimetre and for Avogadro’s number a value
above 440,000,000,000 trillions (44 x 102).

48.—VAN DER WAAL’s Equation.—As a matter of fact,
the upper limit thus set tothe size of the molecules must be
fairly near to the actual value, as may be shown by the line
of reasoning employed by van der Waals, of whose work I
wish to give some account.

We know that fluids obey the gas laws only when beyond
a certain degree of rarefaction (oxygen under a pressure of
500 atmospheres does not obey Boyle’s law at all). The fact
is that under such conditions certain influenecs, which are

80 ATOMS

negligible in the gaseous state, become of great importance.
In order to derive the law of compressibility for condensed
fluids, it is, in the opinion of van der Waals, only necessary
to correct the theory as applied to gases on the two points
following :-—

In the first place, in pileeiatine: the pressure due to mole-
cular impact, it is assumed that the volume of the molecules
(more accurately, the volume of the spheres of impact) is
negligible compared with the volume of the space in which
they move. Van der Waals, taking this circumstance into
account, obtained by a more complete analysis the equation

p (v — 4B) =

where B stands for the volume of the spheres of impact of
the N molecules in a gramme molecule occupying the volume
v under a pressure p at an absolute temperature T. This
equation, however, only has the above simple form if B,
without being negligible, is nevertheless small compared
with v (we may take it that it must be less than one-twelfth
of v).

In the second place, the molecules in a fluid attract one
another, and this diminishes the pressure that the fluid would
exert if its cohesion were nil. Taking this second circum-
stance into account, a simple calculation gives the following
equation, which is applicable to fluids in general :—

(9+ $)0- 4 =

where a is a factor expressing the fluid’s cohesion, which
exerts its influence in proportion to the square of the density.
This is van der Waals’ equation."

This well-known equation agrees sufficiently well with
experiment so long as the fluid is not too condensed (it holds
roughly even for the liquid state). In other words, for
every fluid two numbers can be found which, substituted
for a and B, render the equation very nearly exact for all
corresponding values of p, v, and T. (The two values for
a and B can be determined by assuming that the equation

1 It is more usual to write b instead of 4B.

MOLECULAR AGITATION 81

holds accurately for the fluid under two given sets of condi-
tions and thus obtaining two equations in a and B.)

Once B is known, we can get the swrface of impact and the
volume of impact of the N molecules in a gramme molecule
from the equations

6 ,

which will give us all the magnitudes we are seeking (1873).

49 —MoLecuLaR Maanitupes.—N has been worked
out for oxygen and nitrogen, a value very nearly equal
to 45 x 10% being obtained, (to be precise, taking the
diameters to be about 3 x 10-8, we get 40 x 10%" for
oxygen, 45 x 10** for nitrogen, 50 x 107? for carbon mon-
oxide, a degree of concordance sufficiently remarkable). The
substances chosen are not those best suited to the calculation,
since we are forced to calculate the “ diameters ”’ of mole-
cules that are certainly not spheres. A monatomic substance
only, such as argon, can give a trustworthy result. Employ-
ing the data available for this substance, it is found that the
volume B of the spheres of impact, for 1 gramme molecule
(40 grammes), is 7-5 cubic centimetres. This leads to a dia-
meter of impact for the molecule equal to 2-85 x 1078, so
that

2°85

D= 100,000,000 centimetres.

and to a value for N equal to 62 x 10”, or
N = 620,000,000,000,000,000,000,000.

The mass of any atom or molecule whatever follows. For
: ; 2
instance, the mass of the oxygen molecule will be ve
52 x 10°-*4; similarly the mass of the hydrogen atom will
be 1-6 x 10~*4, or

or

1°6
1,000,000,000,000,000,000,000,00
Such an atom would be lost in our body almost as com-
pletely as our body wouid be lost in the sun.

A. G

9 gramme.

Cd

82 ATOMS hy

3 R
The energy of motion ; oN: T of a molecule at the tempera-

ture 273° A. of melting ice will be -55 x 10° ergs ; in other
words, the work developed by the stoppage of a molecule
would be sufficient to raise a spherical drop of water ly in
diameter to a height of nearly Ip. .
Finally, the atom of electricity (30), which is the quotient

=A of a taraday by Avogadro’s number, will have the value

4-7 x 10-10 (C. G. S. electrostatic units), or, if it be preferred,
1-6 x 10-*°coulombs. This is very nearly the one thousand-
millionth of the quantity that can be detected by a good
electroscope.

_ The probable error, for all these numbers, is roughly 30
per cent., owing to the approximations made in the calcula-
tions that lead to the Clausius-Maxwell and van der Waals
equations.

In short, each ‘nolsiae of the air we breathe is moving with
the velocity of a rifle bullet ; travels in a straight line between
two impacts for a distance at nearly one ten-thousandth of a
millimetre ; is deflected from its course 5,000,000,000 times
per second, and would be able, if stopped, to raise a particle
of dust just visible under the microscope by its own height.
There are thirty milliard milliard molecules in a cubic centi-
metre of air, under normal conditions. Three thousand
million of them placed side by side in a straight line would
be required to make up one millimetre. Twenty thousand
million must be gathered together to make up one Shoo
millionth of a milligramme.

The Kinetic Theory justly excites our admiration. It
fails to carry complete conviction, because of the many
hypotheses it involves. If by entirely independent routes
we are led to the same values for the molecular magnitudes,
we shall certainly find our faith in the theory considerably
_strengthened. |

CHAPTER III
THE BROWNIAN MOVEMENT—EMULSIONS

History AND GENERAL CHARACTERISTICS.

50.—THe Brownian Movement.—Direct perception of
the molecules in agitation is not possible, for the same reason
that the motion of the waves is not noticed by an observer
at too great a distance from them. But if a ship comes in
sight, he will be able to see that it is rocking, which will enable
him to infer the existence of a possibly unsuspected motion
of the sea’s surface. Now may we not hope, in the case of
microscopic particles suspended in a fluid, that the particles
may, though large enough to be followed under the micro-
scope, nevertheless be small enough to be noticeably See
by the molecular impacts ?

It is possible that an inquiry on the above lines might have
led to the discovery of the extraordinary phenomenon which
microscopical observation first brought within our ken and
which has given us such a profound insight into the

properties of the fluid state.
To our observations on the usual scale, all portions of
a liquid in equilibrium appear to be at rest. On placing
any denser object in the liquid it sinks, vertically if it is
spherical, and we know, of course, that once it has got to the
bottom of the containing vessel it will stay there and will not
attempt to rise to the surface by itself.

Though these are quite familiar points, they nevertheless
are valid only on the dimensional scale to which we are accus-
tomed. We have only to examine under the microscope a
collection of small particles suspended in water to notice at
once that each one of them, instead of sinking steadily, is
quickened by an extremely lively and wholly haphazard
movement. Each particle spins hither and thither, rises,

G 2

84 ATOMS

sinks, rises again, without ever tending to come to rest. This
is the Brownian movement, so called after the English botanist
Brown, who discovered it in 1827, just after the introduction
of the first achromatic objectives.!

This remarkable discovery attracted little attention.
Those physicists who mentioned the agitation likened it, I
think, to the movements of the dust particles to be seen with
the naked eye dancing in a sunbeam under the influence of
air currents produced by small inequalities in pressure and
temperature. But in this case neighbouring particles move
in approximately the same direction as the air currents
and roughly indicate the conformation of the latter. The
Brownian movement, on the other hand, cannot be watched
for any length of time without it becoming apparent that the
movements of any two particles are completely independent,
even when they approach one another to within a distance
less than their diameter (Brown, Wiener, Gouy).

The agitation cannot, moreover, be due to vibration of
the object glass carrying the drop under observation, for
such vibration, when produced expressly, produces general
currents which can be recognised without hesitation and
which can be seen superimposed upon the irregular agitation
of the grains. The Brownian movement, again, is produced
on a firmly fixed support, at night and in the country, just as
clearly as in the daytime, in town and on a table constantly
shaken by the passage of heavy vehicles (Gouy). Again, it
makes no difference whether great care is taken to ensure
uniformity of temperature throughout the drop ; all that is
gained is the suppression of the general convection currents,
which are quite easy to recognise and which have no connec-
tion whatever with the irregular agitation under observation
(Wiener, Gouy). Great diminution in the intensity of the
illuminating light or change in its colour is without effect
(Gouy).

Of course, the phenomenon is not confined to suspensions
in water, but takes place in all fluids, though more actively

1 Buffon and Spallanzani knew of the phenomenon but, possibly owing to the
lack of good microscopes, they did not grasp its nature and regarded the
“* dancing particles ” as rudimentary animalcule (Ramsay: Bristol Naturalists’
Society, 1881).

THE BROWNIAN MOVEMENT—EMULSIONS ~— 85

the less viscous the fluid.t Thus it is just perceptible in
glycerine and extremely active, on the other hand, in gases
(Bodoszewski, Zsygmondy).

Incidentally, I have been able to observe it with minute
spheres of water supported by the “black spots” on
soap bubbles. The spherules were 100 to 1,000 times
thicker than the thin film which served to support them.
They thus bore to the black spots very nearly the same
relationship that an orange bears to a sheet of paper. Their
Brownian movement, which is negligible in the direction
perpendicular to the pellicule, is very active in the plane of
the latter (almost as active as if the spherules were in a gas).

In a given fluid the size of the grains is of great importance,
the agitation being the more active the smaller the grains.
This property was pointed out by Brown at the time of his
original discovery. The nature of the grains appears to
exert little influence, if any at all. In the same fluid two
grains are agitated to the same degree if they are of the same
size, whatever the substance of which they are composed
and whatever their density (Jevons, Ramsay, Gouy). Inci-
dentally, the absence of any influence exerted by the nature
of the grains destroys any analogy with the displacements
of large amplitude undergone by specks of camphor when
thrown upon water ; the moving fragments moreover finally
come to rest (when the water has become saturated with
camphor).

In fact—and this is perhaps its strangest and most truly
novel feature—the Brownian movement never ceases.
Inside a small closed cell (so that evaporation may be
avoided) it may be observed over periods of days, months,
and years. It is seen in the liquid inclusions that have
remained shut up in quartz for thousands of years. It is
eternal and spontaneous.

All these characteristics force us to conclude, with Wiener

* The addition of impurities (such as acids, bases, and salts) has no influence
whatever on the phenomenon (Gouy, Svedberg). That the contrary has been
maintained, after a superficial examination, is due to the fact that impurities
cause the small particles to stick to the glass when they happen to touch the
sides of the containing vessel; the movement of the remainder, however, is

unaffected. We might as well say that the motion of the waves is stopped when
we fasten a wave-tossed plank against a quay.

86 ATOMS

(1863), that “ the agitation does not originate either in the
particles themselves or in any cause external to the liquid,
but must be attributed to internal movements, characteristic
of the fluid state,’”’ movements which the grains follow more
faithfully the smaller they are. We are thus brought face to
_ face with an essential property of what is called a fluid in
equilibrium ; its apparent repose is merely an illusion due to
the imperfection of our senses and corresponds in reality to a
permanent condition of uncoordinated agitation.

This view agrees completely with the requirements of the
molecular hypotheses, which indeed find in the Brownian
movement such confirmation as was looked for above.
Every granule suspended in a fluid is being struck continu-
ally by the molecules in its neighbourhood and receives
impulses from them that do not in general exactly counter- -
balance each other; consequently it is tossed hither and
thither in an irregular fashion.

51.—THE BRownIAN MOVEMENT AND CaARNOT’S PRIN-
CIPLE.—We have therefore to deal with an agitation that
continues indefinitely and is without external cause. Clearly
the agitation cannot go on in contradiction to the principle
of the conservation of energy. This condition is satisfied if
every increment of velocity acquired by a grain is accom-
panied by the cooling of the fluid in its immediate neighbour-
hood, and similarly if every diminution in velocity is accom-
panied by local heating. It merely becomes apparent that
thermal equilibrium is itself only a statistical equilibrium.

But it must be remembered (Gouy, 1888) that the
Brownian movement, which is a fact beyond dispute, pro-
vides an experimental proof of those conclusions (deduced
from the molecular agitation hypothesis) by means of which
Maxwell, Gibbs, and. Boltzmann robbed Carnot’s principle of
its claim to rank as an absolute truth and reduced it to the
mere expression of a very high probability.

The principle asserts, as we know, that in a medium in
thermal equilibrium no contrivance can exist capable of
transforming the calorific energy of the medium into work.
Such a machine would, for example, allow of a ship being
propelled by the cooling of the sea water; and because of

THE BROWNIAN MOVEMENT—EMULSIONS 87

the vastness of such a reserve of energy, this would be of
practically the same advantage to us as a machine capable
of “ perpetual motion.’ That is to say, it would be doing
work without taking anything in exchange and without
external compensation. But this perpetual motion of the
second kind is held to be impossible. —

Now we have only to follow, in water in thermal equili-
brium, a particle denser than water, to notice that at certain
_ instants it rises spontaneously, thus transforming a part of
the heat of the medium into work. If we were no bigger
than bacteria, we should be able at such moments to fix the
dust particle at the level reached in this way, without going
to the trouble of lifting it and to build a house, for instance,
without having to pay for the raising of the materials.

But the bulkier the particle to be raised, the smaller is the
chance that molecular agitation will raise it to a given
. height. Imagine a brick weighing a kilogramme suspended
in the air by a rope. It must have a Brownian movement,
though it will certainly be very feeble. As a matter of fact
we shall shortly be in a position to calculate the time we
would have to wait before we had an even chance of seeing
the brick rise to a second level by virtue of its Brownian
movement. (That time! will be found to be such that by
comparison the duration of geological epochs and perhaps of
our universe itself will be quite negligible.) Common sense
tells us, of course, that it would be foolish to rely upon the
Brownian movement to raise the bricks necessary to build a
house. Thus the practical importance of Carnot’s principle
for magnitudes and lengths of time on our usual dimensional
scale is not affected ; nevertheless we shall evidently gain a
better understanding of the ultimate significance of that law
of probability by stating it as follows :—

On the scale of magnitudes that are of practical interest to us,
perpetual motion of the second kind is in general so insignificant
that it would be foolish to take it into consideration. |

It would, moreover, be incorrect to say that Carnot’s
principle is incompatible with the conception of molecular

10
* Considerably more than 10" years ; an inconceivably long period of time.

88 | ATOMS

motions. On the contrary, it follows as a consequence of
that motion, though in the form of a law of probability. In
order to escape the restrictions imposed by that law and to
transform at will all the energy of motion of the molecules
in a fluid in thermal equilibrium into work, it must be
possible to coordinate, or to make parallel, the velocities of
all of them.

52.—Wiener’s researches and conclusions might have
exercised a considerable influence on the mechanical theory
of heat, then in process of development; but, embarrassed
by confused ideas as to the mutual actions of material atoms
and ‘‘ether atoms,’ they remained unknown. Sir W.
Ramsay (1876), and afterwards Professors Delsaulx and
Carbonelle, arrived at a clearer understanding of the manner
in which molecular motion is able to produce the Brownian
movement. According to them, “‘ the internal movements
which constitute the heat content of fluids is well able to.
account for the facts.’ And, going more into detail, “in the
case of large surfaces, molecular impacts, which cause pres-
sure, will produce no displacement of the suspended body,
because taken altogether they tend to urge the body in all
directions at once. But, if the surface is smaller than the
area necessary to ensure that all irregular motions will be
compensated, we must expect pressures that are unequal and
continually shifting from point to point. These pressures
will not be made uniform by the law of aggregates and,
their resultant being no longer zero, they will vary con-
tinuously in intensity and direction. . . .” (Delsaulx and
Carbonelle). !

The same conclusion was reached by Gouy, whose
exposition of the question was particularly brilliant (1888),
by Siedentopf (1900), and finally by Einstein (1905),
who succeeded in formulating a quantitative theory of
the phenomenon ; I shall give an account of his work
later.

However seductive the hypothesis may be that finds the
origin of the Brownian movement in the agitation of the
molecules, it is nevertheless a hypothesis only. As I shall
explain later on, I have attempted (1908) to subject the ques-

THE BROWNIAN MOVEMENT—EMULSIONS 89

tion to a definite experimental test that will enable us to
verify the molecular hypothesis as a whole.

If the agitation of the molecules is really the cause of the
Brownian movement, and if that phenomenon constitutes
an accessible connecting link between our dimensions and
those of the molecules, we might expect to find therein some
means for getting at these latter dimensions. This is indeed
the case, and we have moreover a choice of methods we may
employ. I shall discuss first the one that seems to me the
most illuminating.

STATISTICAL HQUILIBRIUM IN EKMULSIONS.

53.—EXTENSION OF THE GAS Laws TO DILuTE EMULSIONS.
—We have seen (para. 26) how the gas laws were extended
by van’t Hoff to dilute solutions, where osmotic pressure
(exerted on a semi-permeable membrane which stops the
passage of the dissolved substance but allows the solvent to
pass through) takes the place of pressure in the gaseous state.
At the same time (para. 26: note) we saw that this law of
vant Hoff’s holds for all solutions that obey Raoult’s laws.

Now Raoult’s laws are applicable indiscriminately to all
molecules, large or small, heavy or light. The sugar molecule,
containing as many as 45 atoms, and the quinine sulphate
molecule, containing more than 100, exert no greater or less
effect than the active water molecule, which contains 3 atoms
only.

Is it not conceivable, therefore, that there may be no limit to
the size of the atomic assemblages that obey these laws? Is it
not conceivable that even visible particles might still obey them
accurately, so that a granule agitated by the Brownian movement
would count neither more nor less than an ordinary molecule
with respect to the effect of its impact upon a partition that stops
iw? In short, is it impossible to suppose that the laws of perfect
gases may be applicable even to emulsions composed of visible
particles ?

I have sought in this direction for crucial experiments that
should provide a solid experimental basis from which to
attack or defend the Kinetic Theory. In the following para-

90 ATOMS

graph I shall describe the one that appears to me to be the
simplest.

54.—DISTRIBUTION OF EQUILIBRIUM IN A _ VERTICAL
CoLuMN oF Gas.—It is vrell known that the air is more rare-
fied in the mountains than at sea level and that, in general
_ terms, any vertical column of gas is compressed under its
own weight. The rarefaction has been given by Laplace
(who obtained it when working out the connection between
altitude and barometric indications).

In order to obtain his law, let us consider a thin horizontal
cylindrical element, of unit cross-sectional area and of
height h; slightly different pressures p and p’ will be
exerted on the two faces of the element. There would be no
change in the condition of the element if it were to be
enclosed between two pistons held in position by pressures
equal to pand p’; the difference (py — p’) between them must
balance the force gm due to gravity which tends to pull the
mass m of the element downwards. This mass m, moreover,
is to the gramme molecular mass M of the gas as its volume
(1 x h) is to the volume v occupied by the gramme molecule
under the same mean pressure, so that

: : M
p—-p=g.—.h.

And since the mean pressure differs very little from p, so
that we may substitute (from the equation for perfect gases)

RT ,
—— for v, we may write

P
pi Mog h
& aE
; 1—-M.g.h
: yan (O=Byed

Clearly, when the thickness / of the element is fixed, the
ratio between the pressures on its two faces is fixed, whatever
the level of the element. For example, in air, at the ordinary
temperature, the pressure falls by the same relative amount
as we mount each step on a staircase (by about z5,450 of its
value if the step is 20 centimetres high). If p, is the pressure

THE BROWNIAN MOVEMENT---EMULSIONS 91

at the foot of the stairs, the pressure after mounting the first

step is p, (" ss = : 3 ; it is again lowered in the same ratio

after the second step and becomes p, : Fine “h

=p 100
the hundredth step it will be p, eee =: “4 and so on.

) ey

Moreover, it does not matter from what level the staircase
starts. Hence, since it is clear that when we rise to the same
height starting from the same level the fall in pressure does
not depend on the number of steps into which we divide that
height, it appears that the pressure will fall in the same ratio
each time we rise through the height H, no matter from what
level we start. In air (at the ordinary temperature) we find
that the pressure becomes halved each time we rise through
6 kilometres. (In pure oxygen, at 0° C., 5 kilometres is suffi-
cient to halve the pressure. )

Of course, since the pressure, being proportional to the
density, is therefore proportional to the number of molecules

in unit volume, the ratio . between pressures can be replaced

by the ratio _ between the numbers of molecules at the two
oO

levels considered.

But the elevation required to produce a given rarefaction
varies with the nature of the gas. It is apparent from the
formula that the ratio between the pressures does not change
if the product Mh remains constant. In other words, if the
gramme molecular weight of a second gas is 16 times lighter
than that of the first, the elevation required to produce the

1 1f the staircase had q steps, the ratio * between the pressures at the top
and at the bottom would be "
SE Ten Ea
Po RT ) :

The calculation is simplified by taking logarithms of the two sides, which
gives (using ordinary logarithms to base 10) by a simple transformation
« 5 Po _M.g.H
2°3 log — = RT?
where H is the distance between the higher and lower levels and is regarded as
being divided into a very large number gq of steps each of height h.

92 ATOMS

same rarefaction will be 16 times greater in the second gas
than in the first. Since it is necessary to rise to a height of
5 kilometres in oxygen at 0° C. before its density is halved, a
height 16 times greater (or 80 kilometres) will be necessary
in hydrogen at 0° C. to produce the same result.

Below are represented three gigantic vertical gas jars (the
largest being 300 kilometres high), containing the same
number of molecules of hydrogen, helium, and oxygen

; Uae

Fi4. 3.

respectively. Assuming the temperature to be constant, the
molecules will distribute themselves as shown in the figure ;
the heavier the molecules, the more are they collected
together at the bottom.

55.—EXTENSION OF THE THEORY TO EmuULSIOoNS.—The
preceding arguments are clearly applicable to emulsions, if
they obey the gas laws. ‘The particles composing the emulsion

THE BROWNIAN MOVEMENT—EMULSIONS 93

must be identical, as are the molecules of a gas. The pistons
introduced into the argument must be “ semi-permeable,”’
stopping the particles but allowing water to pass through.
The “ gramme molecular weight’ of the particles will be
Nm, where N is Avogadro’s number and m is the mass of a
particle. Moreover, the force due to gravity acting on each
particle will not be the weight mg of the particle, but its
effective weight ; that is, the excess of its weight over the up-
thrust caused by its liquid surroundings. The up-thrust will

be equal to m 3 g, if D is the density of the material of which

the particles are composed, and d that of the liquid. A small
elevation h will therefore change the concentration of the
particles from n to n’ according to the equation
1 — pam (1-5). 98,

which gives at once, as in the case of gases,! the degree of -
rarefaction corresponding to any height H whatever. H
may be considered to be subdivided like a flight of stairs into
' g small steps of height h.

Thus, once equilibrium has been reached between the.
opposing effects of gravity, which pulls the particles down-
wards, and of the Brownian movement, which tends to
scatter them, equal elevations in the liquid will be accom-
panied by equal rarefactions. But if we find that we have
only to rise #5 of a millimetre, that is, 100,000,000 times less
than in oxygen, before the concentration of the particles
becomes halved, we must conclude that the effective weight
of each particle is 100,000,000 times greater than that of an
oxygen molecule. We shall thus be able to use the weight of
the particle, which 1s measureable, as an intermediary or con-

1 As with columns of gases, the calculation may be simplified by using
logarithms, which gives the following form to the equation for the distribution

of the particles :
s Z No 1, 4 N < d
2°3 log = apm p) 74

or, if we wish to introduce the volume V of a particle :
Tit a
2°3 log aie m: V-(D d)gH.

94 ATOMS

necting link between masses on our usual scale of magnitude and
the masses of the molecules.

56.—THE PREPARATION OF A SuITABLE Emutsion..—My
attempts to use the colloidal solutions usually studied
(arsenic sulphide, ferric hydroxide, etc.) were unsuccessful.
I have, however, been able to use emulsions composed of
gamboge and mastic.

Gamboge (which is prepared from a dried vegetable latex)

when rubbed with the hand under water (as if it were a piece -

of soap) slowly dissolves giving a splendid yellow emulsion,
which the microscope resolves into a swarm of spherical
grains of various sizes. Instead of using the natural grains,
it is also possible to treat the gamboge with alcohol, Lice
completely dissolves the yellow matter (which makes up $ by
weight of the crude material). This alcoholic solution,
which looks like a bichromate solution, changes abruptly,
on the addition of much water, into a yellow emulsion com-
posed of tiny spheres, that appear to be identical with the
natural ones.

All resins may be precipitated from alcoholic solution in
this way, but often the grains produced are composed of
a viscous paste and gradually become stuck together.
Out of six other resins tried mastic alone appeared suit-
able. This resin (which gives no natural grains) yields
when treated with alcohol a solution that is transformed
by the addition of water into a white emulsion, like milk,
composed of granules of a colourless, transparent, glassy
substance.

57.—FRACTIONAL CENTRIFUGING.—The emulsion having
been obtained, it is subjected to an energetic centrifuging (as
in the separation of the red corpuscles and serum from
blood). The spherules collect together and form a thick
sediment ; above the sediment is an impure liquid which is
decanted. The sediment is treated with distilled water,
which brings the grains into suspension once more and the
centrifuging process is repeated until the intergranular liquid
is practically pure water.

_ But the purified emulsion contains grains of very various
sizes, whereas a uniform emulsion (containing grains equal

-
EE

THE BROWNIAN MOVEMENT—EMULSIONS 95

in size) is required. The process I use to prepare such
emulsions may be likened to fractional distillation. Just as
during distillation the fractions evaporating first are richer
in volatile constituents, so during the centrifuging of a pure
emulsion (made up of grains of the same material) the first
layers of sediment formed are richer in large grains, which
gives us a means for separating the grains according to size.
The technique is easy to imagine and need not be described
in detail. I have used rotational speeds of the order of
2,500 revolutions per minute, which produces a centrifugal
force 15 centimetres from the axis about 1,000 times that
due to gravity. I need scarcely point out that, as in all
other kinds of fractionating work, a good separation is a
lengthy process. In the most careful of my fractionations
I treated 1 kilogramme of gamboge and obtained after
several months a fraction containing a few decigrammes of
grains having diameters approximately equal to the diameter
I wished to obtain.

58.— DENSITY OF THE GRANULAR MATERIAL.—I have
determined this in three different ways :

(a). By the specific gravity bottle method; as for an ordi-
nary insoluble powder. The masses of water and emulsion
that fill the same bottle are measured ; then, by desiccation
in the oven, the mass of resin suspended in the emulsion is
determined. Drying in this way at 110° C. gives a viscous
liquid, that undergoes no further Joss in weight in the oven
and which solidifies at the ordinary temperature into a trans-
parent yellow glass-like substance.

(0). By determining the density of this glassy substance,
which is probably identical with the material of the grains.
This is most readily done by placing a few fragments of it in
water, to which is added sufficient potassium bromide to
cause the fragments to remain suspended without rising or
sinking in the solution. The density of the latter can then
be determined.

(c). By adding potassium bromide to the emulsion until on
energetic centrifuging the grains neither rise nor sink and
then determining the density of the liquid obtained.

The three methods give concordant results : for example,

96 ATOMS

the same lot of gamboge grains gave the three values 1-1942,
1-194, and 1-195 respectively.

59—TuHE VOLUME OF THE GRAINS.—Here again, as with
the density, it is possible, on account of the smallness of the
grains, to place confidence only in results obtained by several
different methods. I have made use of three.

Fia. 4.

A. Direct measurement of the Radius in the Camera Lucida.
—Considerable error is involved in the measurement of
isolated grains (owing to the magnification by diffraction
that occurs in the images of small objects). This source of
error is very considerably minimised if it is possible to
measure the length of a known number of grains in a row.
I therefore allowed a drop of very dilute emulsion to evaporate

THE BROWNIAN MOVEMENT—EMULSIONS 97

onan uncovered object-glass. When evaporation is nearly
complete, the grains are seen to run together, under the
influence of capillary forces, and to collect together into
groups a single grain in depth and more or less in rows, in the -
same way that the shot are arranged in a horizontal section
through a pile of shot. It then becomes possible, as can be
seen from the photograph reproduced above, to count either
the number of grains lying in a row of measured length or
the number to be found side by side within a regularly

covered area. |

At the same time a general check upon the uniformity of
the grains sorted out by the operation of centrifuging is
obtained. The method gives numbers that are perhaps a
little too high (the rows not being quite perfect) ; but owing
to its being so direct it cannot be affected by large errors.

B. Direct Weighing of the Grains.—In the course of other
researches I noticed that, ina feebly acid medium (;$5 normal),
the grains collect on the walls of the glass without adhering
to each other. At any measureable distance from the walls
the Brownian movement is not modified. But as soon as a
grain chances to reach the wall it becomes fixed, and after a
few hours all the grains in a miscroscopical preparation of
known thickness (equal to the distance between the slip and
cover-glass) become fixed. It then becomes possible to count
at leisure all the grains to be found between the ends of an
arbitrary right cylinder (the superficial area of the end being
measured in the camera lucida). Further counts are made
in specimens taken from various parts of the preparation.
Several thousands of grains having been counted in this way,
the concentration of the grains is known for a droplet with-
drawn, immediately after agitation, from a given emulsion.
If the strength of the emulsion is known (by désiccation in
the oven) the mass and volume of each grain follows by
simple proportion.

C. Application of Stokes’ Law.—Suppose that, at constant
temperature, a tall vertical column of the emulsion under
consideration is allowed to stand by itself. Equilibrium

* With my best emulsion I have obtained the value -373. for the radius by
the first method (from 50 rows of 6 to 7 grains) and -369u by the second (about
2,000 grains distributed over 10—° square centimetres).

A, H

98 ATOMS

distribution will be so far from having been reached that the
grains will sink like the minute drops in a mist ; we may leave
out of account the question of reflux due to the accumulation
of grains in the lower layers. The liquid will therefore
become gradually clearer in its upper layers. This may
_ readily be observed with an emulsion contained in a capillary
tube placed in a thermostat. The edge of the cloud of grains.
as it sinks will not be very sharply defined, for as a result of
the fortuitous fluctuations due to the molecular agitation,
the grains will not all fall from the same height ; however,
by taking the “‘ middle ”’ of the zone, it is possible to evaluate
to within nearly ;45 the mean value of the distance fallen (it is
of the order of a few millimetres per day) and the mean
velocity of fall can consequently be obtained. |

Furthermore, Stokes has shown (and his conclusions are
borne out by experiment, in the case of spheres of directly
measurable diameter; 1 millimetre for example) that in a
fluid of viscosity ¢ the frictional force opposing the motion
of a sphere of radius a moving with velocity v is 67(av.
Hence, when the sphere falls with a uniform motion under
the sole influence of its effective weight, we have

6rGav = : ra (D—d)g.

Applying this equation to the velocity of descent of the
cloud of grains in an emulsion, we have another means for
obtaining the radius of the grains (to a degree of accuracy
double that attained for the velocity of descent). we

The three methods give concordant results, as is shown in
the following table, in which the numbers in the same hori-
zontal line give, in microns, the values indicated for the
grains in the same emulsion :—

Rows. Weight. Velocity of fall.
I “50 = 49
=k 46 “46 45
III 371 3667 3675
IV — -212 213
V — 14 15

THE BROWNIAN MOVEMENT—EMULSIONS 99

Agreement is obtained up to ultra-microscopic magni-
tudes. The determinations with emulsions III. and IV.,
which were particularly carefully prepared, show a mean
variation of less than 1 per cent. Each of the radii -3667
and -212 was obtained by counting about 10,000 grains.

60.—EXxTENSION oF SToKES’ Law.—Incidentally, these
experiments remove the doubt that had been expressed, with
justice (J. Duclaux), as to the propriety of extending Stokes’
law to the velocity of falling clouds. Stokes’ law expresses
the real velocity of a sphere with respect to a fluid, but in the
ease under consideration it is applied to a mean velocity
unconnected with the real velocities of the grains; these
latter velocities are incomparably greater and are constantly
varying.

It cannot now be doubted, in the face of the concordant
results given above, that in spite of the Brownian movement
the extension of the law is legitimate. But the experiments
refer only to liquids.1 In gases, as I shall show later, Stokes’
law ceases to be applicable, not on account of the agitation
of the granules, but because the size of the granules becomes
comparable with the mean free path of the molecules of the
fluid.

61.—METHOD OF OBSERVING AN EmuLsion.—Successful
observations with the emulsions I have used cannot be made
through heights of several centimetres or even millimetres ;
heights of less than the tenth of a millimetre only are suitable.
Their investigation has therefore been carried out under the
microscope. A drop of emulsion is placed in hollow slide
(Zeiss hollow slide, having a depth of -1 millimetre), and is
given a plane surface by means of a cover-glass ; the edges
of the latter are treated with paraffin to prevent evaporation.
Two arrangements are possible (Fig. 5).

The preparation may be vertical and the microscope hori-
zontal ; it is then possible to see at a single observation the
distribution of the emulsion throughout its height. I have
made several observations in this way, but no measurements

1 A further condition is necessary (Smoluchowski) ; the cloud, which must
extend to the sides of the tube in which it is sinking (this condition is fulfilled,

in our case, in a capillary tube), must not be able to descend as a whole (liquid
flowing back up the sides) as such a cloud would do in the atmosphere.

H 2

100 ATOMS

up to the present. Fig. 6 is reproduced from a photograph
taken in my laboratory by M. Constantin, using the above
arrangement.
The preparation may also be horizontal, with the micro-
scope vertical. The objective used, which is of high power,
has a small depth of field, and only those grains in a very thin
horizontal section, of the order of a micron in thickness, can
be seen sharply defined at any given moment. As the micro-
scope is raised or lowered, the grains in other sectional layers
become visible.
’ Following either procedure it is shown that the distribution
of the grains, which is very nearly uniform after the initial
disturbance caused by getting the preparation into position

Microscope
Objective

=

>}—___—4

‘al Cover Sa Choe
Hollow Slide KS

i
Emulsion
Fig. 5.

has subsided, soon ceases to be so, the lower sections becom-
ing richer in grains; the process of enrichment, however,
gradually slackens until a permanent condition is realised in
which the concentration diminishes with the height. Fig. 7
was obtained by placing one above the other diagrams
showing the distribution of the grains at a given moment at
five equidistant levels in a particular emulsion. The analogy
between Figs. 6 and 7 and Fig. 3, which represents the dis-
tribution of the molecules in a gas, is evident.

The next step is to obtain measurements. We have
already the radius a of each grain and its apparent density
(D —d), which is the difference between D, the density of the
grain and d, the density of water or other intergranular
_ liquid. The vertical distance H between two sections
sp cessively examined will be obtained by multiplying the

THE BROWNIAN MOVEMENT—EMULSIONS 101

vertical displacement H’ of the microscope + by the index of |
refraction of the medium separating the slide and cover-

glass.2 But we have still to determine the ratio a between

the concentrations of the grains at two different levels.
62.—MrtHop or CouNTING THE GRAINS.—This ratio is
obviously equal to the mean ratio between ;
the numbers of grains visible under the Ps
microscope at two levels. But the counting ~~. .,”’
of the grains is a difficult matter; when one se
sees several hundreds of grains, agitated in :
all directions, continually disappearing and ¢ . , .
reappearing, it is impossible to estimate — |
their number, even roughly. par
The simplest procedure is certainly to “4... °-
take instantaneous . *- 2+. ¢,
photographs and 23° ¥")¢
then to count at JF UINe eee
leisure the sharp —_ ye : a et
images of the grains :.:: °. REE SE, :
on the plates. But ey a ca
owing tothe magni- '": os Me y, Pe
fication necessary +3. % ae
and the short time “!-~. 0/8.

available for ex- .- 3:™ A met’
posure, an intense ws this: jee
i wae, rte. te

lightisrequired,and = 4t4g3../ >"
withgrainsless than ‘ceeSs-33)
half a micronindia- ;** 2a sy. wa ts
meter I have never ‘2847 "035% a
succeeded inobtain- “'*7* "8°" "=
Fia@. 6. Fie. 7
ing good images.

I therefore reduced the field of vision by placing in the focal
plane a diaphragm consisting of an opaque disc of foil having
a very small round hole pierced in it by a needle. The field
now visible becomes very restricted and the eye is enabled

+ Read directly on the graduated head of the micrometer screw of the Zeiss
microscope used.

* More often I have used water emulsions for these experiments, with a
water immersion objective. In that case H is simply equal to H’.

102 : ATOMS

to estimate at once the exact number of grains to be seen at
any given moment. The number must be less than 5 or 6.
By placing a shutter in the path of the rays that illuminate
the preparation they can be allowed to pass at regular
intervals, the number of grains perceived on each occasion
- being noted, thus :—

236. 3°93 he Le a

Starting again at another level, a similar series of numbers
will be obtained, such as :—

2.1. 07 0:1,.45'3, 40 Se.

Owing to the absolute irregularity of the Brownian move-
ment, 200 readings of this kind will clearly be equivalent to
one instantaneous photograph embracing a field 200 times
as large.} ; :

63.—STATISTICAL EQUILIBRIUM IN A COLUMN OF EMULSION.
—It is now easy to establish accurately that the distribution
of the grains reaches ultimately a permanent condition of
dynamic equilibrium. We have only to determine every
hour the ratio ” between the concentrations at two fixed
levels. This ratio, which is at first nearly 1, increases and
tends towards a limit. For a difference in level of -1 milli-
metre, with water as the intergranular liquid, the limiting
distribution was practically reached after one hour (I have

found exactly the same values for “ after three hours and

: after fifteen days.

The limiting distribution constitutes a reversible equilibrium,
for if it is displaced, the system returns to its original condi-
tion of its own accord. One way of displacing it (7.e., of
causing too many grains to accumulate in the lower sections)
is to cool the emulsion, which causes an increase in the con-
centration in the lower layers (I shall return immediately to

1 By either method uncertainty will arise as to some of the grains observed,

which, though barely visible, are sufficiently so for their presence to be guessed
at. But such uncertainty affects n, and n to the same degree. Thus, two

different observers, determining = by means of the spots in a reduced field of

vision found the values 10-04 and 10-16 respectively.

THE BROWNIAN MOVEMENT—EMULSIONS _ 103_

the consideration of this phenomenon), and then allowing
it to return to its original temperature ; the distribution
then becomes what it was before.

64.—THE LAW ACCORDING TO WHICH THE CONCENTRATION
Decreases.—I have sought to discover whether the distri-
bution of the grains, like that of an atmosphere under the
action of gravity, is indeed such that equal elevations are
associated with equal rarefactions, so that the concentration
falls off in geometric progression.

A series of experiments was carried out with the greatest
care, using gamboge grains of radius -212u (using the reduced
field of vision method). Cross readings were taken in a cell
100 deep on four horizontal equidistant planes across the
cell at the levels

5p, 3d, 65, I5y.

The readings gave at these levels, from a count of 13,000
grains, concentrations proportional to the numbers

100, 47, 22-6, 12,
which are approximately equal to the numbers
100, 48, 23, 11-1,

which are in geometrical progression. *

Another series was obtained using larger grains, of mastic
(radius -52u). Photographs taken at four equidistant levels,
one above the other and with 6y distance between them,
show respectively

1880, 940, 530, 305

images of grains ; these numbers differ but little from
1880, 995, 528, 280

which decrease in geometrical progression.

In this latter case, the concentration at a height of 96u
would be 60,000 times less than at the bottom. Hence,
when permanent equilibrium has been reached, grains will
hardly ever be found in the higher layers of such preparations.

Other series might be quoted. In short, as was expected,
the rarefaction law is obeyed exactly. But does it lead to
these values for the molecular magnitudes that we look for ?

104 ; ATOMS

of such
a kind that an elevation of 6 is sufficient to halve their con-
centration. ‘To reach the same degree of rarefaction in air,
we have seen that a distance of 6 kilometres, which is nearly
10,000 million times as great, is necessary. If our theory is
correct, the weight of an air molecule should therefore be one
ten thousand-millionth of the weight, in water, of one of the
grains. The weight of the hydrogen atom may be obtained
in the same way, and it now only remains to be seen whether
numbers obtained by this method are the same as those
deduced from the kinetic theory.?

It was with the liveliest emotion that I found, at the first
attempt, the very numbers that had been obtained from the
widely different point of view of the kinetic theory. In
addition, I have varied widely the conditions of experiment.
The volumes of the grains have had values distributed
between limits which were to each other as 1 is to 50. I
have also varied the nature of the grains (with the aid of
M. Dabrowski), using mastic instead of gamboge. I have
varied the intergranular liquid (with the help of M. Niels
Bjerrum) and studied gamboge grains suspended in glycerine
containing 12 per cent. of water, the mixture being 125 times
more viscous than water.” I have varied the apparent

density of the grains, in ratios varying from | to 5; in’

glycerine it becomes negative (in which case the influence of
the changed sign of their weight accumulated the grains in
the upper layers of the emulsion). Finally, M. Bruhat has,
under my direction, studied the influence of temperature and
observed the grains first in swper-cooled water (— 9° C.) and
then in hot water (60° C.) ; the viscosity in the latter case
_was half what it was at 20° C., so that the viscosity varied in
the ratio of 1 to 250.

1 The calculations are simplified if the distribution equation given in the
note to para. 56 is used.

2 The Brownian movement, though much abated, is nevertheless perceptible ;
several days are required before a permanent equilibrium is reached. I should
have liked to study the distribution in an even more viscous medium, but, when
less than 5 per cent. of water was added to the glycerine (very feebly acid),
the grains collect upon the sides and permanent equilibrium could no longer be
observed. I have subsequently made use of this circumstance in extending
the gas laws to these viscous emulsions (para. 79),

ee

THE BROWNIAN MOVEMENT—EMULSIONS 105

In spite of all these variations, the value found for Avo-
gadro’s number N remains approximately constant, varying
irregularly between 65 x 107? and 72 x 107%. Even if no
other information were available as to the molecular magni-
tudes, such constant results would justify the very suggestive
hypotheses that have guided us, and we should certainly
accept as extremely probable the values obtained with such
concordance for the masses of the molecules and atoms.

But the number found agrees with that (62 x 102) given

by the kinetic theory from the consideration of the viscosity
of gases. Such decisive agreement can leave no doubt as to the
origin of the Brownian movement. To appreciate how parti-
cularly striking the agreement is, it must be remembered
that before these experiments were carried out we should
certainly not have been in a position either to deny that the
fall in concentration through the minute height of a few
microns would be negligible, in which case an infinitely
small value for N would be indicated, or; on the other hand,
to assert that all the grains do not ultimately collect in the
immediate vicinity of the bottom, which would indicate an
infinitely large value for N. It cannot be supposed that,
out of the enormous number of values a priori possible,
values so near to the predicted number have been obtained
by chance for every emulsion and under the most varied
_ experimental conditions.
The objective reality of the molecules therefore becomes
hard to deny. At the same time, molecular movement has
not been made visible. The Brownian movement is a faith-
ful reflection of it, or, better, it is a molecular movement in
itself, in the same sense that the infra-red is still light. From
the point of view of agitation, there is no distinction between
nitrogen molecules and the visible molecules realised in the
grains of an emulsion,! which have a gramme molecule of the
order of 100,000 tons.

Thus, as we might have supposed, an emulsion is actually
a& miniature ponderable atmosphere; or, rather, it is an

__ atmosphere of colossal molecules, which are actually visible.

* Of course, such grains are not chemical molecules, in which all the cohesive
forces are of the nature of those tiniting the carbon to the four hydrogen atoms
in methane.

106 ATOMS

The rarefaction of this atmosphere varies with enormous
rapidity, but it may nevertheless be perceived. In a world
with such an atmosphere, Alpine heights might be repre-
sented by a few microns, in which case individual atmospheric
molecules would be as high as hills.

66.—THE INFLUENCE OF TEMPERATURE.—1 wish specially
to discuss the way in which temperature variation influences
the equilibrium distribution ; briefly, its effect proves that
Gay-Lussac’s law applies also to emulsions. We have seen
that equilibrium in a column of emulsion, as in a column of
gas, is reached between the opposing tendencies due on the_
one hand to gravity (which urges all the grains in the same
direction), and on the other to molecular agitation (which
constantly tends to scatter them). The feebler the agitation,
that is, the lower the temperature, the more marked will be
the subsidence of the column under its own weight.

This subsidence when the temperature falls and expansion
when it rises can be accurately verified without actually
causing the temperature to vary very much. This is possible
because verification in this case does not necessitate the
exact determination, which is always difficult, of the radius
of the grains in the emulsion. Let T and T, be the tempera-
tures (absolute) of experiment. According to the rarefaction
law (note to para. 55) the elevations H and H, corresponding
in each case to the same rarefaction should be such that

H-8)-B (8)

(It appears that if the densities do not change, equivalent
elevations should be proportional to the inverse ratio between
the temperatures. ) |

M. Bruhat, working in my laboratory, undertook, at my
request, to realise the necessary experimental conditions
under which verification could be sought, and has succeeded
admirably.

The drop of emulsion is placed on the upper surface of a
thin, transparent cell in which the temperature is maintained _
at a fixed value ¢° C. (measured by a thermo-electric couple)
by means of a liquid (hot water or cold alcohol) that flows

THE BROWNIAN MOVEMENT—EMULSIONS | 107

through it. For cover-glass he used the bottom of a vessel
full of liquid (hot water or a non-freezable solution of the
same index of refraction as cedar oil) into which he dipped
the objective used (water or cedar oil immersion). This
liquid was raised to the temperature ¢° C. (measured by a
second thermo-electric couple) by means of a copper tube
that traversed it; a branch stream of the regulating liquid
flowed through the tube. Imprisoned in this way the pre-
paration necessarily reaches the temperature ¢° C.

Counts made under these conditions have verified, to
within about 1 per cent., the conclusions reached above,
which shows to what degree of exactness the gas laws can be
extended to dilute emulsions.

_67.—ExacT DETERMINATIONS OF THE MOLECULAR MAentI-
TUDES.—We- have pointed out that the theory of gases,
applied to their viscosity, gives the size of the molecules
with an approximation of perhaps 30 per cent. Refinements
introduced in the actual measurements with gases do not
lessen this degree of uncertainty, which is really connected
_ with the simplifying hypotheses introduced in the theory.
This is not so in the case of emulsions; with them the
results have the same degree of precision as the experiments
upon which they depend. By studying emulsions we are
really able to weigh the atoms and not merely to estimate
their weights approximately.

A series of careful measurements (radius of grain -212y ;
number of grains counted at different levels, 13,000) had
already given me the value 70-5 x 10?2for N. The uniformity
of the grains, however, did not appear to me to be sufficiently
good. I therefore commenced operations afresh, and a more
accurate series (radius -367u to within 1 per cent., obtained
after prolonged centrifuging ; number of grains counted at
various elevations, 17,000) gave for Avogadro’s number the
probable mean value

68-2 x 1022,

from which it follows that the mass of the hydrogen is, in
grammes,
h 1°47

tie =) OND AND NND AND Ee ee — 241
1,000,000,000,000,000,000,000 ‘— 147 * 107").

108 | ATOMS

The other molecular magnitudes follow at once. For
instance, molecular energy of translation, which is equal to
3 R
oN:
melting ice.

The atom of electricity will be (in C.G.8. electrostatic
units)

T, is very nearly (3) x 107 at the temperature of

4:25 x 10— 10.

The dimensions of the molecules, or, more accurately, the
diameters of their spheres of impact, can be obtained, now
that N is known, from Clausius’s equation (para. 48)

v
Lyv2

by first calculating the mean free path L for a gramme mole-
cule of the substance occupying the volume v in the gaseous
state. |

For example, at 370° C. (643° absolute) the mean free path
for mercury, under atmospheric pressure (v is equal to

22.400 x °4?
273

a N.D? =

), can be deduced from the viscosity 6 x 10~ 4

of the gas by means of Maxwell’s equation (para. 47), which
gives the value 2:1 x 10~ 5 for L. This gives 2-9 x 107 8 (or
-29 millimicrons very nearly) for the required diameter.

I have calculated in this way the following diameters :—

Helium. oN eer
Argon 23 ia ee
Mercury . tee
Hydrogen : oo. ae SE ORS
Oxygen . ; ; os hae On oe
Nitrogen . ; 5c egy Ok caer
Chlorine . : ; oo ae ee

These determinations (particularly for the polyatomic
molecules), depending as they do upon the definition of pro-
tecting spheres, do not carry the same degree of precision
that is possible in the case of masses.

CHAPTER IV
THE LAWS OF THE BROWNIAN MOVEMENT

EINSTEIN’s THEORY.

68.—DISPLACEMENT IN A GIVEN TimeE.—It is in conse-
quence of the Brownian movement that equilibrium distribu-
tion is reached in an emulsion ; the more active the. move-
ment, the more rapidly does this occur. But the degree of
activity, whether high or low, has no influence on the final
distribution, which is always the same for grains of the same
size and the same apparent density. We have therefore
confined ourselves up to the present to the study of the
permanent condition of equilibrium, without bothering about
the mechanism by which it is reached.

This mechanism has beén subjected to a detailed analysis
by Einstein, in an admirable series of theoretical papers.!
The approximate but very suggestive analysis given by
Smoluchowski 2 certainly deserves to be mentioned also.

Kinstein and Smoluchowski have defined the activity of
the Brownian movement in the same way. Previously we
had been obliged to determine the “‘ mean velocity of agita-
tion ” by following as nearly as possible the path of a grain.
Values so obtained were always a few microns per second for
grains of the order of a micron.*

But such evaluations of the activity are absolutely wrong.
The trajectories are confused and complicated so often and
so rapidly that it is impossible to follow them ; the trajectory
actually measured is very much simpler and shorter than the
real one. Similarly, the apparent mean speed of a grain

1 Ann. de Phys., Vol. XVII., 1905, p. 549, and Vol. XIX., 1906, p. 371.
A complete account of Einstein’s theory will be found in my memoir “ Les
preuves de la realité moléculaire”? (Brussels Congress on the Theory of
Radiation and Quanta, Gauthier-Villars, 1912).

2 Bulletin de l Acad. des Se. de Cracovie, July, 1906, p. 577.

® Incidentally this gives the grains a kinetic energy 100,000 times too small.

110 ATOMS

during a given time varies in the wildest way in magnitude
and direction, and does not tend to a limit as the time taken
for an observation decreases, as may easily be shown by
noting, in the camera lucida, the positions occupied by a
_ grain from minute to minute, and then every five seconds,
or, better still, by photographing them every twentieth of a
second, as has been done by Victor Henri, Comandon, and
de Broglie when kinematographing the movement. It is
impossible to fix a tangent, even approximately, at any
point on a trajectory, and we are thus reminded of the con-
tinuous! underived functions of the mathematicians. It
would be incorrect to regard such functions as mere mathe-
matical curiosities, since indications are to be found in
nature of “ underived ”’ as well as “ derived ”” processes.

Neglecting, therefore, the true velocity, which cannot be
measured, and disregarding the extremely intricate path
followed by a grain during a given time, Einstein and
Smoluchowski chose, as the magnitude characteristic of the
agitation, the rectilinear segment joining the starting and
end points ; in the mean, this line will clearly be longer the
more active the agitation. The segment will be the dis-
placement of the grain in the time considered. Its projection
on to a horizontal plane, as perceived directly in the micro-
scope under ordinary conditions (microscope vertical), will
be its horizontal displacement.

69.—THE ACTIVITY OF THE BROowNIAN MOvVEMENT.—In
accordance with the conclusions arrived at from qualitative
observation, we shall regard the Brownian movement as
completely irregular in all directions at right angles to the
vertical.2. This is scarcely a hypothesis; moreover, we
shall verify all its consequences.

This being granted, and without any further hypothesis
whatever, it can be proved that the mean displacement of a
grain is doubled when the time is increased fourfold; it
becomes tenfold when the time is increased a hundredfold
and so on. More precisely, it is proved that the mean

1 Continuous because it is not possible to regard the grains as passing from
one position to another without cutting any given plane having one of those

positions on each side of it. : ;
2 It is not so in a vertical direction, on account of the weight of the grains.

LAWS OF THE BROWNIAN MOVEMENT 111

square e? of the horizontal displacement during the time ¢
increases in proportion to that time.

The same result holds for half this square or the mean
square x? of the projection of the horizontal displacement
along an arbitrary horizontal axis.1_ In other words, for a
given kind of grain (in a given fluid) the quotient : is
constant. Clearly greater the more actively the grain is.
agitated, this quotient characterises the activity of the
Brownian movement for any particular grain.

It must be borne in mind, however, that this result ceases
to be exact when the times become so short. that the move-
ment is not absolutely irregular. This must necessarily be
so, otherwise the true velocity would be infinite. T'he mini-
mum time within which trregularity may be expected is probably
of the same order as the time required by a granule, shot
into the liquid with a velocity equal to the true mean agita-
_ tional speed, before the frictional effect due to viscosity
reduces its initial energy practically to zero. (The same
time, moreover, elapses between successive molecular
impacts.) We find in this way, for a spherule 1 micron in
diameter in water, that the minimum period of irregularity
is of the order of the hundred-thousandth of a second. It
would be only 100 times greater, or one-thousandth of a
second, for a spherule | millimetre in diameter, and 100 times
smaller for a liquid 100 times more viscous. Lengths of time
such as these fall far short of the periods during which it has
been possible to observe the movement up to the present.

70.—TueE Dirrusion or EmMutstons.—We would expect
that, when pure water is left in contact with an aqueous
emulsion composed of equal sized grains, diffusion of the
grains, due to their Brownian movement, would take place
into the water by a mechanism quite analogous to that which
causes the diffusion, properly so called, of dissolved sub-
stances. It is moreover evident that such diffusion should
occur the more rapidly the more active the Brownian move-

* By resolving each displacement along two horizontal axes perpendicular
to each other, and applying the theorem as to the square on the hypothenuse
and taking the mean, we get at once e? = 22°.

112 ATOMS

ment of the grains. Making the single supposition that the
Brownian movement is completely irregular, Einstein’s
rigorous analysis shows that an emulsion diffuses like a
solution,’ and that the co-efficient of diffusion D is simply
equal to half the number that measures the activity of
agitation,
= 2 ee
ae 9° t’

Again, we are familiar with the idea that, in a vertical
column of emulsion, the permanent distribution is main-

D

* Consider a cylinder parallel to Oz, of unit area in cross section, and filled
with solution. Suppose that the concentration has the same value at all points
in the same transverse section (which will be the case when pure water is care-
fully superimposed upon a solution of sugar). The loss of dissolved substance
I across a section will be, at each instant, the mass of dissolved substance that
traverses it in one second from regions of high towards those of low con-
centration. The fundamental diffusion law states that this loss will be the greater
the steeper the fall of concentration across the section :—

Fé Pe

zs’ -x

The co-efficient D, which depends on the nature of the dissolved substance, is

the co-efficient of diffusion. Pig instance, taking the case of sugar, the state-
: 33
ment that D is equal to 36.400 ©XPresses the fact that, for a concentration
>’

gradient maintained equal to 1 gramme per centimetre, -33 gramme of sugar
passes across the transverse section considered in one day, or 86,400 times less
in a second.

Bearing this in mind, I can indicate a line of reasoning (due also to
Einstein) which, although not a rigid proof, is at any rate approximate and
which leads to the formula in question.

In a horizontal cylinder, let »’ and n” be the concentrations of the grains in
two sections s’ and s” separated by a distance X. The concentration gradient
throughout the intermediate section s will be ~ a. and a number of grains

n’—n” : : : °
equal to D me aS t will traverse the section s during time ¢. Further, assuming

that this result is produced by each grain suffering, during the time t, the dis-
placement X either towards the right or towards the left, we find that ; n’ X

4” 1 uv . . .
traverse s towards s” and 5” X towards s’, which gives, for the total drift

towards s” :—

We therefore have

! ”

n-n
Et

‘a 4 ah

or, better,
A= 2D ee
which is Einstein’s equation.

——

LAWS OF THE BROWNIAN MOVEMENT 113

tained by the equilibrium between two opposing actions,
namely gravity, which constantly drags the grains towards
the bottom of the containing vessel, and the Brownian
movement, which continually scatters them. We may give
precise expression to this conception by stating it in the
following form: for any given section, the loss by diffusion
towards the region of low concentration balances the influx
caused by gravity into the regions of high concentration.

In the special case where the grains are spheres of radius a,
to which we can attempt to apply Stokes’ law (para. 59) (I
have shown that the law holds for microscopic spherules
(para. 60) ), and assuming moreover that at equal concentra-
_ tions grains or molecules produce the same osmotic pressure,
we find that

i ie ts 1
D= N° Crag’

where € is the viscosity of the fluid, T its absolute tempera-
ture, and N Avogadro’s number. Since the coefficient of
diffusion is half the activity of the Brownian movement, we
can give the equation the equivalent form

fie SO TER
tN >” Saal’
i 2
in which we can, moreover, replace (para. 35) N by ; of the

mean molecular energy w.

Thus the activity of the agitation (or the rate of diffusion)
should be proportional to the molecular energy (or to the absolute
temperature), and inversely proportional to the viscosity of: the
liquid and to the dimensions of the grains.

71.—RoratTionAL Brownian Movement.—Up to the
present we have considered only changes in the positions of
the grains or their translational Brownian movement. But
it is known that each grain spins in an irregular fashion
during its displacement. Einstein has succeeded in estab-
lishing an equation, for this rotational Brownian movement,
comparable with the one given above, for the case of spherules
of radius a. If A? represents the mean square in time t of

A. I

114 ATOMS

the component of the angle of rotation about a given axis,
2
the quotient - , which is fixed for a given grain, characterises

the activity of the rotational Brownian movement and
should follow the equation

A* RFE 1

tN aaate
the activity of the rotational agitation being, as for the trans-
lational activity, proportional to the absolute temperature
and inversely proportional to the viscosity. It varies, how-
ever, inversely with the volume and not inversely with the.
dimensions of the grain. A sphere of diameter 10 will have.
translational agitation 10 times and rotational agitation
1,000 times more feeble than a spherule of diameter 1.

It is not possible to indicate here the way in which this
equation is derived ; we may point out, however, that it
implies, for a given granule, equality between the mean trans-
lational and mean rotational energies, as was predicted by
Boltzmann (para. 42). We shall verify this when we succeed
in verifying Einstein’s equation.

EXPERIMENTAL VERIFICATION.

Such, in its broad outlines, is the remarkable theory we
owe to Einstein. It is well adapted to accurate experimental
verification, provided we are able to prepare spherules of
measureable radius. Consequently, ever since I became,
through M. Langevin, acquainted with the theory, it has
been my aim to apply to it the test of experiment. As
we shall see, the experiments that I have carried out
myself or supervised in others demonstrate its complete
accuracy.

72.—TuHE COMPLICATED NATURE OF THE TRAJECTORY OF A
GRANULE.—We have assumed that the Brownian movement
(at right angles to gravity) is entirely irregular and have
seen that this assumption is the basis of Einstein’s theory.
However probable this may be, it is important that it should
~ be established on an exact basis.

LAWS OF THE BROWNIAN MOVEMENT 115

We will deal first of all with the measurement of the
successive displacements (horizontal) undergone by the
same grain. To accomplish this we have only to note in the
camera lucida (under known magnification) the positions
occupied by a grain after successive equal time intervals.
In the adjoining figure three diagrams are shown, the scale
being such that sixteen divisions represent 50 microns.
These diagrams were obtained by tracing the horizontal
projections of the lines joining consecutive positions occu-

| wal
ze) (mms:
| eH 1
\ a Se
re at Ne
Vina | 4 YA La
es: { \ A
\ V
“\\ a TN
K P a SS f
it N A \ ater yi I
ea \ =_—
\ ASS
} ly TNA,
4 — - 3 NN
Say
. Deg
F
Fie. 8,

pied by the same mastic grain (radius equal to -53 w); the
positions were marked every 30 seconds. It is clear from
these diagrams that the projection of each segment along any
horizontal axis whatever can readily be obtained (being
given by the abscissze or ordinates as measured by the squares
on the paper). ,

As a matter of fact diagrams of this sort, and even the
next figure, in which a large number of displacements are
traced on an arbitrary scale, gives only a very meagre
idea of the extraordinary discontinuity of the actual tra-

12

116 ATOMS

jectory. For if the positions were to be marked at intervals
of time 100 times shorter, each segment would be replaced
by a polygonal contour relatively just as complicated as the
whole figure, and so on. Obviously it becomes meaningless
to speak of a tangent to a trajectory of this kind.

73.—THE CoMPLETE IRREGULARITY OF THE AGITATION.—
If the movement is irregular, the mean square X? of the

Fig. 9.

projection onto an axis will be proportional to the time.
And as a matter of fact the record of a large number of
positions has shown that this mean square is, for a length of
time of 120 seconds, very nearly twice what it is for 30
seconds.

1 It is not even necessary to follow the same grain, or to know its size. For
any one series of grains we need only know the displacements d and d’ relative
,

to the lengths of time 1 and 4. The quotient d has the mean value 2.

LAWS OF THE BROWNIAN MOVEMENT 117

The possibility of an even more complete verification is
suggested by an extension of the line of reasoning developed
by Maxwell (para. 35) in connection with molecular speeds, to
the displacements of granules. His arguments should apply
equally well in either case. |

Thus projections of displacements along any axis, like
projections of velocities (considering equal spherules during
equal times) must be distributed about their mean value
(which by symmetry. is zero) according to Laplace and
Gauss’ law of probability.*

M. Chaudesaigues, working in my laboratory, has made
the necessary calculations from a series of positions observed
in one of my gamboge preparations (a == -212y). The
number n of displacements having projections lying between
two successive multiples of 1-7 » (corresponding to 5 milli-
metres on the squared paper used) are indicated in the
following table :—

oe : First series. Second series.
Projections (in 4) = ie a * =
lying between:— =|» Found. jCalculated.| » Found. jn Calculated.
0 and 1-7 38 48 48 4k
7, 84 44 43 38 40
ey ol 33 40 36 35
ei; 68 33 30 29 28
Go“), °°. 85 35 23 16 21
mo... 10-2 ll 16 15 15
nee sy LED 14 11 8 10
E1:D ,, .18°6 6 6 7 5
ee. 15:8 5 + 4 +
lors ,, 17-0 2 2 + 2

Another and still more striking verification, which was
suggested to me by Langevin, is obtained by shifting the

* That is to say, out of #1 segments considered,
x2

pS Oe 4 TE Se dx,
1 VQn xX

will have a projection lying between 2; and x2 (the mean square X* being
measured as above).

118 ATOMS

observed horizontal displacements in directions parallel to
themselves, so as to give them all a common origin.! The
extremities of the vectors obtained in this way should dis-
tribute themselves about that origin as the shots fired at a
target distribute themselves about the bull’s-eye. This is
~ seen in the figure given below (Fig. 10), on which 500 of my
observations with grains of radius -367 are recorded ;
positions of grains were noted every 30 seconds. The mean

Fie. 10.

square e? of these displacements was equal to the square of
7-84. The circles marked in the figure have radii

€ 2e 3e |

Pe ae eee

Here again we have a quantitative check upon the theory ;

the laws of chance enable us to calculate how many points
should occur in each successive ring. In the table on the
following page, alongside the probability P that the end —
point of a displacement should fall in each of the rings, are
given the numbers 7 calculated and found for 500 displace-
ments observed. |

. etc.

-1 This comes to the same thing as considering only grains starting from the
same point,

LAWS OF THE BROWNIAN MOVEMENT 119

Displacement between :— 3 pits aoe n Calculated. nm Found.
é

0 and i: 0638 32 34
é e
:. 25 ‘ ‘167 83 - 78
é é

27 PY 37 ‘214 107 106
é €

35 x AG ‘210 105 103
é é

45 * oF : ‘150 75 75
é €

oF 3 67: ‘100 50 AQ
e e :

67 e TF: ‘054. 27 30
é é

% ms 87° ‘028 14 17
é é

87 a IF: ‘014 fe 9

A third verification is to be found in the agreement estab-
lished between the values calculated and found for the

quotient : of the mean horizontal displacement d by the

mean quadratic displacement e. By a Ime of reasoning
quite analogous to that which gives the mean speed G in
terms of the mean square U? of the molecular speed, it is

shown that d is very nearly equal to 5° As a matter of
fact, for 360 displacements of grains of radius -53 yu, I found
“ equal to -886 instead of -894 required by the theory.

Further verifications of the same kind might still be
quoted, but to do so would serve no useful purpose. In
short, the irregular nature of the movement is quantitatively
rigorous. Incidentally we have in this one of the most
striking applications of the laws of chance.

120 : ATOMS

74.— EARLY VERIFICATIONS OF EINSTEIN’S THEORY (FOR
DIsPLACEMENTS).—When his formule were first published
Einstein pointed out that the order of magnitude of the
Brownian movement apparently fitted in completely with
the requirements df the kinetic theory. Smoluchowski,
from his point of view, came to the same conclusion after a
searching analysis of the data then available (the fact that
the phenomenon is independent of the nature and density
of the grains, qualitative observations on the increase in the
agitation as the temperature rises or the radius becomes
smaller, rough measurements of displacement for grains of
the order of 1 micron). |

From this it was undoubtedly possible to conclude that
the Brownian movement is certainly not more than 5 times
more active and certainly not more than 5 times less active
than the degree of agitation predicted by theory. This
approximate agreement in order of magnitude and qualita-
tive properties immediately gave considerable support to the
kinetic theory of the phenomenon, as was clearly brought
out by the authors of that theory.

Until 1908 we do not find in the published literature any
verification or attempt at verification that adds anything
to the information embodied in the conclusions of Einstein
and Smoluchowski.t About this time a very interesting
though partial verification was attempted by Seddig.2 This
author compared, at various temperatures, the displacements
undergone in suctessive tenths of a second by ultra-micro-
scopic grains of cinnabar, which were supposed to be very
nearly equal in size. If Einstein’s formula is correct, the

1 I cannot even except the work published by Svedberg on the Brownian
movement | Zeit. fiir Electrochemie, ¢. XII., 1906, pp. 853 and 909; Nova Acta
Soc. Sce., Upsala, 1907] for the following reasons :—.

(i.) The lengths given as displacements are 6 or 7 times too great, which,
even supposing they were correctly defined, would not contribute any particular
advance, particularly to Smoluchowski’s discussion of the subject.

(ti.) Svedberg believed, which is a much more serious matter, that the
Brownian movement becomes oscillatory for ultramicroscopic grains. He
measured the wave length (?) of this motion and compared it with Einstein’s
displacement. It is obviously impossible to verify a theory on the basis of a
phenomenon which, supposing it to be correctly described, would be in contra-
diction to that theory. 1 would add that the Brownian movement does not show
an oscillatory character on any-dimensional scale.

2 Physik. Zeitschr. Vol. IX., 1908, p. 465,

s

LAWS OF THE BROWNIAN MOVEMENT 121

mean displacements d and d’ at the temperature T and T’

(viscosities € and £') should be to one another in the ratio :—

¢./E JE

or, for the temperature interval 17°—90° C.,

d FASE Lets BE ROSY) ny ers

dad “ 273417" V ‘0032

Experiment gives 2-2. The discrepancy is well within the
possible error.

Seddig’s approximate measurements bring out the influence
of viscosity much more than that of the temperature (the
effect of the latter in the example quoted is 7 times smaller
than the viscosity influence, and it would be difficult to
make it very apparent).

Having in my possession some grains of accurately known
radius, I was able, at about the same period, to undertake
absolute? measurements and to inquire whether the quotient
= BEER Mh
X2 . 3naké
according to Kinstein’s equation, has actually a value
independent of the nature of the emulsion and sensibly equal
to the value found for N.

That such is actually the case appeared at the time to be
far from certain.? An attempt by V. Henri to settle the
question by a kinematographic experiment, in which for the
first time precision was possible,* had just led to results dis-
tinctly unfavourable to Einstein’s theory. I draw attention
to this fact because I have been very much struck by the
readiness with which at that time it was assumed that the
theory rested upon some unsupported hypothesis. I am
convinced by this of how limited at bottom is our faith in

which should be equal to Avogadro’s number N —

+ It is often stated that the Brownian movement may be seen to beeome
more active as the temperature is raised. Actually, from mere inspection, we
could affirm nothing if the viscosity did not diminish.

7 pe hae Rendus, Vol. CXLVII., 1908; Ann. de Ch. et Phys, Sept.,
, ete.

% Compare, for instance Cotton, Revue de Mois (1908).

* Comptes Rendus, 1908, p. 146. The method was quite correct and had the
merit of being then used for the first time. I do not know what source of error
falsified the results.

122 ATOMS

theories ; we regard them as instruments useful in discovery
rather than actual demonstrations of fact.

As a matter of fact, after the completion of the first series
of measurements of displacements it became clear that
Kinstein’s formula is accurate.

75.—CALCULATION OF THE MOLECULAR MAGNITUDES FROM
THE BROwNIAN MoveMent.—l have carried out personally,
or directed in others, several series of measurements, varying
the experimental conditions as much as I was able, parti-
cularly the viscosity and the size of the grains. The grains
were picked out in the camera obscura,! the microscope
being vertical, which gives the horizontal displacements
(measured in a micrometer objective). The positions of the
erains were generally marked off at 30-second intervals, four
positions being obtained for each grain.

I have worked out the method with the help of M. Chaude-
saigues, who wished to undertake (series IT and III.) measure-
ments with the grains (a = :212 u), which had given me a
good value for N from their vertical distribution. He used
a dry objective (Cotton and Moulton’s ultra-microscopic
arrangement). ‘The following series were obtained with an
immersion objective, which permits of a better control of the
temperature of the emulsion (temperature variations are
important because of the viscosity changes they cause). I
obtained the values in series IV. (mastic) in collaboration
with M. Dabrowski; series VI. (in which the liquid was very
viscous, X being of the order of 2 y in five minutes) in colla-
_ boration with M. Bjerrum. Series V. refers to two very large
mastic grains (obtained in a manner to be described later) ;
their diameters were measured directly in the camera lucida
and they were suspended in a urea solution of the same
density as mastic.

The following table, in which is given, for each series, the
mean value of the viscosity €, the radius a of the grains,
their mass m, and the approximate number n of the displace-
ments recorded, summarises the experiments described
above :—

- 1 Tt is a matter of real difficulty not to lose sight of the grain as it incessantly
rises and sinks, Vertical displacements weré measured in series VI. only.

‘ree ido

sare ae

LAWS OF THE BROWNIAN MOVEMENT 123

Radius Mass Displace-
100 Nature of the Emulsion. of the ments cama
; Grains. |” * 10”.| Recorded. | !¥*:
a
1 | I. Gamboge grains . 3 ‘50 600 100 80
1 | IL. Gamboge grains. ; -212 48 900 69-5
4to 5| III. The same grains in sugar
solution (35 per cent.)
(temperature only
roughly known) . : 212 48 400 55
1 |IV. Mastic grains : : 52 650 1,000 72-5
1-2| V. Very large grains (mastic)
in urea solution (27 per
cent.) . - : ; 5-50 | 750,000 100 78
125 | VI. Gamboge grains in gly-
cerine (34; water) . 385 290 100 64
1 | VIL. Gamboge grains of very
uniform equality . : 367 246 1,500 68-8

It may be seen from the table that the extreme values of
the masses bear a ratio to one another of more than 15,000
to 1, and that the extreme values of the viscosities are in the
ratio of 1 to 125. Nevertheless, whatever the nature of the

N
1022
remains in the neighbourhood of 70, as in the vertical distri-
bution experiments. This remarkable agreement proves
the rigorous accuracy of Einstein’s formula and in a striking
manner confirms the molecular theory.

The most accurate measurements (series VII.) refer to the
most equal set of grains that I have prepared. The prepara-

intergranular liquid or of the grains, the quotient

1 To these results might be added Zangger’s measurements [Zurich, 1911],
. which were published later. They were obtained from measurements of the
lateral displacements of mercury droplets sinking through water. The
measurements are of interest in that they could be made to refer to a single
drop, the radius of which could be obtained from its rate of fall. But thig
application of Stokes’ law to a liquid sphere falling through a liquid is not

permissible without a correction that affects the result found for — Toz (60 to 79),
and which, according to a calculation by Rybezinski, increases that result by
about 10 units,

124 ATOMS

tion and the objective (immersion) were surrounded by water,
thus enabling temperature (and consequently viscosity) to
be measured accurately. The illuminating beam, of suffi-
ciently feeble intensity, was filtered through a trough of
water. The emulsion was very dilute. The microscope
was focussed upon the level (6 4 above the bottom) at the
height h such that a grain of the size under consideration
had the same probability of being above or below it. In
order not to be tempted to choose grains which happened to
be slightly more visible than the rest (those, that is to say,
which were slightly above the average size), which would
raise the value of N a little, I followed the first grain that
showed itself in the centre of the field of vision. I then dis-
placed the preparation laterally by 100 1, once more followed
the first grain that showed in the centre of the field at the
height h, and so on. The value obtained, 68-8, agrees to
within nearly 1 per cent. with that derived from the dis-
tribution of the grains in a vertical column of emulsion
(para. 68).
I shall therefore adopt for Avogadro’s number, the value

68-5 x 10”,
which gives for the electron (in electrostatic units) the value
4-2 x 10°10

and for the mass of the hydrogen atom (in grammes) the
value
1:47 > 102*,

76.—MEASUREMENTS OF THE ROTATIONAL BROWNIAN
MoveMEnT (LARGE SPHERULES).—We have seen that Ein-
stein’s generalised theory is applicable to the rotational
Brownian movement, in which case the formula becomes

AS RD

t. N. “ 4nat?’

where A? stands for the third of the mean square of the angle
of rotation in time f. :
In verifying this formula we check at the same time the

LAWS OF THE BROWNIAN MOVEMENT 125

estimates of probability that figure in its demonstration and
which we meet with whenever we require to establish
equipartition of energy ; in this particular case this means
equality between the mean energies of rotation and of trans-
lation. The same difficulties that were met with above (41
to 43), with regard to the limits of applicability of such
equipartition, increase the desirability of a verification.

The formula, however, indicates a mean rotation of about
8 degrees in the one-hundredth of a second, for spheres 1
in diameter; such rotation is too rapid to be perceived
(more especially as no distinguishing marks can be noticed
on such small spherules), much less to. be measured. And
as a matter of fact, this rotation has never been made the
subject of any experimental study, even qualitative.

I have overcome the difficulties in the way by preparing
very large gamboge and mastic spherules. This was done
by precipitating the resins from alcoholic solution, not in the
usual way by the sudden addition of a large excess of water
(which produces grains of diameter generally less than 1
micron), but by causing the precipitating water to penetrate
slowly and progressively into the resin solution. This was
managed by very slowly running pure water from a funnel
with a very slender spout under an alcoholic solution of resin
(dilute), which is steadily forced up by it. A zone is estab-
lished between the two liquids across which they diffuse into
each other, and the grains that are formed in the zone have
diameters of quite a dozen microns. They therefore soon
become so heavy that they sink, in spite of their Brownian
movement, passing downwards through the pure water,
where they are washed, to the bottom of the apparatus,
from which they can be recovered after decantation of the
supernatent liquid. In this way I have precipitated all the
resin in alcoholic gamboge and mastic solutions in the form
of spheres having diameters as high as 50. These large
spheres look like glass balls, yellow with gamboge, colourless
with mastic, which are readily broken up into irregular frag-
ments. They often appear to be perfect and, like lenses,
they produce real, recognisable images of the source of light
which illuminates the preparation (an Auer mantle, for

126 ATOMS

instance). Frequently, however, they contain inclusions 1
by means of which the rotational Brownian movement may
easily be perceived.

Unfortunately the weight of these grains keeps them always
very near the bottom of the vessel, where their Brownian
movement may possibly be affected by cohesion pheno-
mena. I have therefore tried, using solutions of various
suitable substances, to render the intergranular liqu'd of the
same density as the grains themselves. With nearly all the
substances, however, a complication arose, in that the con-
centration necessary to keep the grains just suspended with-
out rise or fall was sufficient to cause the grains to coagulate —
into grape-like clusters. This provides a very pretty illus-
tration of the phenomenon of coagulation and its mechanism,
which is not very easily demonstrated with ordinary colloidal
solutions (in which the grains are ultra-microscopic). With
urea alone does coagulation not take place. |

I have thus been able to follow the agitation of the grains -
in water containing 27 per cent. urea (series [V. in the pre-
ceding table). At the same time it has been found possible
to measure, more or less roughly, their rotation. In doing
this I marked, at equal intervals of time, the successive
positions of particular granular inclusions. This enabled me
subsequently to fix the orientation of the spheres at each
instant and to calculate approximately their rotation from
one instant to another. Calculations based on about 200
measurements with spheres 13 yu in diameter gave me, by the
application of Einstein’s formula, the value 65 x 10? for N,
the probable exact value being 69 x 107%. In other words,
starting from the latter value for N, we should expect to
find for »/A?, in degrees per minute, the value 14°; by
experiment we find 14:-5°.

1 These inclusions do not appreciably affect the density of the grains ; in an
aqueous urea solution mastic grains remain in suspension, in solutions containing
equal quantities of urea, whether they do or do not contain inclusions. I have
also investigated the nature of the inclusions, which probably consist of a viscous
paste containing traces of alcohol.

In exceptional cases a grain is sometimes found to be made up of two spheres
united about a small circle, an effect clearly due to the fusion of two spheres
whilst they are still growing from their respective nuclei. The dual question of
the initial formation of the nuclei and their rate of growth has an interest
outside the scope of the present inquiry.

LAWS OF THE BROWNIAN MOVEMENT 127

The discrepancy is well below the possible error introduced
by the somewhat loose approximations used in connection
with the measurements and in making the calculations. ‘The
agreement is still more striking because a priori we know
nothing even of the order of magnitude of the phenomenon.
The masses of the grains observed were 70,000 times greater
than those of the smallest studied in the determination of
vertical distribution.

77.—Tue Dirrusion oF LARGE MOLECULES.—To carry out
our intention of establishing the various laws deduced by
Einstein on an experimental basis, it only remains to study
the diffusion of emulsions and to see whether the value of N
derived from the equation

agrees with that already found.

In this connection it is proper to refer to the application by
Einstein himself of his formula to the diffusion of sugar into
water. In applying the formula to this particular case it is
assumed—(i.) that sugar molecules may be regarded as very
nearly spherical, and (ii.) that Stokes’ law is applicable to
them. (It is therefore not surprising that the value expected
was not obtained.)

Making these assumptions, the equation in question,
applied to the case of sugar at 18° C., becomes !

aN = 3-2 x 1616

We do not, however, know what radius may be assigned to
the sugar molecule, for we cannot calculate it by the process
available for volatile substances.

It may be pointed out, as has been done above (para. 47),
that we obtain some indication of the “ true’’ volume

e 7a®N) of the molecules making up a gramme molecule

1 For we know (para. 37: note) that R is equal to 83°2x 10°, that D is
equal toga sayy (para. 71: note), and that T is equal to (273 + 18) °C. Moreover,
the viscosity at this temperature of the pure intermolecular water, to which the
reasoning applies (and not the total viscosity of the sugar solution), is -0105
(para. 48: note).

128 ATOMS

of sugar by measuring the volume (208 c.c.) occupied by that
quantity of sugar in the crystalline state. Einstein has very
neatly overcome the difficulties in the way by calculating
this volume from the viscosity of the sugar solution. He
did this by showing, from the laws of hydrodynamics, that
an emulsion of spherules should be more viscous than the
pure intergranular liquid and that the relative increase in

viscosity © . ~ is proportional to the quotient of the ~

volume V of the emulsion into the true volume v of the
spherules present therein. The first calculations actually

/

indicated pure and simple equality between it and si

Extrapolating this theory, once established for emulsions,
to the case of a sngar solution, Einstein obtained in an
approximate manner the true volume of the molecules
making up a gramme molecule of sugar. Using the value
already obtained for the product aN, he found (1905) the
value 40 x 10? for the number N.1

A few years later M. Bancelin, working in my laboratory,
set himself to verify the formula given for the relative
increase in viscosity (which promised to be easy with gamboge
or mastic emulsions). It was at once apparent that the
increase predicted by the formula was too small.

On hearing of this lack of agreement Einstein noticed that

1 The results of a subsequent verification of the diffusion formula by Svedberg
[Zeit. fir Phys. Chem., Vol. LXII., 1909, p. 105] may be compared with this.
Svedberg used colloidal gold solutions, the grains being invisible under the
microscope. The diameter of the grains, calculated, according to Zsygmondy’s

method, to be -5 x 10-’, and the co-efficient of diffusion (equal to = that ‘of

the sugar solutions), should give about 66 x 10” for N. The high degree of
uncertainty involved in the measurement (and even in the definition) of the
radii of invisible granules (which are probably sponge-like bodies of widely
differing bulk) renders these results on the whole less convincing than those
deduced by Einstein from the diffusion of molecules that were not invisible,
very much less massive, and identical among themselves.

Svedberg has also carried out certain relative measurements, wherein he
compares the diffusions of two colloidal gold solutions, the grains in the one
being (on the average) 10 times smaller than the grains in the other; from
colorimetric measurements he drew the conclusion that 10 times as many more
small grains than large pass through identical membranes in the same time.
This is just what would be expected from the formula (supposing always that
the pores in the parchment were sufficiently large).

LAWS OF THE BROWNIAN MOVEMENT 129

an error had occurred, not in the reasoning, but in the calcula
tion, and that the correct formula should be
Ce OS eee,
“a = 25 ye
which agrees with the measurements. The corresponding
value for N is now found to be

65 x 102,

which agrees remarkably well with the accepted value. This
forces us to take the view that sugar molecules possess a
more or less compact structure, even if they are not spherical,
and that Stokes’ law is, moreover, applicable to molecules
which are certainly relatively large, although their diameters
do not exceed the thousandth of a micron.

78.—FinaL EXPERIMENTAL PRooF: THE DIFFUSION OF
VISIBLE GRANULES.—As he himself demonstrated, Einstein’s
- diffusion equation .

which can be only approximate for molecules, happens to
be rigorously obeyed by emulsions. In fact, since this equa-
tion is the necessary consequence of Stokes’ law and the
vertical distribution law, it may be regarded as verified in
the domain in which I have shown that these laws apply.

Direct measurements of diffusion, however, if carried out
in such a way as to extend that domain, have a certain
interest.

When, therefore, M. Léon Brillouin made known to me his
wish to complete the experimental verification of Einstein’s
theory by studying the diffusion of emulsions, I suggested to
him the following method, which makes use of the obstacle
that prevented my studying permanent equilibrium in pure
glycerine, in which the grains stick to the glass walls of the
containing vessel when they chance to come in contact with
it (para. 66: note).

Consider a vertical glass partition enclosing an emulsion,
initially of uniform distribution, composed of gamboge grains

A K

130 ATOMS

in glycerine, the number of grains per unit volume being n.
The partition, which behaves as though it were a perfect
‘‘ absorber,” captures all grains brought by chance Brownian
movements into contact with it, so that the emulsion
becomes steadily weaker by diffusion towards the glass,
- while the number J? of the grains collected by unit surface
steadily increases. The variation of #2 with the time will
determine the coefficient of diffusion.

The absorbing partition observed will be the lower surface
of the object-glass confining a preparation maintained verti-
cally at an absolutely constant temperature. The thickness
of the preparation will be sufficiently great to ensure that
during observations extending over several days the absorp-
tion by the cover-glass will be throughout what it would be
if the emulsion extended to infinity.?

The following approximate line of reasoning enables us to
deduce the coefficient of diffusion D from the measurements
taken. -

Let X? throughout be the mean square (equal to 2D?)

of the displacement during time ¢ that elapses from the
beginning of an experiment. No great error will be intro-
duced if we assume that each grain has. undergone, either
towards the absorbing partition or in the opposite direction,
the displacement X. The number J2 of the grains stopped
by unit surface during the time ¢ is then clearly

a
VN — 5 nX,
from which we get, replacing X by /2Df, -
pep)”
Bree
2 sp?
or DS oat es

which is the required coefficient of diffusion.

1 The grains, being slightly less dense than glycerine, slowly rise (about 1
millimetre in two weeks at the temperature of oxperiaey This fact has no
influence on #2 if the preparation is deep enough to ensure that the surface
studied always remains above the lower layers that are impoverished by this
rising of the grains.

a) re

‘
i i i i

LAWS OF THE BROWNIAN MOVEMENT 131

M. Léon Brillouin carried out the experimental work and
obtained measurements—a work of considerable difficulty—
with much skill. Gamboge grains, equal in size, (radius -52 p)
freed by desiccation from intergranular water, were treated
for a long time with glycerine, a dilute uniformly distributed
emulsion containing 7-9 <x 108 grains per cubic centimetre
being obtained (the volume of the grains thus did not come

within — of that of the emulsion). Diffusion took place
in a thermostat constant at 38-7° C., at which temperature
the viscosity of the glycerine
employed was 165 times that
of water at 25°C. Twice a
day the same portion of the /” 7
partition to which the grains | "3
were adhering was_ photo-
graphed and the grains
counted on the negatives. The
diffusion was followed in six
preparations, each during the
course of several days.!

_ Examination of the series
of negatives showed that the
square of the number of grains
fixed. is roughly proportional
to the time, so that, plotting ,
the results so that the abscissz Vumber of 6)

—

€
(ea hogrs Z

/ -

= =

So

: VA Diff sion\ of Sb herdles
j in Brownian

movement

mM © KR wm DD YN &
ae

represent the values of Sand 0 0 200 300 400 <0 600 700
the ordinates the time /t, the sa th ier

points representing the measurements fall roughly on a
straight line passing through the origin, as is shown in the

adjoining figure. The coefficient D, equal to 4 follows

* M. Brillouin has examined qualitatively preparations kept at the melting
point of ice, at which temperature the viscosity of glycerine becomes more than
3,000 times that of water. The Brownian movement, which is quite difficult
to perceive with the viscosity at its initial value, now appears to be completely
arrested. It occurs, nevertheless, and successive photographs show that grains
diffuse slowly towards the partition’, the number of grains which happen to
adhere to it increasing with time in the right way, although it was not possible
to wait long enough for accurate measurements to be taken.

K 2

132 ATOMS

at once. It is found to be equal to 2-3 x 10-1 for the grains
employed, deduced from the fixation of several thousand
grains ; this corresponds with a rate of diffusion 140,000
times slower than that of sugar in water at 20° C.

To verify Einstein’s diffusion equation, it only remains to
A bee
D 6naé
a matter of fact, it is equal to 69 x 10” to within + 3 per
cent.

79.—SumMAryY.—The laws of perfect gases are thus applic-
able in all their details to emulsions. This fact provides us
with a solid experimental foundation upon which to base the
molecular theories. The field wherein verification has been
achieved will certainly appear sufficiently wide when we
remember :

That the nature of the grains has been varied (gamboge,
mastic) ;

That the nature of the intergranular liquid has been
varied (pure water, water containing 25 per cent. urea or 33
per cent. sugar; glycerine, containing 12 per cent. water,
pure glycerine) ;

That the temperature varied (from — 9°C. to +
58° C.) ;

That the apparent density of the grains varied (between

—-03 and + -03) ;

That the viscosity of the intergranular liquid varied (in the
ratio of 1 to 330) ;

That the mass of the grains "paxded (in the enormous ratio
of 1 to 70,000) as well as their volume (in the ratio of 1 to
90,000).

From the study of emulsions the following values have

gee whether the number is near 70 x 107%. As

been obtained for a

68-2 deduced from the vertical distribution of grains.
68-8 deduced from their translatory displacements.
65 deduced from observations on their rotation.

69 deduced from diffusion measurements.

If we wish we may express our results by stating that the

LAWS OF THE BROWNIAN MOVEMENT © 133

mass of the hydrogen atom, in terms of trillionths of trillionths
of a gramme, has the values 1-47, 1-45, 1-54, and 1-45 respec-
tively.

As we shall see later, other facts imply a discontinuous
structure for matter, and, like the Brownian movement,
enable us to estimate the masses of the structural units.

[ y \ 2

CHAPTER V
FLUCTUATIONS

SMOLUCHOWSKIS THEORY.

THE molecular agitation of which the Brownian movement
is the direct manifestation can be inferred from other sets of
phenomena that include a constant succession of variable
inequalities in microscopic portions of matter in equilibrium.

80.—Density Fiuctuations.—We have already indicated
one of these phenomena in speaking of the definite though
very feeble thermal inequalities which are produced spon-
taneously and continuously in spaces of the order of a micron,
and which are, indeed, a second aspect of the Brownian
movement itself. These thermal fluctuations, of the order
of a thousandth of a degree for such volumes,! seem in
practice to be inaccessible to our measurements.

The density of a fluid in equilibrium, like its temperature
or molecular agitation, should vary from point to point. A
cubic micron, for example, will contain sometimes:a larger
and sometimes a smaller number of molecules. Smolu-
chowski has drawn attention to these spontaneous
inequalities, and has been able to calculate the fluctuation
n

in density, —, n being the chance number of molecules

in a volume v of fluid which in the case of rigorously constant
and uniform concentration would contain py@ molecules.

To begin with he showed, by a simple statistical argument,
that the absolute mean value of this fluctuation for a gas or a

dilute solution should be equal to V :s ace If the density of

T No

_ the gas is the so-called normal density, we see that the mean

1 According to a calculation by Einstein, based, like the formule that have
already been verified, on the kinetic theory of emulsions.

: ) FLUCTUATIONS 135

variation, for volumes of the order of a cubic centimetre, is
ef the one thousand-millionth order only. It becomes of the
order of one-thousandth for the smallest cubes resolvable by
the microscope. Whatever the density of the gas, the varia-
tion will be about 1 in 100 if the volume considered contains
6,000 molecules and 10 in 100 if it contains 60. |

Sixty molecules in a cubic micron, for fluorescein, would
make a solution of 1 part in 30,000,000 ; I do not consider it
impossible for us to succeed in observing fluorescein in such
volumes and at such dilutions, and thus for the first time to
perceive fluctuations in composition directly.

81.—CriTicaAL OPALESCENCE.—No longer confining him-
self to the case of rarefied substances, Smoluchowski
succeeded a little later, in a most remarkable memoir,! in
calculating the mean density fluctuation for any fluid what-
ever, and proved that, even with condensed fluids, the
fluctuations should become noticeable in spaces visible
under the microscope when the fluid is near the critical
state.2 He thus succeeded in explaining the enigmatic
opalescence ® which is always shown by fluids in the neigh-—
bourhood of the critical state.

This opalescence, which is absolutely stable, indicates a
permanent condition of fine grained heterogeneity in the
fluid. Smoluchowski explains it as being due to the magni-
tude of the compressibility (infinite at the critical point
itself) which enables contiguous regions of notably different

1 Acad. des Sc. de Cracovie, December, 1907.

2 It is known that for every fluid there is a temperature above which it is
impossible to liquefy it by compression ; that temperature is the critical tempera-
ture (31° C for carbon dioxide). Similarly, there is a pressure above which a
gas cannot be liquefied by cold ; that pressure is the critical pressure (71 atmo-
spheres for carbon dioxide). A fluid is in the critical state when it arrives at

its critical temperature under its critical pressure. At the point representing
the critical state in a p, v, T diagram the isothermal shows a point of inflexion,

the tangent at that point being parallel to the volume axis (at this point oP is
nothing and the compressibility is infinite).

® A liquid is opalescent if the path of a beam of light is visible in it, as in
soapy water or air charged with smoke. The light thus seen is distinguished
from fluorescent light in that, when analysed in the spectroscope, it contains
no colours that are not found in the illuminating beam, altaough its tint is
generally more bluish owing to change in the distribution of intensities (it is
also-distinguishable by the fact that, being completely polarised, it fails to reach
an eye observing it at right angles to the pencil through a suitably orientated
analyser).

136 ATOMS

density to be nevertheless almost in equilibrium with each
other. Hence, owing to the molecular agitation, the forma-
tion of dense swarms of molecules, diffuse in contour, will be
facilitated. These swarms will break up but slowly, while
at the same time others will be forming elsewhere and will
produce opalescence by causing lateral deviation of the
light.

The quantitative theory shows how the density fluctua-

tions increase as the compressibility rises.1_ Thus at the

critical point we find, in a volume which contains n molecules
in the case of uniform distribution, that the mean fluctuation
is very nearly the inverse fourth root of that number, what-
ever the fluid, which gives a value of 2 per cent. in a cube
containing 100,000,000 molecules. For most liquids in the
critical condition the side of such a cube is of the order of
the micron. The heterogeneity is thus very much more
accentuated than in a gas, and we may conceive that the
opalescence, always existing more or less, would become
very marked under such conditions.

82.— EXPERIMENTAL VERIFICATION OF THE THEORY OF
OPALESCENCE.—Smoluchowski’s theory, amplified by
Keesom, is confirmed by the results recently obtained at
Leyden by Kamerlingh Onnes and Keesom. The intensity
of the opalescence can be calculated by making use of earlier
work ? which gives the quantity of light deviated laterally
(for an illuminating pencil of given intensity and colour)

1 Smoluchowski’s statistical thermodynamical reasoning gives, for the mean

square of the fluctuation in volume ¢, an expression which, except in the im-
mediate neighbourhood of the critical point, is sensibly equal to

__ RT I
eile. 5
Pov.

v, being the specific volume corresponding to uniform distribution and SP the

0Vo
y 2
compressibility (isothermal). At the critical point, where .b and Z P’ fail,
Vo Vo
“8
the third differential as must be introduced. (See Conseil de Bruxelles,

p. 218.)

2 Rayleigh, Phil. Mag., Vol. XLI., 1881, p. 86; and Lorenz, Oeuvres L.,
p- 496 (see Conseil de Bruxelles, p. 221).

Ain "om ‘i ;
ee

FLUCTUATIONS 137

by a very small transparent particle (of fixed volume) placed
in a medium of different refrangibility. This quantity of
light is found, moreover, to be the greater the more refrangible
the incident light (that is, the smaller its wave length). Thus
for incident white light the light diffused laterally will be
blue (the blue and violet being diffused more by the particle
than the yellow or red). And the opalescence actually is
bluish.

More accurately, as long as the dimensions of the illu-
minated particle may be regarded as small compared with
the wave length of the incident light, the intensity of the
diffused light is inversely proportional to the fourth power
of that wave length, but directly proportional to the square
of the volume of the particle and to the square of the relative
difference in refractive index.!

If, as actually happens in the case of density fluctuations,
the particle which deviates the light is composed of the same
substance as the surrounding medium, this relative variation
in index is proportional to the relative variation in density,?

n—N,

that is, to the fluctuation , the mean quadratic value

of which has been given by ‘Smoluchowski. Summing all
the intensities thus separately due to the small sections
composing a perceptible volume of fluid, we find that the
intensity 7 of the light diffused by a cubic centimetre at right
angles to the incident rays is
eee. - Eee 1
—_ nis eS 2 Phe 2 et is pe ee Ae
5. Ce.” aS (Ho 1) (1, ot: 2) ‘ Op
oy Sn aes
Ov,
where p, is the refractive index (mean) of the fluid for the

* At right angles to the incident light, this intensity is given by the expression

2r? . - Fas PSs :
At, K, ?
¢ being the volume, A, the wave length in the medium outside the particle,

and « and u the refractive indices in that medium and in the particle.

: 3 This follows from the law of refraction (Lorentz), according to which
we],
a‘ ui28 constant for any fluid.

138 ATOMS

light used of wave length A (in “free space’ or vacuo),

v, the specific volume of the fluid, and = its compressibility
(isothermal). fe

All the quantities in the above equation are measurable
except. N ; a comparison of the value of N derived thus
with the value obtained already will therefore enable us to
check the theories of Smoluchowski and Keesom.

An examination of the fine series of measurements recently —
carried out on ethylene will be found to provide the required
test. The critical temperature (absolute) was 273 + 11-18° ;
the opalescent light was quite blue even at 11-92°. At this
temperature the ratio of the intensities of opalescence for
incident light of the same intensity in the blue and yellow
(lines F and D) was 1-9, but little different from the ratio
2-13 of the fourth powers of the vibration frequencies of the
two colours. 3

At the same temperature measurements in yellow light
gave, per centimetre cube illuminated and for incident light
of intensity 1, an intensity of opalescence varying between
‘0007 and -0008. The compressibility is known from
Verschaffelt’s measurements. Keesom’s formula then gives,
for Avogadro’s number N, a value in the neighbourhood of
75 X 1072 with a possible error of 15 per cent., which is in
very good agreement with the probable value.

Analogous considerations can be applied to the opalescence
always shown by liquid mixtures (water and phenol, for
example) in the neighbourhood of the point of critical
miscibility.1| Opalescence in this case indicates a permanent
condition of fluctuation in composition from one point to
another in the mixture. The theory of these fluctuations,
which is a little more difficult than in the above case, has been
given by Einstein (using the conception of work done in

1 At all temperatures below 70° C. the mutual solubilities of water and phenol
are limited ; two layers of liquid are produced, containing unequal amounts ~
of phenol. As the temperature rises, the difference between the two layers
becomes less and less, until at 70° C. the concentration of phenol becomes
equal to 36 per cent. throughout ; the dividing surface then disappears and the
point of critical miscibility is reached. At all higher temperatures miscibility
is complete and two layers of different composition can no longer remain in
equilibrium in contact with each other.

FLUCTUATIONS 139

separating the constituents instead of the idea of work
done in compression). The equation! he has obtained,
assuming it to be exact, again allows us to find N from
measurable quantities, but in this case the determination
has not yet been carried out.

-83.—TuEr BLUENESS OF THE Sky.—We have applied the
formule of Smoluchowski, Keesom, and Einstein in the
neighbourhood of the critical point. They are equally
applicable to the case of a gaseous substance. We will
suppose that the gas is pure, or at least, if it is a mixture,
that its components have the same refracting power (which is
sensibly the case for air), so that fluctuations in composition
will have a negligible influence in comparison with density
fluctuations. In this case, making use of Boyle’s law, the

]
product [ve") becomes equal to o further, the refrac-

tive index being very nearly equal to 1, we can replace
(u2 + 2) by 3, and Keesom’s equation becomes

The quantity of light thus emitted laterally by 1 cubic
centimetre of gas is extremely small, because of the feeble
refractive power of gases (u; is very little greater than 1).
But the total emission produced by a very large volume may
become noticeable, and in this way the blue light which
comes to us from the sky in the daytime can be explained
(Einstein). We thus arrive at a result obtained by Lord
Rayleigh 2 previous to the more general theories I have just
summarised.

We know that a beam of light has a visible track when
traversing a medium charged with dust. To this lateral
diffusion is due the visibility of a ‘‘ sunbeam ”’ in the air.
The phenomenon still persists as the dust particles become
increasingly smaller (and it is this fact that makes ultra-
microscopic observation possible), but the diffracted

1 Ann, der Phys., Vol. XVI., 1910, p. 1572.
2 Phil. Mag., Vol. XLI., 1871, p. 107, and Vol. XLVIL., 1899, p. 375.

140 ATOMS

opalescent light turns to blue, light of shorter wave length
thus undergoing the greater diffraction. It is, moreover,
polarised in the plane passing through the incident ray and
the eye of the observer.

Rayleigh supposed that the molecules themselves behave
like the dust particles just visible under the microscope
and that the origin of the colour of the sky lies in them. In
agreement with this hypothesis, it is found that the blue
light from the sky, when observed in a direction perpen-
dicular to the sun’s rays, is strongly polarised. It is, more-
over, difficult to believe that it is a question of actual dust
particles, for the blueness of the sky is not diminished in the
slightest at the height of 2,000 or 3,000 metres, which is well

above most of the dust that contaminates the air near the ~

earth. We may therefore conclude that we have here a
means of counting the diffracting molecules which enable
us to see a given portion of the sky and in consequence a
means for obtaining N.

Rayleigh did not restrict himself to this merely qualitative
conception, but calculated, while developing the elastic
theory of light, the relation that should, on his hypothesis,
obtain between the intensity of the direct solar radiation
and that of the light diffused by the sky. Let us suppose
that we are observing the sky in a direction the zenith-
distance of which is a and which makes an angle 6 with the
solar rays ; the illuminations e and E obtained in the field
of an objective pointed successively towards this region of
the sky and towards the sun should be, for each wave
length \, in the ratio :—

oon 2, Se aya

eo ae cosa d ES Ne
where © represents the apparent semi-diameter of the sun,
p and g the atmospheric pressure and the acceleration.due to
gravity at the point of observation, M the gramme-molecular

2.
weight of air (28-8 gramme), ze 7 : the refractive power of

air (Lorentz), and N Avogadro's constant. Langevin
obtained the same equation (with p? replaced by the dielec-

;
P
1

FLUCTUATIONS 141

tric constant K) during the course of development of a
simple electro-magnetic theory. In each case the _pre-
ceding formula was obtained by summing the intensities
of the light diffracted by the individual molecules (assumed
to be distributed in an entirely irregular manner).

Identically the same formula is obtained (for 8 = 90°) by
applying Keesom’s equation, as was shown by Einstein.

It follows that the extreme violet of the spectrum should
be 16 times more diffracted than the extreme red (the wave
length of which is twice as great), and this is well borne out
by the actual colour of the sky (which no other hypothesis
has succeeded in explaining).

The above formula takes no account of the light reflected
by the earth. The brightness of the sky would be doubled
by a perfectly reflecting earth (which would be equivalent
to the illumination of the atmosphere by a second sun).
The reflecting power of the earth entirely covered with snow
or by clouds would be little different from -7, and the bright-
ness of the sky would be 1-7 times that due to the sun alone.

An experimental verification should be possible at a
height sufficient to avoid perturbations due to dust (smoke,
small drops of water, etc.). The first indication of such a
verification was obtained by Lord Kelvin from the early
experiments of Sella, who, at the summit of Monte Rosa,
compared the brightness of the sun at a height of 40° and
the brightness of the sky at the zenith at the same instant
and obtained a ratio equal to 5,000,000. This gives for
N x 10~ * (allowing for the absence of precision with regard
to wave length) a value between 30 and 150. Roughly, the
correct order of magnitude was attained.

Bauer and Moulin! have constructed an apparatus for
making the necessary spectrophotometric comparison and
have made some preliminary measurements on Mont Blanc,
with, unfortunately, a not very favourable sky.2 Their
comparisons give (for green light) numbers between 45 and
75 for N x 10-22,

1 Comptes Rendus, 1910.

* The presence of water droplets made the value found for N too small, and

their effect was intensified by the fact that the wave length used for comparison
was too large.

142 ATOMS

A long series of measurements has, however, just been
completed with the same apparatus on Monte Rosa by
M. Leon Brillouin, and a provisional scrutiny (gauging of
the absorbing plates and comparison of the negatives) gives
numbers in the neighbourhood of 60. There is thus no
doubt that Lord Rayleigh’s theory is verified and that the
familiar blue colour of the sky is one of the phenomena
through which the discontinuous structure of matter is
made manifest to our observations on the usual dimensional
scale.

84.—CHEMICAL FLUcTUATIONS.—-Up to the nome we
have not attempted to formulate a kinetic theory of chemical
reaction ; without going deeply into the matter at this early
stage, a few simple remarks may be pertinent.

We will limit ourselves to the consideration of two par-
ticularly important and simple types of reaction, types
which, in fact, by addition or repetition make up all classes
of chemical reaction. On the one hand we have dissociation
or the splitting up of one molecule into simpler molecules or
into atoms (I, into 21; N,O, into 2NO,; PCl, into PCI,
+ Cl,, ete.), which is expressed in general form by

A —> A’ + A’;

on the other hand we have the inverse phenomenon or the
building up of a molecule, expressed by

A <«— A’ + A”.

If at a given temperature two inverse transformations

exactly counterbalance one another :—

A A’. -+ A’, .
so that on our scale of observation the quantities of the com-
ponents remain constant throughout the system, we say
that chemical equilibrium has been reached and that no
further change will take place.

Actually both reactions are taking place, and at each
instant an enormous number of molecules are breaking up at
certain points while at others an equivalent amount of A-
is being re-formed. I have no doubt that in microscopic
spaces we should be able to see, at a sufficient magnifi-

FLUCTUATIONS 143

cation, an incessant fluctuation in chemical composition.
Chemical no less than physical equilibrium in fluids is merely
an illusion that masks a continuous cycle of compensating
transformations.

A quantitative theory of this chemical Brownian move-
ment has not yet been developed. But, though only qualita-
tive, the kinetic conception of equilibrium has rendered
great services. It is the real basis of the whole of chemical
mechanics that is concerned with velocities of reaction
(Law of Mass Action).

85.—FLuctuaTions IN MOLECULAR ORIENTATION.—The
remarkable phenomenon discovered by Mauguin during the
course of his splendid work on liquid crystals falls into the
same group of phenomena as the Brownian movement and
the fluctuations of density and composition.

It has been known, since Lehmann’s famous investiga-
tions, that there are some liquids which exhibit when in
equilibrium the optical symmetry of uniaxial crystals, so
that when a film of one of them is examined under the
microscope between a polariser and analyser set at the
extinction point illumination is re-established, except where
the crystalline orientation of the liquid is parallel to the ray
of light traversing it. When, however, the light is very
intense, we notice that extinction is not absolute for such
orientations and that an incessant scintillation, like the
swarming of a luminous ant heap, is visible at all points in
the field, producing a feeble light that varies rapidly from
place to place and from instant to instant.1 Mauguin at once
connected this phenomenon with the Brownian movement,
and, indeed, it seems difficult to explain it except on the
supposition that the molecular agitation continuously tends
to strain the molecular axes from their positions of equili-
brium. Analogous fluctuations should occur during the
magnetisation of ferro-magnetic bodies, and undoubtedly
the theories of ferro-magnetism (P. Weiss) and of liquid
crystals will reduce to a common basis.

1 This is well shown by para-azoxyanisol, spread out in a thin film between
two accurately plane glass surfaces (the crystalline axes being thus fixed per-
pendicularly to the plane surfaces) and maintained at temperatures lying between
130° C. and 165° C. (outside these temperature limits change of state occurs).

CHAPTER VI
LIGHT AND QUANTA

Buiack Boptzs.

86.—ANny CAVITY COMPLETELY ENCLOSED BY A MATERIAL
AT A UNIFORM TEMPERATURE IS FULL OF LiauT IN STATIs-
TICAL EqQui~iprium.—When a fluid fills an enclosure,
molecular agitation, which is the more active the higher the
temperature, gradually transmits from point to point all
thermal actions and the degree of agitation gives a measure
of the temperature once equilibrium is established. But we
know that, even in the absence of all intermediary matter, |
the temperature of the space inside an isothermal enclosure
(an enclosure in which, that is to say, the temperature is
uniform) has a definite physical significance ; we know that
a thermometer always ends by giving the same indication
(it arrives, that is to say, at the same final state) at any
point whatever in an opaque enclosure surrounded with
boiling water, whether the enclosure contains any fluid
whatever or whether it is absolutely empty. The effect
upon the thermometer in the latter case is produced solely
by radiation from the various points of the enclosing
medium. |

This radiation is visible or not according to the tempera-
ture of the enclosure (an ice-house, an oven, or an incan-
descent furnace), but its visibility, which is of importance to
us alone, has no claim to be regarded as an essential charac-
teristic of the radiation, which is light in the general sense
of the word and traverses space at the invariable velocity
of 300,000 kilometres per second.

When we say that the enclosure is sealed and that it is
opaque, we mean that no thermal influence can be exerted
by radiation between two objects, one of which is inside and

LIGHT AND QUANTA 145

the other outside the enclosure. This is the reason why a
thermometer inside the enclosure reaches and persists in a
definite invariable state. This does not mean, however,
that no subsequent change takes place in the region wherein
the indicating thermometer is placed. That region is
constantly receiving radiation emitted by the various parts
of the enclosure ; the fixed indication shown by the receiving
_ instrument (thermometer), however, proves that it under-
goes no further change in property, but maintains itself in
a stationary condition.

This stationary state in space traversed continually and
in all directions by light really represents a permanent
condition of extremely rapid changes. Details of them
escape us, in spaces and times on our usual dimensional scale,
just as the agitation of the molecules in a fluid in equilibrium
cannot be perceived, although the latter phenomenon is of a
much higher order of magnitude. ‘In fact, the thermal
equilibrium in fluids, which has already been studied at
length, and the thermal equilibrium of light are in many
respects comparable. I now propose to define our concep--
tions of the latter equilibrium.

I have pointed out that a thermometer invariably registers
the same temperature, at all points inside a closed cavity
with walls at a fixed temperature, that it would show in con-
tact with the walls themselves. This remains true whether
the enclosure is made of porcelain or of copper, whether it
is large or small, prismatic or spherical. More generally,
whatever the means of investigation employed, we shall
find that absolutely no influence is exerted by the nature
of the enclosure, its size or shape, on the stationary condition
of the radiation at each point ; this state completely deter-
mines the only temperature to be recorded within the
enclosure.

It follows from this that all directions passing through a
given point are equivalent. No arrangement of lenses or
mirrors, in the interior of an incandescent furnace, would

1 It is obviously possible to concentrate light of external origin by means of
lenses upon a thermometer suspended within a cavity in a transparent block of
ice and to make it indicate any desired temperature.

A. L

146 ATOMS

produce the slightest effect; neither temperature nor
colour would be altered in the least and no images would be
formed. Expressed differently, the point image of a point
on a wall would not be distinguishable by any property
whatever from any other point inside the furnace. An eye
capable of existing at the temperature of the furnace would
not be able to distinguish any particular object or outline
and would perceive merely a general uniform illumination.
Another necessary consequence of the existence of a
stationary régime is that the density W of the light (quantity
of energy contained in 1 cubic centimetre) will have a defi-

nitely fixed value for each temperature. Similarly, if we —

consider within the enclosure a flat closed contour 1 square
centimetre in area, the quantity of light passing across the
contour in one second, say from the left towards the right
of an observer lying along the edge of the contour and looking
towards its interior, is at each instant equal to the quantity
of light passing in the same time in the opposite direction
and has a perfectly definite value E, which is proportional
to the density W of the light in equilibrium at the given
temperature. More precisely, if c stands for the velocity
of light, it appears, as the result of a simple integration,

that E is equal to ek It is clear, moreover, that strictly

speaking the quantities of light E or W undergo fluctuations
(which are negligible on the dimensional scale with which we
are concerned).

87.—Biack Bopiges: StEeran’s Law.—A knowledge of
the density of the light in equilibrium in an isothermal
enclosure is gained in a simple manner by contriving a small
aperture in. the walls of the enclosure and studying the
radiation that escapes through it. If the aperture is
sufficiently small, any disturbing effect upon the internal
radiation will be negligible. The quantity of light that
escapes per second through an orifice of area 8 is then simply
the quantity (S x E) that happens to strike in the same time
on any equal surface of the wall.
_ Naturally there will be no privileged direction for escape.
If therefore, as may easily be done, we look through the

-_e e? ee

LIGHT AND QUANTA 147

aperture, we shall not be able to distinguish any details
within the enclosure, the sole impression received being one
of a luminous pit, in which nothing definite can be perceived.
And the well-known fact is that if one looks through a small
opening into a crucible of dazzling molten metal, the surface
of the metal cannot be seen. It is not only at low tempera- —
tures that nothing can be distinguished within a furnace.

It is, moreover, no more possible at high than at low
temperatures to illuminate noticeably the inside of the
furnace (in such.a way as to make its shape visible) by a
beam of light passing from the outside in through the small
aperture. Such auxiliary light having once entered, it will
be dissipated by successive reflections from the walls and
will have no chance of getting out again through the aper-
ture in any noticeable quantity. The aperture may be said
to be perfectly black, if we regard the fact that it reflects
none of the light it receives as the essential characteristic
of a black body. With regard to the emissive power of a
black body thus defined, we see that it will be given by the
product SE referred to above.

It is not now very difficult to understand how it is possible,
by placing two black bodies of this kind face to face, their
temperatures being T and ?#, and one of them functioning as a
calorimeter, to measure the excess of energy sent from the
hot into the cold source of heat over that sent by the cold
into the hot source. In this way it may be proved that
the emissive power of a black body is proportional to
the fourth power of the absolute temperature T* (Stefan’s
law),

B= ol,
the co-efficient « being “ Stefan’s constant.” :

It is clear that the emissive power increases rapidly as the
source of heat gets hotter ; when the temperature is doubled,
the radiated energy is multiplied 16 times.

The above law has been verified over a wide temperature
interval (from the temperature of liquid air to that of melting
iron); on theoretical grounds, which are too long to be
discussed here, we are inclined to regard it as rigorously

exact and not merely an approximation.
L 2

148 ATOMS

The value of Stefan’s constant may readily be obtained by
making use of the fact that within an enclosure surrounded
by melting ice each square centimetre of black surface at the
temperature of boiling water loses in one minute very nearly
1 calorie more than it receives (more exactly, 1-05 calories or
1:05 x 4:18 x 10’ ergs in sixty seconds). In C.G.S. units
this gives

1:05 x 4:18 x 10°
60

= o (3734—2733),

or very nearly 6-3 x 10~ ° for the value of o. |

The density of the light in thermal equilibrium, at the
temperature T, being proportional to the emissive power E,
is consequently proportional to T+; or, more precisely, it is

o 6-3 x 1075
equal to (4 a. T!) ,or 4 x S10 ot, or 8-4X 10-15. T4.

Though extremely small at the ordinary temperature, it
rises very rapidly. Finally, the specific heat of space (the
heat required to raise by 1° the temperature of the radiation
in 1 cubic centimetre) increases in proportion to the cubes
of the absolute temperature.*

88.—THE COMPOSITION OF THE LIGHT EMITTED BY A
Biack Bopy.—The complex light that escapes through a
small aperture contrived in an isothermal enclosure may be
received on a prism, or, better, on the slit of a spectroscope.
It is then seen that such light always behaves as if it were
made up by the superposition of a continuous and infinite
series of simple monochromatic lights, each having its own
particular wave length and each producing an image of the
slit. The sequence of images (or spectral lines) shows no
interruption and forms a continuous luminous band, which
is the spectrum of the particular black body. (This spectrum,
of course, is not limited to the part that is visible, but
includes an infra-red and ultra-violet part.)

By means of screens it is then easy to cause only the energy
corresponding to a narrow band of the spectrum, in which
wave lengths lie between \ and i’, to enter a black receiving

_ 1 Itis, in fact, the differential of T with respect to W, being equal to 33-6 x
10— T?; at a temperature of 10,000,000° (the centre of the sun?) it would be
of the order of the specific heat of water at the ordinary temperature.

LIGHT AND QUANTA 149

body that acts as a calorimeter. The quantity of energy
Q received, divided by (\’ — A), tends towards a limit I as
the band becomes narrower and \’ tends towards ». This
limit I defines the intensity of the light of wave length A in
the spectrum of the black body. Plotting wave lengths as
abscisse and this intensity as ordinates, a curve will be
obtained that shows the distribution of total energy of the
spectrum as a function of the wave length. In this way it has
long been established that the intensity, which is negligible
for the extreme infra-red and extreme ultra-violet, always
shows a maximum that varies in position according to the
temperature, being displaced towards the region of small
wave length (towards the ultra-violet, that is) as the tempera-
ture of the black body under consideration is raised.

The above are qualitative considerations only. A precise
law has been formulated by Wien, who has succeeded in
showing that the principles of thermodynamics, although
they do not give the actual distribution law required,
nevertheless narrow down considerably the number of forms
a priori possible for it. According to this line of argument,
an account of which would lead me into too great a digres-
sion, the product of the intensity by the fifth power of the
wave length depends only on the product AT of that wave
length by the absolute temperature

1 >
T= Gf at),

f being a function as yet indeterminate. From this it
follows that if the distribution curve shows a maximum at a
certain temperature, it will show one at all other temperatures
and that the position of the maximum will vary inversely
with the absolute temperature :

Ay T = A’, 1’ = constant.

Experiment shows that the product A, T is constant and
that A, T = -29 very nearly, so that, at 2,900° C. (a tempera-
ture little lower than that of the electric arc), the maximum
intensity corresponds to a wave length of one micron and
still lies in the infra-red. At twice that temperature, at

150 ATOMS

about 6,000° C. (the temperature of the black body that,
put in place of the sun, would send us as much light as the
latter), the maximum lies in the yellow.

The position of the maximum is thus fixed. It follows,
moreover, from Wien’s equation, that the maximum
intensity is proportional to the fifth power of the absolute
temperature, being 32 times greater at 2,000° C., for example,
than at 1,000° C.

It remains to determine the form of the function f. Many
physicists have attacked the problem without success.
Planck, however, has finally derived an expression that
agrees accurately with all measurements! in the domain
between 1,000° and 2,000° absolute of temperature and
between 60u and -5u in wave length. Planck’s equation may
be written

C,; ]
PS iS G, ,
e€ AT _y

Where C, and C, are two constants and e is the base of the
Napierian logarithms (very nearly 2-72).

89.—QuanTa.—The publication (in 1901) of Planck’s
formula marks an important epoch in the history of physics.
It has introduced certain very novel and at first sight very
strange ideas into our views on periodic phenomena.

The rays emitted by a black body are, as we have seen,
identical with those which, in the isothermal enclosure,
traverse a section equal in area to the aperture. From this
it follows that in finding the spectral composition of the
light emitted, the composition of the light in statistical
equilibrium that fills an isothermal enclosure has at the same
time been determined. ;

In arriving at a theoretical knowledge of this composition
we must bear in mind that, according to a hypothesis
discussed but little nowadays, all monochromatic light is
composed of electromagnetic waves sent out by the oscilla-

1 Lummer, Kurlbaum, Paschen, Rubens (extreme infra-red), Warburg, and
others have carried out these beautiful and difficult measurements.

LIGHT AND QUANTA 151

tory displacements of electric charges in matter.’ An
electric oscillator (wherein a mobile electric charge may be
caused to vibrate by the electric fields due to the waves that
successively impinge upon it) can reciprocally and by
resonance absorb light having exactly the same period as the
oscillator.

Let us imagine, within an isothermal enclosure, a large
number of identical oscillators vibrating lineally (for
example, sodium atoms, such as: those regarded as causing
the well-known yellow light given by an alcohol flame
impregnated with salt). The period of oscillation thus fixed,
the light that fills the enclosure must be in statistical
equilibrium with these resonators, giving them during the
very short period of each oscillation as much energy as it
receives from them. If E stands for the mean energy of the
oscillators, Planck found that, as a consequence of the laws
of thermodynamics, the density w of light for wave length
A is proportional to E, the relationship being expressed
more precisely by the equation

_ 8a

Wh 5 aE

consequently, in order to reconcile this result with the experi-
mental fact that the radiation density becomes. infinitely
small for very short wave lengths, it must follow that the
mean energy of the oscillators will become extremely small
when the frequency becomes very high.

Now oscillators in thermal equilibrium with radiation
must also be in thermal equilibrium with any gas that fills
the enclosure at the given temperature. In other words,
the mean oscillatory energy must be what it would be it it
were sustained solely by the impacts of the gaseous mole-
cules. In the case where the oscillatory energy can vary
continuously, the kinetic energy of oscillation will, as we

have already had occasion to point out, be equal in the mean

to : = . T, or to one-third of the kinetic energy of a mole-

1 The electric and magnetic fields at a point on the wave front are always in
the plane tangential to the wave (light vibrations are transverse) and are per-
pendicular to each other.

152° | ATOMS

cule of the gas ; it will, that is to say, be independent of the
period. Radiation should therefore be infinite for very
small wave lengths, which is certainly not the case.

We must therefore assume that the energy of each oscil-
lator varies in a discontinuous manner. Planck supposes
- that it varies by equal quanta, in such a way that each
oscillator always retains a whole number of atoms or grains
of energy. The value E of this grain of energy should be
independent of the nature of the oscillator, but should
depend on its frequency v (number of vibrations per second)
and be proportional to it (being 10 times greater, for instance,
when the frequency becomes 10 times greater); E should
therefore be equal to hy, h being a universal constant
(Planck’s constant). | |

If we accept these hypotheses, which at first sight appear
extraordinary (and which will therefore possess all the more
importance if they can be verified), it will no longer be
at all accurate to regard the mean energy E of a linear }
oscillator as equal to one-third of the energy possessed on the
average by a gaseous molecule. Statistical enumeration of
all the possible cases 2 shows that in order to arrive at
statistical equilibrium through the impacts between gaseous
molecules and the oscillators, we must have

hy
pares PE
é Br —1

N being Avogadro’s number; or, remembering that the
velocity c of light is equal to » times the wave length A
corresponding to the frequency vp,

ch 1
E=7- Ni ase

ee CAT 2 9

1 An oscillator having 3 degrees of freedom would possess a mean energy 3
times as great.

2 Nernst shortens the calculation very considerably by assuming that the
number of oscillators that possess, for instance, energy 3 € is equal to the number
that would possess energy lying between 3 € and 4 « if their energy varied con-
tinuously. (The number in complete repose is therefore equal to the number
that in the case of continuity would possess energy less than «.)

LIGHT AND QUANTA 153

from which we get finally, for w, equal to a
__ 8ach 1
Nae Sy eb =
4 OAD

this is the very equation that has been found to agree with
experiment (para. 88), for the density w, is simply equal to
the emissive power I, divided by the fourth part of c.

The theory I have just outlined has achieved a great
success, in that it has led to the discovery of the law that
determines the composition of isothermal radiation at each
temperature. But a still more striking verification lies in
the agreement found between the values already obtained
for Avogadro’s number and the value that can be deduced
from Planck’s equation.

90.—THE RADIATION EMITTED BY A BLACK BODY ENABLES
us TO DETERMINE THE MoLEcuLAR MagnitupEs.—Clearly
everything in the above equation is either measurable or
known except the number N (which expresses the fact of
molecular discontinuity) and the constant / (which expresses
the discontinuity of the oscillatory energy). These numbers
N and h/ can therefore be determined if two reliable measure-
ments of the emissive power can be obtained for different
values of the wave length A or the temperature T (it will
naturally be better to use in the determination all the
available reliable measurements and not two only). Making
use of the data that appear most trustworthy at the present
moment, we arrive at the following value for h

h== 62 x 107?"
and for N
N = 64 x 10”,
the probable error being more or less 5 per cent.
The agreement between this value and those already found
is indeed marvellous. And at the same time we have

acquired yet another means for determining accurately the
molecular magnitudes.

154 ATOMS

EXTENSION OF THE THEORY OF QUANTA.

91.—Tue Speciric Heat or Sotips.—By a bold extension
of Planck’s idea Einstein has succeeded in accounting for the
influence of temperature on the specific heat of solids. His
_ theory, to which allusion has already been made (para. 44),
depends upon the assumption that each atom in a solid body
is urged towards its position of equilibrium by elastic forces
in such a way that, if it be slightly displaced, it will vibrate
with a fixed period. As a matter of fact, since neighbouring
atoms also vibrate, the frequency will not be rigidly fixed,
and we ought rather to consider a series of frequencies more
analogous to a band than to a spectral line. Nevertheless,
as a first approximation we may confine ourselves to the
consideration of the case of a single frequency.

With this limitation, Einstein supposes that although the
oscillator set up by each atom is not necessarily an electrical
one, its energy must be a whole number multiple of hy as
with Planck’s oscillators. Its mean energy at any tempera-
ture has therefore the value _

3hv
N ?
on: Av
Z RT —l

with reference, as has been pointed out above, to an oscillator
capable of undergoing displacement in all directions. The
energy contained in a gramme atom will be N times greater
and the increase of this energy per degree, or the specific heat
of the gramme atom, can be calculated.1_ The expression
found in this way for the specific heat tends towards zero,
in agreement with Nernst’s results, as the temperature falls,
and towards 3R or 6 calories as it rises, in agreement with
Dulong and Petit’s law (the latter limit is reached the more
quickly the smaller the value of the characteristic fre-
quency »). In the interval between these limits the above
expression represents in a remarkable manner the run of the
specific heats, though not without systematic error explicable
by the approximations employed (we have noticed that the

1 It will be simply the differential with respect to temperature of the energy
contained in a gramme atom.

LIGHT AND QUANTA 155

frequency cannot be defined satisfactorily). It also defines
the frequency » of the atomic vibration, if it is unknown.

_ It is worthy of notice that frequencies calculated in this
way agree with those to be expected from the consideration
of other phenomena. The absorption of light of long wave
length by bodies such as quartz or potassium chloride
(Rubens’ experiments) is a case in point. This kind of
absorption, as well as “ metallic” reflection, is explicable
if the light is in resonance with the atoms of the body and
consequently possesses a frequency deducible from the
latter’s specific heat. This is found approximately to be the
case (Nernst).

At the same time it is. conceivable (Einstein) that the
elastic properties of solid bodies might provide a means for
predicting the frequency of the vibrations of an atom
displaced from its equilibrium position. An approximate
calculation has been made by Einstein with reference to com-
pressibility ; applied to silver the predicted value for the
atomic frequency is 4 x 102, that obtained from its specific
heats being 4-5 x 10!%. I must content myself with these
brief allusions and refer to the splendid work of Nernst,
Rubens, and Lindemann ! for more ample details.

92.—DisconTINvITy IN RotatTionaL VeLociry.—If we
remember that we have already been forced to assume, with
Nernst (para. 45), that the rotational energy of a molecule
varies discontinuously, we shall perhaps be readier to extend,
keeping the same value for the universal constant h, the
law of discontinuity which holds for the energy of oscillators
to the case of rotation (molecular). There is, indeed, analogy
of a certain kind between the rotation of a body about itself
and the oscillation of a pendulum (or the path of a planet),
since periodicity is characteristic of both cases. An obvious
difference is that the pendulum (or planet) has a well-defined
proper period, whereas so long as a ball is at rest, it is not
possible to predict a definite period-of rotation for it. How-
ever, generalising Planck’s result, we may say :

1 The latter worker deduced the proper frequency from the melting point of
the solid ; he supposes that a body liquefies when the amplitude of the atomic
oscillation becomes sensibly equal to the mean distance between atoms.

156 ATOMS

When a body rotates at the rate of v revolutions per second,
its energy is equivalent to a whole number of times the product
hy.

Since 27v is the angular velocity of rotation (angle de-
_ scribed per second), the kinetic energy of rotation is further-

more equal to the product len)? where I stands for the

moment of inertia ! of the body (about its axis of rotation).
Hence it follows that, » being a whole number,

51. 4x4 = p.v.h,

h

or ae fe ae oF)

so that the number of revolutions per second must necessarily
be either once, twice, or 3 times a certain value ¢ equal to
pes! Intermediate speeds of rotation should be impossible.

93.—UNSTABLE RotTaTion.—The above result is sur-
prising ; it appears, moreover, to be inconceivable that the
number of revolutions can pass from the value ¢ to the
value 2¢ or 3t without taking up the intermediate values. I
would suggest that the intermediate velocities are unstable,
and that when, for instance, the body while rotating receives
an impulse that communicates to it an angular velocity
corresponding to 3-5 times ¢ revolutions per second some
effect due to friction or radiation as yet unknown? at once
operates to reduce the number of revolutions per second to
exactly 3 times t, after which the rotation can persist indefinitely
without loss of energy. The result of this will be that, out of
a large number of molecules, very few will be in the un-
stable condition, and we may take it as a first approximation
that, for any one molecule taken at random, the rotation in
one second is either no revolutions or ¢ or 2¢ or 3¢, ete. The
occasional molecules having rotational energy in process of

1 We know that the rotational energy of a solid revolving with an angular
velocity w is ; Iw? (which may be used to define its moment of inertia).

2 Connected perhaps with the colossal value of the acceleration (or centri-
fugal force), which is at least a trillion times greater than any value reached in
our centrifugal machines and turbines.

eS ee ee -

eS. ee

~

LIGHT AND QUANTA 157

change may be neglected, just as we may neglect the few
molecules in a gas that are actually undergoing impact and
whose energy is in process of changing.

94.—THE MATERIAL PART OF AN ATOM IS CONCENTRATED
ENTIRELY AT ITS CENTRE.—We are now possibly in a position
to understand why the molecules of a monatomic gas (such
as argon) do not produce rotation when they impinge (or,
more exactly, why they communicate no rotational energy
to each other), with the result that the specific heat c of the
_ gas is equal to 3 calories (para. 39). If the material part of
the atom is concentrated closely about its centre, its moment
of inertia will be very smal], its minimum possible rotation

h
(its frequency v being =a) will be extremely rapid and the
7

quantum hy of rotational energy consequently will be large.
If this quantum is large by comparison with the energy of
translation possessed on the average by the molecules (at
the temperatures we have available), it will practically never
happen that a molecule that strikes another molecule will be
able to communicate to it even the minimum rotation ; and,
conversely, a molecule possessing that rotation will have
every chance of losing it during an impact. In short, at any
particular instant, rotating molecules will be extremely few
in number. »

Since argon, to take a particular case, retains its specific
heat 3 up to about 3,000° C., it follows that even at that
high temperature the molecular translational energy is still
well below the quantum of energy corresponding to the
minimum possible rotation. Let us assume that the trans-
lational energy is less than half the quantum, which is
certainly a very low estimate. Further, since it is propor-
tional to the absolute temperature, it will be approximately
10 times greater than at the ordinary temperature and hence

1
very nearly equal to 5 * 10~12; the quantum hy being

2

expressible in the form —.., this gives us
271
2

1 ea
Mas —12 i, rae B et
5 X 10° Ng X OnE

158 ATOMS

Substituting for h its value 6 x 10~?’, it becomes possible
to deduce from this inequality some interesting results with
regard to the frequency and moment of inertia.

In the first place, if hy is greater than ; x 107 1%, we see

at once that » is certainly higher than 101: A
The slowest stable speed of rotation corresponds to more than
10,000,000,000 revolutions in one hundred-thousandth of a
second.
With regard to the moment of inertia, we see that it is
less than 2 x 10~*. If the mass m of the argon atom
(equal to 40 times the mass 1-5 x 10~ 24 of the hydrogen

atom) occupies a sphere of diameter d with uniform density,
2

md* . |
its moment of inertia would be 0” and, from the inequality

given above, we get
de®. KOE EE,

Remembering (para. 67) that what we ordinarily call the
diameter of the argon molecule (which is really its radius of
protection) is. 2-8.x 107%, we see that the material part of
the atom is condensed within a space, of dimensions at least
50 times smaller, where the real density (which varies
inversely as the cube of the dimensions) is certainly well
above 100,000 times the density of water.

We have assumed that the molecular kinetic energy is only
less than half the rotation quantum. If it were one-eighth
of the latter quantity (which is still a modest estimate) we
should obtain a diameter d one-half of the above. In short,
I feel sure that we shall be well within the mark if we assume
that the material part of the atom is condensed into a
volume at least one million times less than the apparent
volume occupied by atoms in a cold solid body.

In other words, if we can imagine the atoms of a solid
body under such a high magnification that their centres
appear distributed in space like the centres of the shot
in a pile of shot, each 10 centimetres in diameter, then the
actual matter in each ‘“ atomic shot ”’ will occupy a sphere
less than 1 millimetre in diameter ; we should perceive them

LIGHT AND QUANTA 159

as minute grains of lead at a mean distance of 20 centi-
metres apart. In air, viewed under the same magnification,
the “ grains of lead’ would have a mean distance of 20
metres.

It is of course possible to regard the atom as possessing
an extremely minute extension from its centre, but we must
always regard the greater part of its mass as collected very
near to its centre.

Matter 1s porous and discontinuous to an extent far beyond
our expectation. —

The radius of protection, or distance between centres at
the moment of impact, may be defined, as we have already
suggested, as the distance at which the material of the atom
exerts an enormous repulsive force upon the material part
of another atom. We shall see, when discussing the rapid
positive rays, that at still smaller distances the force of
repulsion becomes feeble or fails altogether. In other words,
every atom is condensed at the centre of a thin spherical
casing, which is of vast dimensions relatively to the atom
itself and which protects the latter from the approach of
other atoms. This conception of the atom is in other respects
imperfect, seeing that a material casing should impede the
exit of a projectile as well as its entry, whereas actually entry
alone is stopped.

95.—THE ROTATION QUANTUM OF A Potyatomic MoLE-
CULE: THE DISTRIBUTION OF MATTER WITHIN THE MOLE-
CULE.—We can now understand why even a molecule may
cease to spin at very low temperatures, although its moment
of inertia is much greater than that of a single atom. The
only necessary condition is that the molecular energy due

to agitation should be small in comparison with the rotation
2

h |
quantum 22] for the molecule. Naturally this will occur

sooner the smaller the moment of inertia of the molecule,
and we can understand why such a state of affairs has been
realised as yet with hydrogen alone (para. 45).

Let d be the distance between the centres of two hydrogen
atoms making up a molecule H,. Their masses are con-
centrated at the two extremities of d, and the moment of

160 ATOMS

inertia | about an axis passing through the centre of gravity
of the molecule and perpendicular to the line joining the —
centres will be

2
2x v4 x 10-« (2)?

At about 30° absolute (at which temperature the specific
2

heat is very nearly 3) the quantum will certainly be

2721
greater than twice the energy due to molecular agitation,
which, at this temperature, is very nearly : x 10> Bee

follows from this that the distance between the centres is
certainly less than 1-5 x 1078. We shall readily accept this
upper limit when we remember that 2-1 x 10~8 was found
for the diameter of im-
pact of the hydrogen
molecule.

A somewhat more
accurate calculation is
possible if the small
difference between the
actual specific heat at
say, 50° and 3 calories
is known. I therefore
conclude that the mini-
mum speed of rotation
possible for the hydrogen molecule, perpendicular to the
line joining the atomic centres,! corresponds (very roughly)
to 5 x 10~ !* revolutions per second, which gives the value
10 ~— § for the distance d.

It would perhaps not be without interest to draw a
hydrogen molecule to scale and to attempt to express the
above results in what seems to me the most probable form.

Almost the entire substance of the molecule is collected
at the centres H’, H” of the two atoms. About each atom
I have drawn the spherical protecting casings, which must

Fie. 12.

1 Parallel to the line joining the centres the minimum frequency would be
much higher and of the same order of magnitude as for argon, for the moment of
inertia about that line must be extremely small.

LIGHT AND QUANTA 161

overlap to some extent.1. The outer surfaces of these
spheres form the casing A of the molecule, into which no
other atomic centre can penetrate (at least if its velocity
does not greatly exceed the speeds met with in molecular
agitation).

The contour B is the surface defined as the surface of
molecular impact in the kinetic theory (it cannot be pene-
trated by the analogous contour B’ of any other molecule).
If H’ and H” are 10~8 centimetres apart, we find that the
dimensions of this contour have roughly and on the average
the value 2 x 10~°® already assigned to the diameter of
impact of the hydrogen molecule. This provides a verifica-
tion of the theory of rotation quanta.

We have seen how small in reality is the space occupied by
the atoms within the molecular edifice. It would be of the
greatest interest to know the distribution of the field of force
about each atom and particularly to gain precise ideas as to
the nature of the chemical bonds or valencies. As yet
information on this subject is entirely wanting.

In this connection I should like to add a remark with
reference to the strength of the valency bond. When, at
about 2,000° C., the dumb-bell-like hydrogen molecule is
spinning without rupture perpendicularly to its axis with
a frequency but little less than a hundred thousand
milliards of revolutions per second, it is obvious that the
bond or union between the atoms must be resisting the
centrifugal force. A union that would give the same strength
to a dumb-bell would have a tenacity at least 1,000 times
that of steel.

96.—LIGHT MAY POSSIBLY BE THE CAUSE OF MOLECULAR
Dissoctation.—I have indicated (para .84) the possibility
of a kinetic theory of chemical reaction. I should like to
point out that light may possibly play a capital rdle in the
mechanism of chemical change.

This appears to me to be proved by a law ? that is quite
generally recognised, without, I think, a sufficient apprecia-
tion of its really surprising molecular interpretation, which

1 I presume that an atom combined with another is inside its casing.
* As a matter of fact, experiments referring to gases are few in number.

A. M

162 ATOMS

may perhaps show it to be the fundamental law of chemical
mechanics (since all chemical eapeceme presuppose certain
molecular dissociations).

According to this law, the rate of dissociation at constant
temperature, in unit volume of a gas A, for a reaction of the
following kind :

A-—> A’ + A",

is proportional to the concentration of the gas A and cannot
be altered by the addition of other gases to the reacting
system.

In other words, for a given mass of the substance A, the
proportion transformed per second is independent of the
dilution ; if the given mass occupies 10 times more space,
its concentration then being 10 times less, then 10 times
less will be transformed per litre, or just as much in all as
before dilution. Thus, contrary to what might have been
expected, the number of impacts has no influence on the
rate of dissociation. Out of the given N molecules of
the gas A, always the same number will decompose per
second (at a given temperature) whether the gas is
relatively concentrated or mixed with other gases (in
which case impacts will be frequent), or whether it is
dilute (when impacts will be rare).

It seems to me that, for any given molecule, the probable
value for the time that must elaspe before, under the sole
influence of impacts, a certain fragile condition will be reached
must be smaller the more often the molecule receives
impacts per second. Further, supposing this fragile state
to have been reached, the probable value for the time
required for a molecule to receive the kind of impact capable
of rupturing it must again be shorter the more frequent the
impacts. For this double reason, if rupture is to be pro-
duced by molecular impact, it should occur more frequently
(and dissociation should therefore become more rapid) as
the concentration of the gas increases.

Since this is not the case, dissociation cannot be caused
by impact. Molecules do not decompose by striking against
each other, and we may say: T'he probability that any mole-

LIGHT AND QUANTA 163

cule will be ruptured does not depend ss the number of
umpacts vt receives. :

Since, however, the rate of dissociation depends largely
on the temperature, we are reminded that temperature
exerts its influence by radiation as well as through molecular
impact, and are faced with the suggestion that the cause of
dissociation lies in the visible and invisible light that fills,
under stationary conditions, the isothermal enclosure .
wherein the molecules of the gases under consideration are
moving.

The essential mechanism of all chemical reaction is therefore
to be sought in the action of light upon atoms.

CHAPTER VII
THE ATOM OF ELECTRICITY

WE have seen that the properties of electrolytes suggest
the existence of an indivisible electric charge and that every
ion either carries that charge or some whole number multiple
of it. But we have not as yet attempted the direct measure-
ment of this elementary charge and have merely calculated
its value by dividing by Avogadro’s number N the electric
charge (1 faraday) carried during electrolysis by a mono-
valent gramme ion.

Now the direct measurement of very small charges, which
up to the present has not been successfully accomplished in
liquids, is found to be easy with gases, and has in fact shown
that such charges are always whole number multiples of the
same quantity of electricity, which has a value agreeing
with that already calculated. Experiments that I am about
to describe have proved the discontinuous structure of
electricity and have provided yet another means for obtain-
ing the molecular magnitudes. 7

97.—KaATHODE Rays AND X Rays: THE IONISATION OF
GASsES.—Since the time of Hittorf (1869) it has been known
that when an electric discharge passes through a rarefied
gas, the kathode emits rays which show their trajectory by
a feeble luminescence of the residual gas, excite a beautiful
fluorescence on the glass walls at which they are arrested,
and which are deviated by magnets. If, for instance, they
are directed at right angles to a uniform magnetic field,
their trajectory becomes circular and perpendicular to the
field.

As early as 1886 Sir W. Crookes supposed that these
kathode rays are negatively electrified projectiles which,
issuing from the kathode and being repelled by it, have

:
— Is eee ee ee a

THE ATOM OF ELECTRICITY 165

acquired an enormous velocity. But neither he nor Hertz
was able to prove the existence of this electrification, and a
wave theory was favoured for some time, although Hertz
had discovered that the rays can pass through thin plates
several microns thick and Lenard had shown that: it is
possible to allow them to escape from the tube wherein the
discharge takes place through a thin metallic plate strong
enough to support the pressure of the atmosphere. (They
can then be studied in the air, into which they diffuse and
are stopped after a path of a few centimetres.) }

A decided reversion to the emission theory put forward
by Crookes occurred, however, when it was proved ! that
the kathode rays always carry negative electricity along
with them. It is absolutely impossible to separate this
electricity from the rays, even by making them pass through
thin metallic leaf.

Finally, we may remark that any obstacle struck by the
kathode rays emits X rays, the discovery of which by
Roentgen (1895) marked the commencement of a new era in
physics.

Like kathode rays, X rays excite fluorescence of various
kinds and affect photographic plates. They differ pro-
foundly from kathode rays in that they carry no electric
charge and consequently are not deviated either by
electrified bodies or by magnets. It is well known that they
possess a very considerable penetrating power and that they
cannot be reflected, refracted or diffracted, so that they are
of much shorter wave length than the most extreme ultra-
violet light (-l) yet studied.”

It was very soon noticed that X rays “ discharge electrified
bodies.” A careful analysis of the phenomenon ® showed |

1 Jean Perrin (Comptes Rendus, 1895; Ann. de Ch. et Phys., 1897). I
showed that the rays carry negative electricity with them into a completely
closed metallic box and that they are, moreover, deviated in an electric field.

2 The work of Laue, Bragg, etc., has shown that the X ray spectrum extends
from A = ‘00084u to A = 000056 (cf. Moseley, Phil. Mag., April, 1914).—Tr.

8 Jean Perrin: ‘‘ Mécanisme de la décharge des corps électrisés par les
rayons X”’; “Kelairage électrique,’ June, 1896; Comptes Rendus, August,
1896 ; Ann. de Ch. et Phys., August, 1897. Sir J. J, Thomson and Rutherford
have arrived at the same conclusions, from their side, as the result of quite
different experiments. Righi has also reached the same position.

166 ATOMS

that the rays produce, in the gases they pass through,
nuclei charged with positive or negative electricity, or
mobile ions, which soon recombine in the absence of an
electric field, but which move under the influence of such a
field in. opposite directions along the lines of force until
stopped by a conductor, which discharges them (thus
enabling the degree of ionisation of the gas to be determined),
or by a non-conductor, which they charge. Once the
oppositely charged ions have been carried by this two-fold
motion into different regions of the gas, they escape recom-
bination and the two electrified gaseous masses thus obtained
can be manipulated at leisure.

In the same way, by their ionising effect on gases, other
forms of radiation were soon afterwards detected (extreme
ultra-violet rays, Lenard’s kathode rays, the #, 8, and y
rays of radioactive substances) that “‘ discharge ”’ electrified
bodies situated in gases, when they cut the lines of force
emanating from those bodies. The gases issuing from flames
are also ionised, and we may reckon them as conductors as
long as any ionisation persists.

98.—THE CHARGES SET FREE DURING THE IONISATION
OF GASES ARE EQUAL IN VALUE TO THOSE CARRIED BY
MonovaLent Ions DURING ELEcTROLYSIS.—As yet we
know nothing as to the magnitude of the charges separated
during the ionisation of a gas, or whether they bear any
relation to the ions concerned in electrolysis.

That the elementary charges are the same in the two
cases | was first shown by Townsend. Thus let:e’ be the
charge on an ion, situated in a gas of viscosity €. Under the
influence of a field H this ion will be set in motion, and,
being continuously checked by the impacts it receives, it
will be displaced with a uniform motion (on our dimensional
scale) at a velocity wu such that |

He’ = Au,

the coefficient of friction A no longer having the value 67a&

that it takes (para. 60) for a relatively large spherule ;

it is constant, however, which is all we require. As a
1 Phil. Trans., 1900.

THE ATOM OF ELECTRICITY 167

matter of fact, w can be py ae (Rutherford), and it can

be shown that the quotient =, which may be regarded as a

se
mobility, is constant, and furthermore not the same for each
of the two kinds of ion produced. This mobility corresponds
roughly to a velocity of 1 centimetre per second in a field of
1 volt per centimetre.

After the separation of the two kinds of ions s by the electric
field, two gaseous masses are obtained in which ions of the
same kind only are to be found. These ions are in a state
of agitation and diffuse just like the molecules of a very
attentuated gas scattered throughout a non-ionised 1 gaseous
medium.

Hence, making use of Einstein’s argument (para. 70), we
find for the value D for the diffusion coefficient of the ions
under consideration |

RT. 1
ee ie

/

that is, since A is equal to cal
. U

RT
D
This is Townsend’s equation (obtained by him, as a

matter of fact, by a different method).
To obtain the product Ne’ we have only, since the mobility

Ne’ =

8

U
H is known, to measure the diffusion coefficient D. This

Townsend has done. He found, for various gases and the
various kinds of ionising radiation, that the value of the
product Ne’ is in the neighbourhood of 29 x 10'%, which is
the value obtained for Ne from electrolysis.?

A later verification, in connection with the very interesting
case of the ions in flames, follows from Moreau’s ® experi-

1 Neglecting the extremely weak repulsive action that tends to drive these
mobile charges towards the periphery of the enclosure.

2 A small proportion of other kinds of charges (polyvalent ions, for instance)
may have escaped observation ; the degree of uncertainty attached to the

measurements appears to be about 10 per ‘cent.
3 Comptes Rendus, 1909,

168 ATOMS

ments on the mobility and diffusion of such ions. The value
30-5 X 10!8 is obtained for Ne, which is equal to the value
given by electrolysis to within 5 per cent.

Remembering that, on account of the irregularity of
molecular motion, the coefficient of diffusion is always

equal to half the quotient a characteristic of the agitation -

(para. 71), we can re-write Townsend’s equation in the

form
te
Ae — ==
Ne =2RT.y.-
which, though without interest in connection with the

invisible ions dealt with by Townsend in his experiments,

becomes the most interesting form in the case of large ions
(charged powders), if their displacements can be measured.

This has actually been done by de Broglie, using air
charged with tobacco smoke.t In his apparatus the air
and smoke is blown into a small box maintained at a constant
temperature, and luminous rays are caused to converge into it
from a powerful source. At right angles to these rays a
microscope is fixed, which resolves the smoke into globules
that look like brilliant points of light and are agitated by a
very active Brownian movement. If now an electric field
is produced at right angles to the microscope, the globules
are seen to be of three kinds. The first kind move in the
direction of the field and are therefore positively charged ;
others move in the opposite direction and are therefore
negative. Finally, there is a third group, which continue
their agitation without changing their position and are
therefore neutral. In this very striking manner large gaseous
ions were for the first time made visible.

De Broglie has carried out a large number of measure-
ments of X and of uw for ultra-microscopic globules of very
nearly the same brightness (and hence of about the same
size). The mean of these experiments gives the value
31-5 x 10! for Ne’; we thus obtain, with a degree of
accuracy equal to that obtaining in Townsend’s experiments,
a value equal to the product Ne given by electrolysis.

1 Comptes Rendus, Vol. CXLVI., 1908, and Le Radium, 1909.

a ae

Se Es

THE ATOM OF ELECTRICITY 169

More recently Weiss (Prague) has found the same value
of Ne’ for the charges carried by the ultra-microscopic
particles that occur in a spark passing between metallic
electrodes.1 But, instead of taking the means of isolated
readings relative to different grains, he recorded for each
grain enough readings to obtain an approximate value for
Ne’ from those readings alone. It was therefore not necessary
to compare only grains of the same size and shape.

These various facts considerably enlarge the notion of
elementary charge introduced by Helmholtz. Moreover,
whereas electrolysis has not up to the present suggested any
means for measuring directly the absolute value of the
charge e on a monovalent ion, we shall see that it is possible
to measure that charge when it is carried by a microscopic
granule in a gas. In this way we shall obtain, since Ne is
known, a fresh determination of N and of the molecular
magnitudes.

99.—Drrect DETERMINATION OF THE IONIC CHARGE IN
GasEsS.—If an ion in a gas is brought by the molecular
agitation into the vicinity of a speck of dust, it will be
attracted by induction and will-attach itself to the speck,
charging it in consequence. The arrival of a second ion of
the same sign will be checked by the repulsion between the
two charges, and will also be the less likely to occur the
smaller the speck of dust.? The arrival of an ion of opposite
sign will, on the contrary, be facilitated. A number of the
dust particles will therefore either remain neutral or will
become neutral again subsequently, and a permanent
equilibrium will be set up if the ionising radiation continues
to act. This has actually been demonstrated to be the case
for various kinds of smoke particles, neutral to begin with,
when the gas wherein they are suspended is ionised (de
Broglie). 7

Another interesting case is that of an ionised gas, free
from dust particles but saturated with water vapour.

1 Physik. Zeitschrift, Vol. XII., 1911, p 630.

2 More strictly it will rarely happen tha the molecular agitation will impart
a sufficient velocity to an ion to enable it to penetrate into the region where the
attraction of the speck due to induction will overcome the repulsion. The
theory of electric images enables a definite calculation to be made.

170 | ATOMS

C. T. R. Wilson’s experiments (1897) prove that the ions
acts as centres of condensation for the water droplets that
make up the mist that forms when the gas is cooled by an
adiabatic expansion.

Finally, a gas can be charged simply by bubbling through
a liquid (which involves the rupture of liquid membranes).
The formation of charged mists in gases prepared by electro-
lysis, first noticed by Townsend, is probably caused in this
way.

In any one of the above cases the elementary charge e
will be determined if the charge acquired by the drop or
dust speck can be measured. The first determinations of
this charge were made by Townsend and J: J. Thomson
(1898). ‘Townsend worked on the mists carried along by
gases produced in electrolysis, while Thomson used the
clouds formed on the condensation of ionised damp air by
expansion. They determined the total charge e present in
the form of ions in the cloud under investigation, the weight
P of the cloud, and finally its rate of fall v. This latter
measurement gave the radius of the drops (assuming that
Stokes’ law is applicable to them) and hence the weight p
of each. Dividing P by p we get the number of drops n
and hence the number of ions. Finally, the quotient of E by
n gives the charge e. The number obtained in Townsend’s
experiments, which obviously were not very exact, varied
between 1 x 10~—1° and 3 x 10718; Thomson’s varied
between 6:8 x 10~ © (using the negative ions emitted
by zine illuminated by ultra-violet light) and 3-4 x 107
(with the ions produced in a gas by X rays or the rays
from radium). These values approximated well enough as
to the order of magnitude of the expected result, and,
although the agreement was still rather rough, it was ot
great importance at the time.

The method, employed in this way, involved a high
degree of uncertainty. It was assumed, for instance, that
every ion is united with a rome and that each droplet
carries only one ion.

H. A. Wilson simplified the method very considerably
(1903). He confined himself to the measurement of the

THE ATOM OF ELECTRICITY 171

rates of fall of the cloud, first when gravity operates alone,
and then when it is opposed by an electrostatic force. Let
v and v’ be the velocities, before and after the application
of an electric field H, of a droplet bearing a charge e’ and
weighing mg. Making the single hypothesis that these
‘constant velocities are proportional to the operative forces,
we get (H. A. Wilson’s equation) even if Stokes’ law is
inexact : |

He' — mg _ v'
mg v
me eee.)
or c= mM ( 3

Further, during the uniform fall of a drop, the motive force

+
(its weight 379) is equal to the frictional force and hence to

6rakv if Stokes’ law is valid. This gives the radius and hence
the mass m, so that the charge e’ can be calculated.

Under the influence of the field, the charged cloud obtained
by the expansion of air (strongly ionised) divides itself into
two or even three clouds sinking at different rates. Applying
the above equations to the motions of these clouds (regarded
as being composed of identical droplets), values roughly
proportional to 1, 2, and 3 were obtained for the charge e’.
This proves the existence of polyvalent drops. The value
found for the charge e’ for the least charged cloud varied
between 2:7 x 107 and 4-4 x 107 1°, the mean value being
3-1x 10~ 1. The want of precision is thus still great.
Fresh experiments were carried out, using the same form of
apparatus, by Przibram [alcohol droplets], who found
3-8 x 10~ 1°; he was followed by other physicists. The
latest and most trustworthy result (Begeman, 1910) gives
4-6 x 10~— 1° (Stokes’ law being always assumed). We shall
see that the measurements are very greatly facilitated by
studying the charged particles individually.

100.—TuHeE StupDy OF THE INDIVIDUAL CHARGES PROVES
THE ATOMIC STRUCTURE OF ELEcTRICITY.—H. A. Wilson’s
reasoning refers to a single particle. Now, in the experiments
described above, it is applied to a cloud, and it is assumed

172 ATOMS

that the droplets in the cloud are identical, which is certainly
incorrect. All uncertainty of this kind is avoided by working
under the experimental conditions postulated in the theo-
retical treatment of the question ; in other words, by observ-
ing a single spherule, infinitely removed from all other
spherules and from the walls of any enclosure.

Observations on individual charged grains, thus correctly
applying the method invented by H. A. Wilson, were made
independently (1909) by Millikan and by Ehrenhaft.

Ehrenhaft, however, working with dust particles (obtained
by sparking between metals), unfortunately applied Stokes’
law, regarding his particles, without proof, as complete
homogeneous spheres. I am inclined to think that they are
really irregular, spongy bodies having an entirely irregular
and jagged surface ; their frictional effect in gases will be
very much greater than if they were spheres, and the applica-
tion of Stokes’ law to them has no meaning. I regard as
proof of this the fact, pointed out by Ehrenhaft himself,
that many of these dust clouds have no appreciable Brownian
movement, although they are ultra-microscopic. This obser-
vation, which has received little attention, indicates an
enormous frictional effect. And, in fact, Weiss’s recent
measurements referred to above (para. 98) show that dust
particles that, according to Ehrenhaft, should bear very
small charges lying between 1 x 10-18 and 2 x 107 1°, show
displacements which give quite normal values for Ne. These
dust particles therefore carry charges in the neighbourhood
of 4:55 x 107 1.

Millikan, working with droplets that certainly. possessed
a massive, close-grained structure (obtained by atomising
liquids), has carried out experiments that are free from the
objections referred to above. The droplets are carried by
a current of air to the neighbourhood of a pin-hole pierced
in the upper plate of a flat horizontal condenser. A few of
them pass through the hole and, when between the condenser
plates, are illuminated laterally and can be followed by means
of an eye-piece (as in de Broglie’s apparatus); they then
appear as brilliant points of light on a black background.
The electrostatic field, of the order of 4,000 volts per centi-

—s

THE ATOM OF ELECTRICITY 173

metre, acts counter to gravity and generally prevails over it.
It is‘then possible to balance the same droplet for several
hours without losing sight of it, alternately making it rise
under the influence of the field and letting it sink by cutting
it off.1 Since the droplet, being composed of a non-volatile
substance, remains the same throughout, its rate of fall has
always the same constant value v. Similarly its upward
motion takes place at a constant velocity v’. But in the
- course of a long series of observations, it sometimes happens
that the upward velocity suddenly changes, in a discon-
tinuous manner, from the value v’ to another value v,’,
which may be greater or less than v’. The charge on the
droplet has therefore changed, in a discontinuous manner,
from e’ to another value e,’. This discontinuous variation
becomes more frequent when the gas in which the droplet
is moving is subjected to an ionising radiation. It therefore
seems natural to attribute the variation of the charge to the
fact that an ion, when near the dust particle, gets captured
by electric attraction, in the way explained above.

_ Millikan’s remarkable observations demonstrate in a
vigorous and direct manner the atomic structure assumed
for electricity. . Writing down H. A. Wilson’s equation for
the condition of affairs before and after the discontinuous
_ change and dividing the one by the other, we get

e “v+v
é uty,’
or, better,
Se FS ep hy eg

v + y' v + V1 v + Vo, NM Os FO ee ae 7m

for the ratio between the charges e’ and e,’.. The successive
charges borne by the drop must therefore be whole-number
multiples of the same elementary charge e, if the sums
(v + v’), (v+ v1’), ete. are proportional to whole numbers
(if, that is to say, they are equal to the products obtained by
multiplying various whole numbers by the same factor).
Moreover, the whole numbers corresponding to two succes-
sive charges will differ in general by one unit only, correspond-

1 For full details of Millikan’s work, see Physical Review, 1911, pp. 349-397.

174 ATOMS

ing to the addition of one elementary charge (it being
nevertheless possible for a polyvalent ion to be formed).

These conclusions can be checked by means of Millikan’s
figures!. For instance, the successive values of (v + v’),
and consequently of the successive charges, for a certain oil
drop, were to one another as the following numbers :

2-00, 4:01, 3-01, 2-00, 1-00, 1-99, 2-98, 1-00:
that is to say, they are to one another, to within 1 per cent.. as
54 3-2 F Ogg:

For another drop, the successive charges, as indicated by
the velocities, were again proportional to whole numbers ;

5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 5, 4, 6, 5, 4,

with a variation of the order of 1 in 300, which is the limit
of precision set by the accuracy of the measurements of
velocity.

As Millikan points out, such a degree of precision is com-
parable with that which satisfies chemists in verification
of the laws of discontinuity resulting from the atomic
structure of matter. |

The numerical examples just quoted show that the
moments when a given drop is carrying a single elementary
charge can be very quickly recognised. If at such a time

2
the activity = of its Brownian movement is measured by

de Broglie’s or Weiss’s methods (para. 98), the product Ne
can then be derived from Townsend’s equation. This has
been done by Fletcher in Millikan’s laboratory; 1,700
determinations, divided among 9 drops, give 28-8 x 10!
for the product, which agrees to within | part in 200 with
the value given by electrolysis.

In short, Millikan’s experiments demonstrate in a decisive
manner the existence of an atom of electricity equal to the
charge carried by the hydrogen ion during electrolysis.

1 As a matter of fact Millikan presents his results in a different form, giving
at once the absolute values of the charges obtained by combining Stokes’ law
with H. A. Wilson’s equation. In my opinion, it is of greater advantage to
put forward first of all the facts that would be unassailable even if Stokes’ law
were quite inapplicable to droplets falling in a gas.

ee a a

THE ATOM OF ELECTRICITY 175

101.—THE VALUE OF THE ELEMENTARY CHARGE: Dts-
cussion.—The precise value of this elementary charge,
which we now know to exist, has yet to be determined.
To do this the mass m of the droplet must be measured,

| €
since H. A. Wilson’s equation gives the ratio es only, and up

‘to the present no better means has been found than the
application of Stokes’ law. :

mg = 6rakv

with the introduction of suitable corrections.

There is no doubt, in the first place, that the product
6;agv does not accurately express the frictional force
acting upon a microscopic spherule moving in a gas at a
velocity v. The expression holds for liquids (para. 61), but
in that case the radius is very great in comparison with the
mean free path of the fluid molecules ; whereas in gases
it is of the same order of magnitude. The frictional effect
will in consequence be lessened, as will be apparent when
we consider that if L were to become very great, or if, which
comes to the same thing, the gas were to disappear, there
would be no friction at all, although the formula indicates
that the friction is independent of the pressure.’ A very
complete theory, put forward by Cunningham, leads us to
take, for the value of the frictional force, the product
67aév divided by

L ]
1 + 1:63 Pe sae |
f being the ratio of the number of molecular impacts that
are followed by regular reflection (elastic impacts) to the
total number of impacts undergone by the spherule.

Millikan made the single assumption that the frictional

force must be of the form

and worked out the value of the constant « that would give

1 Compare para, 48, note 2.

176 ATOMS

approximately equal values of e for each drop. With «
equal to -81,1 the values of e x 10! for different drops fall
between 4-86 and 4-92 (the uncorrected values lying between
4-7 and 7). Millikan therefore concluded that the value of
eis 4:89 x 10°1°, which gives the value

59 x 1074

for N, which is, after all, in remarkable Seon ee with the
value 68 x 107? given above.

Millikan regards the error in his result as being well below
1 in 2,000, though in my opinion so high a degree of pre-
cision is very doubtful,? on account of the magnitude of the
correction that has to be applied to Stokes’ law before we
can deduce the mass of a spherule from its rate of fall
in air. : ;

M. Roux uhderthok to repeat the experiments in my
laboratory, and to measure, at my suggestion, the rate of
fall of the same spherule in air and in a liquid.? Since Stokes’
law applied in the case of a liquid, measurements in the latter
case give, without correction, the exact radius of the spherule.
_ M. Roux made use principally of spherules of super-cooled,
pulverised sulphur, which were glass-like at the ordinary —
temperature, and found that Cunningham’s formula is
applicable, though with the co-efficient f nearly equal to 1
(the surface of the spherules being therefore polished rather
than rough). He then, like Millikan, followed the same
spherule for several hours under the microscope, as it sank
under the influence of gravity, rose under the influence of
the electric field, and from time to time suddenly gained or
lost an electron.

In this way he found that the value of the charge e lay

1 This value follows from Cunningham’s equation for the case where f is zero
(z.e., for a perfectly rough sphere). This assumption of perfect roughness seems
to me to present difficulties. A shot fired obliquely against a surface itself
made up of shot packed together almost perfectly might rebound back along its
line of incidence, but such a case would be exceptional. Moreover, the mean
direction of reflection, even if it were not the direction of regular reflection,
cannot be far short of it.

2 More especially since Millikan, in a recent publication, has himself raised
the value he proposed for N by 2 per cent.

5 For details of these difficult experiments see Roux, Ann. de Chim. et Phys.,
1913...

THE ATOM OF ELECTRICITY 177

between 4 x 107!° and 4:4 x 107? or, using the equivalent
form, N had the value 69 x 10°? to within + 5 per cent.
This result is practically identical with that obtained from
my study of the Brownian movement. Applying to Milli-
kan’s rough results the corrections which M. Roux’s experi-
ments have made legitimate, we get the value 65 x 10”
for N. I shall adopt 67 x 1072 as the mean value given by
this method.

102.—CorpPuscLEs : Sir J. J. THOMSON’s RESEARCHES.—
Sir J. J. Thomson’s brilliant work has shown that the atom
of electricity, the existence of which has just been established,
is an essential constituent of matter.

The electrical nature of the kathode rays once established,
attention was turned to the solution of the two equations
which, by the application of the known laws of electro-
dynamics, can be shown to express the electrostatic and
magnetic deviations undergone by an electrified projectile
bearing a charge e, of mass m, and moving with velocity v ;

ae 5 ! rene
these deviations remain constant so long as the ratio a

keeps a constant value.

In this way Thomson found values for the velocity of the
order of 50,000 kilometres per second in an ordinary Crookes’
tube ; they depended, moreover, very much on the potential
difference used to produce the discharge (in the same way
that the velocity of a falling stone depends on the height of
its fall).

e
But the ratio Pe is independent of all circumstances what-

ever, and according to the best measurements! is 1,830
times greater than the ratio of the charge to the mass for
the hydrogen ion during electrolysis. The same value is found
whatever the nature of the gas through which the discharge
passes and whatever the nature of the electrode metals ; it
is the same for Lenard’s slow kathode rays (1,000 kilometres
per second) that, without any discharge, are emitted by
metallic surfaces (zinc, the alkali metals, etc.) when acted

' 1 Classen, Cotton and Weiss, etc. I have allowed for the fact that the fara-
day is equal to 96,600 coulombs (and not to 100,000).

A. N

178 ATOMS

on by ultra-violet light. In explanation of this universal
result Thomson suggested (and all subsequent facts have
tended to confirm his theory) that the kathode projectiles
are always identical and that each of them carries a single
atom of negative electricity ; each of them is consequently
about 1,800 times lighter than the lightest. of all atoms.
Moreover, since they can be produced from any kind of
matter, that is to say, from any kind of atom, these material
elements must be a universal constituent common to all
atoms ; Thomson proposed to call them corpuscles.

A corpuscle cannot be considered independently of the
negative charge it carries ; it is inseparable from that charge
and the charge constitutes the corpuscle.

Incidentally, the high conductivity of metals is very
simply explained (Thomson, Drude) if it is assumed that at
least a few of the corpuscles present in their atoms can be
displaced by the action of even the feeblest electric fields,
passing from one atom to another or even moving hither and
thither in the metallic mass as freely as the molecules in a
gas.1 When we remember to what extent matter is really
empty and hollow (para. 95), this hypothesis will not greatly
perturb us. The electric current, which in electrolytes
consists in the movement of charged atoms, becomes in
metals a stream of corpuscles, that can produce no chemical
effect on passing through a zinc-copper junction, since
corpuscles are the same in zinc and copper. The action of a
magnet on a current does not differ fundamentally from the
action of a magnet on kathode rays. Laplace’s law, more-
over, at once gives, at all points, the value and direction of
the magnetic effect produced by a single moving charged
particle.”

As long as we were unable to measure even approximately
the charge on a single projectile, and as long as we hesitated
to accept the proof of the view that such projectiles are

1 A more detailed analysis shows that the other essential properties of the
metallic state can be explained in the same way (opacity, metallic lustre,
thermal conductivity).

& sina

2 Laplace’s classic expression is gives the value cee" for the field
2 e

due to a single particle,

THE ATOM OF ELECTRICITY 179

fragments of atoms, there were certainly grounds for the
objection that the high value found for = might be equally

well explained as being due to the magnitude of the charge
as to the smallness of the mass. But, as we have seen
(para. 99), Thomson obtained accurate measurements of the
charge on the droplets obtained by the expansion of damp
air containing no other ions than the negative corpuscles
detached from a metallic surface by ultra-violet light. If the
charge borne by these corpuscles is equivalent to 1,800
electrons, the charge on a droplet should be at least 1,800
times greater than the charge actually found.!

We are thus forced to the conclusion that the corpuscle
has a mass very much less than that of any known atom.
To be precise, the mass of this natural element, the smallest
we have hitherto discovered, is the quotient of the mass of
the hydrogen atom by 1,835, which in grammes comes to

8 x 10 28,

Thomson has gone a step further and has finally been able
to give us some idea of the corpuscular dimensions. The

kinetic energy sine? of a moving corpuscle must exceed the

magnetic energy produced in space as a result of its motion.
In this way ? it is found that the corpuscular diameter must

be less than : x 10°}; less, that is to say, than the hundred-

thousandth part of the diameter of impact of the smallest
atoms.

For reasons that I cannot enter into here, it is probable
that the upper limit to the magnetic energy is actually
reached ; in other words, the whole of the imertia of the

1 It is easy to prove that the droplets in a cloud have captured all the electric
charges in the gas and, by the application of an electric field, to ascertain that
the droplets themselves are all charged ; those among them that are polyvalent
(H. A. Wilson) would therefore be carrying several times 1,800 electrons.

2

2 We have only to integrate the expression = over all space outside a
corpuscle of radius a, remembering that (Laplace’s law) the magnetic field H is
ev sina

72

equal to at any point.

N 2

180 ATOMS

corpuscle is due to the magnetic effect that accompanies its
motion like a wake. It is possible that this may be the case
for all matter, even when neutral, if its neutrality is simply
the result of equivalence between charges of opposite sign to
be found therein (such charges may even be the sole con-
stituents of matter). All inertia would then be lectro-
magnetic in its origin. I cannot explain here how the
mass of a projectile, which remains sensibly constant
so long as its speed does not exceed 100,000 kilometres per

second, would above that speed actually increase as the

speed increases, slowly at first and then more and more
rapidly, finally becoming infinite at a speed equal to that
of light; which means that no matter can reach that
velocity (H. A. Lorentz).

103.—PositiveE Rays.—Besides the kathode rays and
X rays, a third variety, of equal importance, has been
observed in Crookes’ tubes.

Before the pressure becomes very low (one-tenth of a
millimetre) a luminous sheath (violet in air) surrounds the
kathode but does not touch it (like an equipotential surface).
It moves from the kathode and shades away from it as the
pressure falls ; but its interior contour remains fairly sharp
and is found 1 or 2 centimetres away from the kathode when
the kathode rays have become vigorous.

A second fairly bright luminosity (orange in air) now
becomes visible close up to the surface of the kathode ; it
extends, gradually becoming paler, several millimetres from
the kathode. It is produced by rays coming from the inside
contour of the sheath (for an obstacle inside the contour
_ throws a shadow upon the kathode, no shadow being formed
when the obstacle is without the sheath). The velocity of
these rays probably increases as they pass from the sheath
to the kathode (so that a maximum luminosity is produced
near it).

In order to observe these rays better, Goldstein conceived
the happy idea (1886) of piercing a canal through the kathode
on which they strike. Provided that the kathode divides the
tube into two parts (it has since been found that the essential
condition is that the space lying behind the kathode should

—_ == = a.

THE ATOM OF ELECTRICITY 181

be protected from electrical effects), a ray passing through
the canal is able to travel for several centimetres into the
second part of the tube, and finally to indicate the point
where it arrives on the glass wall by a pale fluorescence.

I have attempted to represent in the diagram the relations
between the three kinds of rays produced in Crookes’
tubes.? } |

The electrical nature of the rays once recognised, it was
natural to inquire whether the rays discovered by Goldstein,

HIG=-ls;

which strike against the kathode instead of starting from it,
are not positively charged.

That this is the case, as seemed very probable from certain
observations of Villard’s, was demonstrated by Wien. He
established the fact that the rays are deviated by an
electric field as though they were a stream of electricity (on
being passed between the two surfaces of a small condenser) ;
they are also deflected by a magnetic field (which is very
much less effective than with kathode rays). Measurements

of the velocity and of the ratio = are therefore possible.

* In addition, kathode rays start only from these points on the kathode_that
are struck by Goldstein rays (Villard). :

182 ATOMS

The velocity, which varies according to the conditions,
is only a few hundred kilometres per second. The ratio of
charge to mass turns out to be of the same order of magnitude
as in electrolysis ; the positive rays are therefore composed
of ordinary atoms (or groups of atoms). :

The measurements mentioned above were rough, for the
positive rays become very indistinct when deviated. This
fact may be accounted for (Thomson) on the supposition
that an atom projected at very high speed may, when it
strikes a neutral molecule, lose (or gain) fresh corpuscles.1
If this happens while the projectile is passing through the
deviating field, the deviation may become anything. By
hermetically sealing the kathode into the wall of the tube,
Thomson arranged matters so that communication between
an observation chamber and the emission chamber was
maintained by means of the canal through which passed
the pencil of rays under examination. This canal was so
long and so narrow that it was possible to maintain a much
higher vacuum in the observation chamber than in the
emission chamber. Encounters during the passage through
the field are thus practically eliminated and, if the (common)
direction of the electric and magnetic fields is perpendicular
to the pencil of rays, projectiles of the same kind (7.e., those

€ eae 4
having the same = but moving with different velocities) must

_ strike a plate placed opposite the canal at various points on
the same parabola. Conversely, each parabola that appears

! eS
on the plate determines (to within 1 per cent.) the ratio ra

for a particular kind of projectile.

Not to mention the unusual molecular types that in this
way can be shown to exist, and which, as Thomson expresses
it, foreshadow a new chemistry, it can be shown that the same
atom may lose (or gain) several corpuscles. The bundles of
positive rays produced in the monatomic vapour of mercury
indicate, for instance, that the mercury atom may lose as
many as 8 corpuscles, without its chemical individuality

1 The fact that the positive rays render the rarefied gases through which they
pass conducting, by leaving ionised molecules in their track, is explained at the
same time.

THE ATOM OF ELECTRICITY 183

becoming lost (since no new simple substance appears as the
result of the electric discharge).

It is very remarkable that positive electrons have never
been isolated ; the various kinds of ionisation always divide
the atom into one or more negative corpuscles of insignificant
mass on the one hand, and a positive ion, relatively very
heavy and containing the rest of the atom, on the other.

The atom is therefore not indivisible, in the strict sense
of the word, but consists possibly of a kind of positive sun,
wherein resides its,chemical individuality, and about which
swarms a cloud of negative planets, of the same kind for all
atoms. Because of the continually increasing electrical
attraction, it will become more and more difficult to remove
these planets one after another.

104.—Maanetons.—This rough model suggests that
revolving corpuscles, which would be equivalent to circular
currents, probably exist within the atoms. Now, a circular
current (solenoid) possesses the properties of a magnet. We
are reminded of the hypothesis that explains magnetism
by supposing that the molecules of a magnetic body are small
magnets (Weber and Ampére).

Langevin regards thermal agitation as the cause that
prevents these little magnets from arranging themselves
parallel to each other in the feeblest magnetic field, in which
case the body would at once reach its maximum magnetisa-
tion (its moment M, per gramme molecule being equivalent
to N times the amount of a single molecule). By assuming
that statistical equilibrium! is reached between the de-
orientation caused by thermal agitation and the orientation
caused by the field H, Langevin has been able to calculate
the maximum magnetisation M, from the observed mag-
netisation, of moment M, produced by that field 2 (1905).

This brilliant theory was put forward for the case of a
feebly magnetic fluid. Pierre Weiss completed it and showed

1 An analogous theory explains electrical double refraction (Kerr) and the mag-
netic double refraction recently discovered by Cotton and Mouton.

2 The observed moment M is, for low values of Le equal to ie (the law

RI 3RT
expressing the influence of temperature, discovered previously by Curie, may
here bé recognised).

184 ATOMS

that his results hold good even for solids. He moreover
explained ferro-magnetism (by the hypothesis of a very
intense internal field, caused by mutual action between
molecules). Finally; he was able to deduce from his experi-
ments the values of the maximum moment M, per gramme
molecule for various atoms. In so doing he made the very
important discovery that these values are whole number
multiples of the same number (1,123 C.G.S. units), so that
the magnetic moment of all atoms should be a whole number
1,123

Nn

A short time previously Ritz had suggested, in order to
explain emission spectra, that in every atom magnets are
to be found, all exactly alike, that are able to set themselves
end to end.! Weiss in his turn also found himself forced
to the conclusion that not only in atoms of one particular
kind, but in all atoms, small identical magnets exist and are
therefore a new universal constituent of matter. These magnets
he called magnetons.2, These magnetons may be arranged in
file or in parallel positions, in which case their moments
would be added together, although they might also oppose
each other in astatic couples having no external effect. This
is the only atomic model that up to the present is able to
account for Pierre Weiss’s law as well as the results of
Balmer, Rydberg, and Ritz relating to line series. The
length of the elementary magnet * should be about one ten
thousand-millionth of a centimetre and hence a hundred
times less than the atomic diameter of impact.

of times

1 Let there be p identical magnets of length a, placed end to end. In the
same straight line, at a distance 2a, let there be an electron assumed to be movable
only in the plane perpendicular to the magnets. When moved from its equili-
brium position, it will oscillate in the magnetic field due to the end poles with
a frequency of the form A [i- a pal: According to the various whole
number values possible for p, the successive spectral lines of the Balmer series,
containing all the ordinary hydrogen lines, may be obtained. It must be as-
sumed that, for a given atom, p is variable.

2 Investigations in the magnetic field have been carried out for the atoms of
Fe, Ni, Co, Cr, Mn, V, Cu.

8 Determined from the twofold condition that it must account for the

: ; ee
frequencies of the hydrogen lines (using the Ritz model) and give . E (equal

to 16-5 x 10- 2) for the moment of the magneton. ,

son See ee

THE ATOM OF ELECTRICITY 185

These magnets, however, would lie in the periphery of
‘the atom ;- in fact, measurements of magnetisation show
that small chemical and physical changes are able to alter
the number of atomic magnetons that are arranged in the
same sense (thus valency change is sufficient to cause
alteration).

Still more fundamental atomic constituents will now be
considered.

CHAPTER VIII
THE GENESIS AND DESTRUCTION OF ATOMS

TRANSMUTATIONS.

105.—Rapioactiviry.—The study of the- electric dis-
charge through rarefied gases has led to the discovery of
three kinds of radiations, all of which possess the common
characteristics of affecting photographic plates, of exciting
various kinds of fluorescence, and of making conductors of |
the gases through which they pass.

Certain substances are known that, without external
excitation, continually emit rays analogous to those obtained
in rarefied gases. This most important discovery was made
in 1896 by Henri Becquerel with respect to uranium com-
pounds and to metallic uranium itself. The wraniwm rays
have a feeble though constant intensity, which is the same
in light or in darkness, at high or low temperatures, at mid-
day or at midnight.1. Their intensity depends only on the
mass of uranium present, and not in the least on its state of
combination. Two different uraniferous substances, spread
out in very thin layers (to prevent absorption in the layer) in
such a way that the same quantity of uranium is present
per square centimetre, will emit the same amount of radia-
tion from equal surfaces. We are therefore dealing with an
atomic property ; wherever uranium atoms are to be found,
there energy is being constantly emitted. In this we find
the first indication that something may be happening in the
interior of the atom and that the atoms themselves are not
immutable (Pierre and Marie Curie). To this atomic property
the name of Radioactivity has been given.’

1 This fact, which was established by Curie, eliminates the hypothesis of an
excited radiation caused by an invisible solar radiation.

2 This word was introduced by Mme. Curie. Of course, a substance is not
radioactive (any more than a Crookes’ tube is) when it emits ionising rays

GENESIS AND DESTRUCTION OF ATOMS 187

It appeared hardly probable»that this property would be
found to be associated with uranium alone. On all sides
systematic examinations of all the known elements were
undertaken. Schmidt was the first to record the radio-
activity of thorium and its compounds, which have an
activity comparable with that of uranium. More recently,
as a result of the high state of perfection reached in the
methods of measurement, it has been possible to demonstrate
radioactivity, definite though about 1,000 times more
feeble, in the case of potassium and rubidium. It is per-
missible to suppose that all atoms are radioactive, though
in very different degrees.

It occurred to Mme. Curie to examine the natural minerals,
as well as the pure substances obtained from them. She
then noticed that certain rocks (notably pitchblende) are
about 8 times more active than their uranium and thorium
content would lead one to suppose, and came to the con-~—
clusion that this fact pointed to the presence of traces of
strongly radioactive unknown elements. It is well known
how this hypothesis was verified and how, by solution and
fractional crystallisation (the successive stages in the purifi-
cation being followed with the electrometer), Pierre Curie
and Mme. Curie obtained, starting from various uraniferous
minerals, products that became, as their purification pro-
-gressed, more and more radioactive, then luminous by
auto-fluorescence, and finally yielded pure salts of a new
alkaline earth metal, radium. The atomic weight of the
new metal is equal to 226-5, and it is analogous to barium
in its spectrum and in its general properties (radioactivity
excepted). It is at least a million times more radioactive
than uranium (1898—1902). In the course of this work -
Mme. Curie had detected, without isolating it, another
strongly radioactive element, polonium, analogous to
bismuth, and shortly afterwards Debierne established the
existence in the same minerals of an element actinium that
accompanies the rare earths in the fractionations.

With the very active preparations that could now be

in a purely temporary manner, as the result, for instance, of a chemical reaction
(glowing metals, phosphorus in process of oxidation, etc.).

188 ATOMS

obtained, the radiation could. easily be analysed ; the three
types of radiation discovered in Crookes’ tubes were found
at once and could be shudieg by similar methods. The three
types are:

a rays or positive rays (Rutherford) composed of posi-
tively charged projectiles, having masses of the same order
of magnitude as the various atomic masses. Their velocity
may exceed 20,000 kilometres per second and they are
consequently much more penetrating than Goldstein’s rays ;
they are nevertheless stopped completely after travelling a
few centimetres in air;

8 rays or negative rays (Giesel, Meyer and Schweidler,
Becquerel), composed of corpuscles moving at speeds that
may exceed nine-tenths that of light. These rays are very
penetrating kathode rays and lose scarcely half their inten-
sity after a path in air of the order of a metre ;

y rays, which cannot be deviated (Villard) and are ex-
tremely penetrating, passing through 1 centimetre of lead
before their intensity is halved. They are very analogous
to X rays, and undoubtedly do not differ from them more
than blue light differs from red.

These three radiations, each having properties varying
according to the nature of the radioactive source, are not
emitted in a constant ratio to each other, and generally
speaking, are not even all emitted by the same element (for
instance, polonium emits practically only « rays).

Pierre Curie discovered (1903) that the total energy
radiated, which is measurable in a calorimeter having
absorbing walls, has an enormous value, and is independent
of the temperature. A closed tube containing radium
liberates, when in radioactive equilibrium, 130 calories per
hour per gramme of radium. Expressed differently, it
liberates in about two days, without appreciable change,
as much heat as would be produced by the combustion of
an equal weight of carbon. It has thus become possible to
trace the origin of the internal heat of the earth and of the
radiation of the sun and stars to radioactive sources.

106.—RADIOACTIVITY AS THE MANIFESTATION OF ATOMIC
DISINTEGRATION.—Pierre and Mme. Curie noticed (1899)

GENESIS AND DESTRUCTION OF ATOMS 189

that solid objects placed in the same enclosure as a salt
containing radium (arranged in such a way that the path
from the salt to the object lay entirely through the air)
appeared to become radioactive also; this induced radio-
activity, which was independent of the nature of the object,
gradually decays when the object is withdrawn from the
influence of the radium and becomes practically nothing
after a day. Rutherford soon after observed the same pro-
perty in connection with thorium, which excites an induced
radioactivity of slightly longer duration.

These induced radioactivities are produced at all points
to which a gas liberated by the original radioactive prepara-
tion might penetrate by diffusion. It occurred to Rutherford
_ that material gaseous emanations might actually be con-
tinuously engendered by radium and thorium. On aspirating
off the air that had remained in contact with a thorium salt,
he found that the air remained a conductor, as though it
were preserving some internal source of ionisation.. This
spontaneous ionisation decreases in geometrical progression,
being reduced to about half its value every minute. The
same effect is noticed in air that has passed over a radium
salt, though the rate of decay is slower, diminution to about
half value occurring after four-day intervals.

Rutherford then assumed that the radioactivity of an
element does not indicate the presence of atoms of the ele-
ment, but their disappearance or transformation into atoms
of another kind. The radioactivity of radium, for instance,
implies the destruction of radium atoms and the appearance
of atoms of emanation; and although a given mass of
radium seems to us to be invariable, this is so only because
our measurements do not extend over a sufficiently long
period of time. The radioactivity of the emanation implies
the destruction of atoms of that gas, at the rate 1 out of 2 in
four days, new atoms appearing that form a solid deposit on
objects that come in contact with the emanation. The
atoms of the deposit die in their turn, at the rate of 1 out of 2
in about half an hour, which explains the induced radio-
activity mentioned above ; and so on.

Rutherford’s ingenious suggestions have been completely

190 ATOMS

verified. A radium emanation can be isolated and is con-
tinuously liberated by radium at the rate of a tenth of a cubic
millimetre per day per gramme. This gas liquefies at — 65° C.
under atmospheric pressure and solidifies at — 71°C. (to a self-
luminous solid). It is chemically inert, like argon, and
hence monatomic (Rutherford and Soddy); its density
(Ramsay and Gray) and its rate of effusion through a small
hole (Debierne) indicate in that case an atomic weight of
about 222. When caused to glow by an electric discharge
it gives a characteristic line spectrum (Rutherford). In
short, it is a definite chemical element, which Ramsay has
proposed to call Niton (shining). It is, however, an element
that decays spontaneously to the extent of one-half in four-
day intervals (more exactly, after intervals of 3-85 days).
For the first time the fact has been established that a simple
substance, and hence an atom, can be born and can die.

We can now scarcely avoid the conclusion that radium
itself also gradually decays, and at the same rate that it
produces niton, or very nearly at the rate of one-thousandth
of a milligramme per day per gramme. In short, we are led
to the conclusion that all radioactivity is the sign of the
transmutation of an atom into one or more other atoms.

These transmutations are discontinuous. We find, for
instance, no intermediate steps between radium and niton ;
we either have radium atoms or niton atoms, and we can
obtain no evidence of any matter that has ceased to be
radium and has not become niton. Similarly, for as long
as it is possible to observe niton, that gas retains its par-
ticular properties, whatever its “age,” and continues to
disappear to the extent of one-half after every four-day
interval. Transmutation must occur atom by atom, sud-
denly and explosively, and it is during the explosions that
the various rays are shot out. When, for instance, we say
that the radioactivity of uranium is an atomic property,
we do not mean that all the uranium atoms present are
concerned, but only those that are actually disintegrating
(the number of the latter being moreover proportional at
each instant to the mass of uranium present). It is only at
the very moment of explosion that an atom is radioactive.

GENESIS AND DESTRUCTION OF ATOMS 191 -

107—TnE Propvuction oF HeEtium.—We should not
perhaps have been so ready to accept Rutherford’s views
were it not that a certain simple substance already known
was found to be produced by transmutation. Ramsay and
Soddy succeeded in proving that heliwm is produced in ever-
increasing quantity in a closed vessel containing radium
(as, indeed, Rutherford and Soddy had predicted would be
the case). This brilliant piece of experimental work removed
all doubt in the minds of physicists as to the possibility of
spontaneous transmutation (1903).

It was known, moreover, that the « particles have masses
of the same order as the atomic masses. More precisely,

the ratio 2 is always very nearly the same from whatever

element the rays are produced, and is equal to about half
the value of the same ratio for the hydrogen ion in electro-
lysis. The « particles might therefore be atoms having a
coefficient equal to 2; but they might also (Rutherford)
be helium atoms carrying two elementary charges each.
That the second alternative is correct was proved directly
by Rutherford and Roys. They enclosed some niton in a
thin-walled glass vessel (the thickness of the walls was of
the order of a hundredth of a millimetre), through which
molecules of a gas possessing the degree of agitation cor-
responding to the ordinary temperature were unable to pass
(this had been demonstrated for helium in particular),
though the « rays emitted by niton could pass through
easily. Under these conditions helium is soon found in the
outer vessel into which the rays have penetrated ; the «
projectiles are helvum atoms expelled at the prodigious speed of
20,000 kilometres per second.

108.—« Rays.—The atomic weight of radium is very —
nearly the sum of the atomic weights of niton and helium.
During transmutation, therefore, the radium atom splits up
into a helium atom and a niton atom, with an explosion that
expels the helium atom to a distance, and which must also
propel the niton atom in the opposite direction, with an
equal quantity of motion (a phenomenon analogous to the |
recoil of a gun). The initial velocity of this niton projectile

192 ATOMS

can therefore be easily calculated, and is found to be
several hundred kilometres per second.

I do not see that there is any reason for drawing a
distinction between the two projectiles ; slow « rays com-
posed of niton (very similar to Goldstein’s rays) must be
taken into account as well as « rays of helium. I shall
return to this point later.

109.—A TRANSMUTATION Is NoT A CHEMICAL REAcTION. —
A first account of the splitting up of radium into helium
and niton raises the question whether that change may be

regarded as a chemical reaction that certainly disengages —

much heat, but which is nevertheless not essentially
different from ordinary reactions. Why cannot radium be
considered as a compound sinlding niton and helium on
- dissociation 2

This attitude cannot be maintained when it becomes
apparent that all the factors that influence chemical reaction
are found to be of no effect in the case of radioactive change.
A rise in temperature of 10° is just about enough to double
the speed of a reaction. At this rate a reaction should become
10,000,000,000 times faster for each elevation of 300° C.
Now, the heat liberated by radium remains absolutely
unaffected by much greater temperature variations.

This behaviour is general. By no means whatever is it
possible to modify the inflexible course of radioactive trans-
formation. Heat, light, magnetic field, high concentration,
or extreme dilution of the radioactive material (that is to
say, intense or negligible bombardment by ~ and f pro-
jectiles) have no effect. Deep within the atom, in the highly
condensed nucleus which has been shown to exist therein,
a disintegration takes place that. is affected as little by
influences we can control as is the evolution of a distant
star. We may add that the explosions of two atoms of the
same kind appear to be absolutely identical, giving exactly
the same velocities to the emitted « projectiles (and also to
the 8 projectiles).

_ 110.—AtToms po not Drcay.—We can go even further
and catch a glimpse of the infinitely complex world within
the nucleus.

Ee ee

GENESIS AND DESTRUCTION OF ATOMS 193

We have seen that whatever the age of a given mass of
niton, half of that mass disappears in four days. The atoms
therefore do not decay, since every atom that escapes
destruction (during any given time) still has an even chance
of survival for the four days following.

Similarly, if two small globes connected by a tube were
to contain a mixture of oxygen and nitrogen in statistical
equilibrium, it might happen that the chances of molecular
agitation would collect all the oxygen molecules to one side
and all the nitrogen to the other; all that then need be
done to keep the two gases separate would be to close a tap
in the connecting tube. The kinetic theory enables us to
calculate the time T (which will be very long if the number
of molecules is large) during which spontaneous separation
of this kind will have an even chance of occurring. Consider
now a very large number of similar pairs of globes. During
each lapse of time T, whatever time has elapsed } already,
spontaneous separation will occur in half the pairs of
globes still effective ; the variation law is the same as for
radioactive elements.

The above illustration makes it clear, in my opinion, that
in each atomic nucleus (comparable with the gaseous
mixture that fills one of our pairs of globes) a statistical
equilibrium must be set up between a large number of
irregularly varying parameters, as in the case of a gas in
equilibrium, or of light filling an isothermal enclosure.

When, by chance, certain conditions that are as yet un-
known are satisfied within the complex nucleus, a funda-
mental upheaval occurs, resulting in a redistribution of the-
matter present according to another stable and permanent
scheme of uncoordinated internal motion. We may suppose
(though it is not certain) that the « and 8 particles pre-exist
within the nucleus and already possess there, before the ex-
plosion, speeds of several thousands of kilometres per second.

I need scarcely point out that the law of chance found for
the radium and thorium emanations is the general law of

* It makes no difference whether or not separation had, at a given instant,
say after one hour, been nearly complete for any pair; in general, such a state
of affairs cannot persist for very long, since a return to a state of mixture is
much more probable for a partially separated system.

A. o

194 ATOMS

atomic disintegration. To each radioactive element cor-
responds a definite period or time during which half of any
measurable mass of the element undergoes transmutation.
This period is about 2,000 years for radium (Boltwood),
so that if a tube containing 2 grammes of it were sealed up
now there would not be more than 1 gramme of radium in

the tube in the year 3914, together with 1 gramme of other
substances yet to be determined (among which will be
helium). As may be shown by a simple calculation, this
may be also expressed by the statement that very nearly
a thousand-millionth (more exactly, 1:09 x 10~™) part of
any given mass of radium disappears per second.

111.—RaDIOACTIVE SERIES.—It has been possible (as was —

(6 x 10° years) . : Lae —— Helium.
(25 days) . : . Uranium X.
(100,000 years) ? . lonium ——> Helium.
(2,000 years) . . Radium ——> Helium.
(3-85 days) : . Niton enh. Haitirk:
(3 connie : . Radium A—— Helium.
(27 minutes). - Radium B.
(20 minutes) : . Radium C—— Helium.
(15 years) . : : Rehick D.
(A few days) . ¢ Re ek E.
(5 months) ‘ - Polonium —— Helium.
| |
(?) (?)

done for niton before it was isolated) to characterise by
their periods no less than thirty new simple substances,
derived from uranium and thorium by successive trans-
~mutations.1. One of these periods is no more than the one

1 Readers who wish for further details are referred to Mme. Curie’s treatises
on Radioactivity (Gauthier-Villars, 1910).

en

~ GENESIS AND DESTRUCTION OF ATOMS 195

twenty-fifth of a second (and there are certainly others even
shorter) ; others exceed 1,000,000,000 years. In the table
above are shown the periods T for a series of elements
derived from uranium by successive internal decompositions
or rearrangements.

Bifurcations are possible, side chains being formed.! In
other words, the same atom may undergo, according to
which of two critical internal configurations happens to occur
first, one or another kind of transmutation. We may
suppose that if, during the same time, a uranium atom had 9
chances out of 10 of undergoing the rearrangement that
gives uranium X, and | chance in 10 of undergoing another
that would give actinium, the whole of any measurable mass _
of uranium would be transformed 2 into uranium X and 10
into actinium.

It will be noticed that helium (which undoubtedly has a
very stable nucleus) is a frequent product of atomic disin-
tegration. This perhaps explains why many of the differences
between atomic weights (lithium and boron, carbon and
oxygen, fluorine and sodium, etc.) are exactly equal to 4,
the atomic weight of helium.

I cannot, however, believe that the element helium is
unique. Other chains of transmutations may show smaller
differences. Moreover, I presume that a radioactive element,
though classed as emitting only 8 and y rays, might very
well project atoms heavier than the helium atom (copper,
for instance), without our becoming aware of it, for reasons
that will be apparent later.” |

112.—Cosmocony.—In all cases lighter atoms are obtained
by the disintegration of heavier ones. If the inverse phe-
nomenon is possible and heavy atoms can be regenerated,

1 One such chain starts from radium C (Fajans and Hahn).

2 For instance, since Ra B, Ra D, Ra E emit no helium, I consider it possible
that the simple substance derived from polonium might have an atomic weight
less than 140. This body is often assumed to be lead, the latter's atomic
weight, 207, being obtained by subtracting 5 times the atomic weight of helium
(there are five emissions of helium from radium to polonium) from the atomic
weight of radium ; and lead is present in radium minerals. This view may be
correct, but a definite proof is indispensable. Analogous remarks apply to
thorium D and actinium C.

0 2

196 ATOMS

the process must take place at the centres of stars, where
the temperature and pressure is enormous and favours
reciprocal penetration between atomic nuclei, accompanied
by energy absorption.* :

The: high value given by analysis for the mean radio-
activity of the earth’s crust appears to me to afford a strong
presumption in favour of this hypothesis. If radioactive
atoms were equally abundant near the centre, the earth
would be more than 100 times more radioactive than is
sufficient to account for the preservation of its central heat.
It has therefore been suggested that such atoms are present
only in the superficial layers. This view appears to me to be
incorrect, for the radioactive atoms, being very heavy,
ought on the contrary to accumulate enormously at the
centre. We are therefore forced to accept a very rapid rate
of cooling for the earth, unless we assume that in the deeper
layers a highly endothermic formation of heavy atoms
occurs.

Slow convection processes would bring these heavy atoms
to the surface, where they would disintegrate; the heat
then radiated away, as well as the total “evaporation ”’ of
the solar system (in the form of positive rays, corpuscles,
fine powders repelled by light, and of light itself), might for
a long time remain compensated in respect both of matter
and energy, by the falling in.of bulky powders formed in
interstellar space out of corpuscles and light atoms, and also,
I presume, at the expense of light itself.2 The universe,
passing always through the same immense cycle, statisti-
cally must always remain identically the same.*

113.—AtTomic ProgectTiLes.—The penetration of « rays
into matter gives us important information about the atoms
and the singular properties they may acquire when pro-
tected at the enormous speeds possessed by these rays.

1 This hypothesis, which Mme. Curie put forward at the same time as myself,
certainly expresses the attitude of many physicists.

2 The principle of relativity (Einstein) forces us to attribute mass and weight
to light.

. We here meet with, in its essentials, a hypothesis put forward by Arrhenius
that explains the stability of the universe by the existence, at the centres of the
stars, of highly endothermic ‘‘ compounds.” A better understanding of how
it may be possible for the stellar universe to persist indefinitely will be obtained
by the perusal of Arrhenius’s book of scientific poetry, “ Worlds in the Making.”

GENESIS AND DESTRUCTION OF ATOMS 197.

The essential fact is that «2 rays pass in straight and
sharply defined lines, without noticeable diffusion, through
layers of air several centimetres thick, and through homo-
geneous thin sheets of aluminium and of mica up to four or
five hundreths of a millimetre in thickness.

Now, taking the atomic diameter in the sense employed
in the kinetic theory (diameter of impact), we find that the
atoms in aluminium or mica are as closely packed together
as the shot in a pile of shot. It cannot be supposed that the
helium projectiles pass through the interstices, and we must
assume that they pierce the atoms, or more accurately the
casings (para. 95), that protect the atoms from molecular
impacts. It is easily shown, from the density of aluminium,
that each « projectile pierces about 100,000 aluminium
atoms before it is stopped. This will not seem so surprising
if it is remembered that the initial energy of such a pro-
jectile is more than 100,000,000 times greater than that of a
molecule in ordinary thermal agitation. Finally, the thin
metallic sheets exposed to this bombardment do not appear
to be altered.

Extrapolation to the case of any kind of atom whatever
is certainly permissible, and we may picture two atoms
colliding at sufficiently high speeds as passing through each
other without mutual effect.1_ This becomes comprehensible
when we remember what has been said as to the extreme
smallness of the volume actually occupied by the material
part of the atom (para. 94). Ifastar happened to be impelled
towards the solar system, regarded as bounded by the orbit
of Neptune, the chances are small that it would hit the sun
itself. If, moreover, the relative motions of the star and the
sun were sufficiently rapid, the forces of attraction would
not have time to do any reasonable amount of work and
neither star nor sun would be deviated perceptibly from
their courses. Similarly, the extreme smallness of the
atomic nucleus certainly makes actual impact between
nuclei extremely rare. But a few peripheral corpuscles that
offer less resistance to being set in motion may get detached,

1 A rifle bullet moving sufficiently rapidly would pass through a man without
hurting him.

- 198 ATOMS

with the result that the projectile leaves a train of ions
behind it. |

In consequence of the ionisation thus produced, « rays
gradually lose their velocity as they pass through matter.
The surprising fact has been established that all their
characteristic properties cease to be shown when their
velocity falls to a certain critical value, which, however, is
still very high (more than 6,000 kilometres per second).

Consider a minute speck of polonium in air; the « rays
emitted suddenly cease to have effect on reaching the cir-
cumference of a sphere of radius 3-86 centimetres with the
grain as centre. About a speck of radium in radioactive
equilibrium (containing, that is to say, the limiting pro-
portions of the successive products of its disintegration),
it is possible to trace five sharply defined concentric spheres
with radii lying between 3 and 7 centimetres.! :

It was at first supposed that this fact established a
difference in nature between « rays and the positive rays
from a Crookes’ tube, wherein the velocity is only a few
hundred kilometres per second, although the particles travel
in straight lines for several decimetres. But a distance of
several decimetres in a Crookes’ tube is not equivalent to a
hundredth of a millimetre in ordinary air. It is now held,
therefore, quite simply, that the penetrating power, being a
function of velocity, falls off very rapidly when the velocity
falls below a so-called critical value (ill-defined), so that
an atomic projectile that cannot do more than, say, 5,000
kilometres per second cannot pass through more than a
quarter of a millimetre of air. Moreover, towards the end
of its path ionisation becomes intense and diffusion con-
siderable, until finally the projectile gets very considerably
slowed down, no longer breaks through the atomic casings,
and rebounds from them like an ordinary molecule.

It is now apparent why I took occasion to point out.
(para. 111) that if an atomic explosion were to project a

1 In minerals the circumferences are microscopic and appear as small round
spots (pleochroic halos), which are observed about minute feebly radioactive
crystals imbedded in certain micas. The extent of the blackening produced
by activity of a known kind enabled Joly and Rutherford to estimate the time
taken by the halo to form as several hundred million years.

GENESIS AND DESTRUCTION OF ATOMS 199

sufficiently heavy atom of some common element we should
not be able to perceive it. In such cases a masked trans-
mutation would occur. For explosive energies of the order
of magnitude established up to the present only the lighter
atoms could acquire sufficient velocity and energy to give
them a noticeable path in air ; a copper atom, for instance,
could not be detected.

CouNnTING ATOMS.

114.—ScInTILLATIONS: THE CHARGE ON THE « PRO-
JECTILE.—Sir William Crookes discovered that the phos-

phorescence excited by the 2 rays in substances that stop
- them is resolved under the magnifying glass into. separate
scintillations, fugitive starlike points of light that are
extinguished as soon as they are kindled. They may be
seen continually appearing and disappearing all over the
screen that receives the stream of projectiles. Crookes at
once suggested that each scintillation marks the point of
arrival of one projectile and thus enables us to perceive, for
the first time, the individual effect of a single atom. Similarly,
although we may not see a shell, we can perceive the con-
flagration that it kindles when it is stopped.

Rutherford, moreover, had measured, in a Faraday cylinder,
the positive charge q radiated per second in the form of «
rays from a given mass of polonium, and (by measuring the
conductivity of the gas) had determined the positive and
negative charges + Q — Q that the same rays liberate in
ionising the atoms they pass through before being stopped
in air. In this way he had found that the liberated charges
Q were equal to very nearly 100,000 times (94,000 times) the
charge q carried by the projectiles.

Combining the two processes, Regener determined the
molecular magnitudes in a new way. He counted one by
one the scintillations produced within a given angle by a
given polonium preparation and from the result calculated
the total number of « particles emitted per second by that
preparation (1,800 in point of fact). He found, moreover,
that in one second these particles liberate -136 electrostatic

200 ATOMS

136
1,800 x 94,000
or 8X 10°! for each « particle. Since the « projectile
carries twice the elementary charge, the value of the latter
must be 4 x 10~1°, which agrees well with the other deter-
minations.

115.—EectricaL MErHops or CountTine.—In spite of
this agreement, it might still be doubted whether the
scintillations are exactly equal in number to the number of
projectiles. Rutherford and Geiger extended and con-
solidated Regener’s brilliant work and devised a second
extraordinarily ingenious method for counting the pro-
jectiles.

In their apparatus the « cngiertiias start from a thin
radioactive layer of known surface (radium C) and are
filtered through a mica diaphragm (thin enough for all of
them to pass through). They then enter a gas at low
pressure between two plates at different potentials, one being
connected with a sensitive electrometer. In the gas each
projectile produces a train of ions which move, according
to their sign, towards one or other of the electrodes.

If the pressure is sufficiently low and the potential
difference sufficiently high, it becomes possible for each ion
to acquire a velocity in the interval between two molecular
impacts fast enough to split up the molecules it meets into
ions, which become ionising centres in their turn. This
multiplies quite a thousandfold the discharge that would be
caused by those ions only that were produced by the pro-
jectiles directly. The discharge is thus made large enough
to be detected by a noticeable deflection of the electrometer
needle.2 Under these conditions,- the radioactive source
being sufficiently far removed and the « radiation that it
sends between the two plates being limited in amount by
its passage through a small aperture, the movements of the
electrometer needle are seen to take place in distinct jerks
irregularly distributed in time (from two to five per minute).

1 This phenomenon was discovered by Townsend and is the basis of the present
explanation of the mechanism of the disruptive discharge (electric spark).

2 Rapid return of the needle to zero is assured by making the insulation im-
perfect.

unit of each signin air. This gives the charge

GENESIS AND DESTRUCTION OF ATOMS 201

This very clearly demonstrates the granular structure of the
radiation.

The jerks can be counted with rather greater precision
than the scintillations, and the numbers obtained by the
two methods are equal. Rutherford found that 1 gramme
of radium in equilibrium (with its disintegration products)
emits 136,000,000,000 helium atoms per second, which
means that radium by itself produces 34,000,000,000
(3-4 x 10?) projectiles per second.

Omitting Regener’s intermediate step, Rutherford and
Geiger then allowed « projectiles, emanating from a thin
radioactive layer and m in number, determined as above,
to fall within a Faraday cylinder (the negative 8 particles,
being much more readily deviated by a magnet, were

removed by a powerful magnetic field). The quotient f of

the positive charge q that gets into the cylinder by the num-
ber of projectiles n gives the charge 9-3 x 10~1° borne by a
projectile, which gives 4-65 x 10°! for the elementary
charge e, and |

62 <°.10*

for Avogadro’s number, with an error of probably not more
than 10 per cent.

116.—Tue NuMBER oF ATOMS THAT GO TO MAKE UP A
KNOWN VOLUME OF HeELIUM.—Since we can count the «
projectiles emitted in a second by a radioactive body, we
know how many atoms there are in the mass of helium pro-
duced during that time. If we determine that mass or the

* Regener has recently carried out determinations of this kind with a rays
from polonium by counting the scintillations produced on a homogeneous flake
ofdiamond. His determination of the charge g seems to me, however, to involve
an uncertain factor, and a short discussion will be of interest.

In this method it is implicitly assumed that the whole of the charge registered
by the receiver is carried by a projectiles. Now, the explosion that propels an
a projectile in one direction also propels the rest of the radioactive atom, a, in
the opposite direction. These @ rays, which have scarcely any penetrating
power cannot have any effect in Rutherford’s apparatus (in which a thin plate
separates the active body from the receiver). Butin Regener’s experiment they
may exert their influence (the ends of the apparatus being open and no thin
screen being used). For it is possible that these a rays do not produce
scintillations ; it is probable that they are positively charged (like all violently
projected atoms) and that they carry two positive charges, like helium. In
short, the value 4-8 x 10~?° obtained cannot be regarded as certain.

202 ATOMS

volume it occupies at a given temperature and pressure,
we shall obtain the mass of the helium atom directly. The
difficulty, by no means small, is to collect all the helium and
to prevent its contamination with other gases.

Measurements carried out by Sir James Dewar and
subsequently improved upon by Boltwood and Rutherford,
indicate. that 156 cubic millimetres are liberated annually |
per gramme of radium. Allowing for the disintegration
products present with the radium, this gives 39 cubic
millimetres for the pure radium alone. Since it projects
34,000,000,000 helium atoms per second, we get 34 x 86,400
x 365 thousand million molecules in that volume. The
number of monatomic molecules N of helium that occupy
22,400 cubic centimetres, and which therefore make up a
gramme molecule, is thus

34 <x 86,400 x 365 x 22,400
‘039

x 10° or 62 « 10??.

Mme. Curie and Debierne subsequently carried out a
similar determination of the quantity of helium liberated
by polonium.*

Projectiles were counted, as in Rutherford and Geiger’s
experiments, by the scintillation method, and by the method
of ‘‘ electrometer jerks.” The latter, made to occur at
considerable intervals (one per minute) so that they should —
not overlap, were recorded on a ribbon, each jerk being -
indicated by a small denticulation in a continuous line.
The denticulations could then be counted at leisure ? (Fig. 14).
The volume of helium liberated was -58 cubic millimetre.
This series of experiments gives for N the value

65 x 1074

1 The choice of polonium offers many advantages, because the radioactive phe-
nomena in connection with it are less complex, polonium being the end product
of its radioactive series (only one transmutation occurs, into helium), and
because, no gaseous emanation being produced in the space where the radio-
active material is mounted, the number of a projectiles that-penetrate into the
glass is negligible ; in this way the difficulties involved in the removal of helium
occluded in the glass are avoided.

2 Taken, for convenience in printing, from some later work of Geiger and
Rutherford, in which the projectiles from radium were counted by this method
with very great accuracy. 2

GENESIS AND DESTRUCTION OF ATOMS 203

which is in remarkable agreement with the values already
obtained.

117.—Tue NuMBER OF ATOMS THAT MAKE UP A KNOWN
Mass or Raprum.—The number of projectiles emitted gives
the number of generative atoms that disappear as well as
the number of helium atoms that make their appearance.
If, therefore, we have any means of finding out what fraction
of a gramme atom of the generative body has disappeared,
we can obtain at once the mass of the atom of that body and
hence the other molecular magnitudes.

All the necessary data are available in the case of radium ;
its gramme atom is known to be 226-5 grammes, and the loss
in « projectiles is 3-4 x 10! per gramme. The gramme
atom therefore emits 226-5 x 3-4 x 10! « projectiles per
second. We know, moreover (para. 110), that out of N

Fig. 14.

radium atoms N x 1:09 x 10°" disappear per second,
which gives N from the equation

226°5 x 34 x 101° = N x 1:09 x 10-1;
we thus get
NaF i) 10";

118.—Tue Kinetic ENERGY OF AN « PROJECTILE.—If
we know, as is the case with radium, the kinetic energy
and speed of the « projectiles, we can obtain, in yet
another way, the mass of the helium atom and the molecular
magnitudes.

The kinetic energy, to within a few per cent. (due to the
penetrating § and y rays) is equivalent to the heat con-
tinually liberated (Curie). Let u,, wy, ws, uy be the initial
velocities (determined by Rutherford) of the four series
of a projectiles emitted by radium in radioactive equili-
brium. Since radium liberates 130 calories per gramme per

204 ATOMS

hour (3,600 seconds), and since the mass of one helium atom

is nw we have, very nearly,

1 4 130 x 4:18 x 10’
5 X N x 3°4 x 10!9[w,? + w.?-+ ws? + u,7] = 3 600 ‘

or a value for N of nearly 60 x 10”.

The individual energy of an « particle is of the order of a
hundred-thousandth of an erg.

119.—TuHE PaTH oF EACH ATOMIC PROJECTILE CAN BE
MADE VISIBLE.—Thanks to the scintillations produced, we
are able to perceive the stoppage of each of the helium atoms
that constitute the « rays.
But the path followed by
each atom is nevertheless
invisible, and we only
know that it is approxi-
mately rectilinear (since
the « rays scarcely diffuse
at all), and that it must
be marked by a train of
ions, liberated from the
atoms passed through.
Now, in an atmosphere
saturated with water
vapour, each ion can act
as the nucleus of a visible drop (para. 99), and C. 'T. R. Wilson,
who discovered this phenomenon, has made use of it, in a
most ingenious manner, to demonstrate the path as a visible
streak.

A minute radioactive speck, placed at the end of a fine
wire, is introduced into an enclosed space saturated with
water. vapour. A sudden expansion increases the volume
and produces supersaturation by cooling. At very nearly
the same instant a spark is produced and lights up the
enclosure. In the form of white rectilinear streaks starting
from the active granule rows of droplets can be seen (and
photographed) along the paths followed by the few particles

Fig. 15.

1 Proc. Roy. Soc. A., Vol. LX XXVIL., 1913.

GENESIS AND DESTRUCTION OF ATOMS 205 .

emitted after the expansion and before the illumination of
the vessel (Fig. 15).

Closer examination, however, shows that the trajectories
are not rigorously straight, but bend noticeably during the
last few millimetres of their path, and even show sharp
angles (several are visible in the figure). Each time the
atomic projectile passes through an atom it undergoes a
deviation, very slight, but. nevertheless not absolutely
negligible ; these deviations, which act cumulatively and in
opposition to one another quite irregularly, explain the
observed tendency to curve. Finally, in very exceptional
cases (owing to the extreme smallness of the atomic nuclei)
it happens that the nucleus into which almost all the mass
of the projectile is condensed strikes the nucleus of another

Fig. 16.

atom ; a considerable deviation is then suddenly produced.
At the same time, the nucleus that has been struck receives
an impulse sufficiently intense to make it become, in its turn,
an ionising projectile, with a trajectory that, although very
short, is nevertheless recorded quite clearly on the plate as a
kind of spur.?

Finally, C. T. R. Wilson has succeeded in making
visible, by the condensation of water droplets, the path
followed by an ionising corpuscle (8 rays and kathode rays).
The phenomenon is particularly interesting in the case of

1 In this we have, I think, a means of estimating the relative dimensions of
the atomic nucleus and the atom. We haveonly to find how many single impacts
occur on the average in the trajectory of an a particle on passing through an
approximately known number of atoms (p. 197). An examination of Wilson’s
photographs seems to me to indicate (very roughly) that one nuclear impact
occurs for every million atoms traversed ; the diameter of the atomic nucleus
should therefore be about a thousand times less than that of the atom.

206 | ATOMS

the secondary rays (having small penetrating power and
diffuse trajectories) produced by the emission of corpuscles
from atoms struck by 7 rays or X rays. Curving of the
trajectory is then very marked. Moreover, since their
ionising power is less than for a rays, the droplets appear
separated from each other and give a visible indication of
each ionising impact. Fig. 16, which is a photograph of
the trajectory in air of a pencil of X rays, shows that the
primary ionisation is of very little importance and that
nearly all the ions are produced along the curvilinear
trajectories of the various secondary rays produced by the
primary ionisation.

The beauty of these brilliant experiments needs no
comment.

CONCLUSIONS.

120.—THE AGREEMENT BETWEEN THE VARIOUS DETER-
MINATIONS.—In concluding this study, a review of various
phenomena that have yielded values for the molecular magni-
tude enables us to draw up the following table :—

Phenomena observed. _—
Viscosity of gases (van der Waal’s equation) . <1
Distribution of grains . . | 68-3
Picteuticn wticwedibud | Displacements. 2 eet Re
Rotations . ; ‘ 65
Diffusion 69

Critieal opalescence 75

Irregular molecular distribution | The blue of the sky | 60 (2)
: ‘ : : 64

Black body spectrum

Charged spheres (in a gas) epese : : ; 68
, Charges produced . ; . |" 625

Radioactivity | Helmminendered ae
Energy radiated . ‘ ; 60

Our wonder is aroused at the very remarkable agreement
found between values derived from the consideration of such

GENESIS AND DESTRUCTION OF ATOMS 207

widely different phenomena. Seeing that not only is the
same magnitude obtained by each method when the condi-
tions under which it is applied are varied as much as possible,
but that the numbers thus established also agree among
themselves, without discrepancy, for all the methods
employed, the real existence of the molecule is given a
probability bordering on certainty.

Yet, however strongly we may feel impelled to accept the
existence of molecules and atoms, we ought always to
be able to express visible reality without appealing to
elements that are still ‘invisible. And indeed it is not
very difficult to do so. We have but to eliminate the
constant N between the 13 equations that have been
used to determine it to obtain 12 equations in which
only realities directly perceptible occur. These equations
express fundamental connections between the phenomena,
at first sight completely independent, of gaseous viscosity,
the Brownian movement, the blueness of the sky, black
body spectra, and radioactivity.

For instance, by eliminating the molecular constant
between the equations for black radiation and diffusion by
Brownian movement, an expression is obtained that enables
us to predict: the rate of diffusion of spherules 1 micron in
diameter in water at ordinary temperatures, if the intensity
of the yellow light in the radiation issuing from the mouth
of a furnace containing molten iron has been measured.
Consequently the physicist who carries out observations on
furnace temperatures will be in a position to check an error
in the observation of the microscopic dots in emulsions !
And this without the necessity of referring to molecules.

But we must not, under the pretence of gain of accuracy,
make the mistake of employing molecular constants in
formulating laws that could not have been obtained without
their aid. In so doing we should not be removing the
support from a thriving plant that no longer needed it ; we
should be cutting the roots that nourish it and make it grow.

The atomic theory has triumphed. Its opponents, which
until recently were numerous, have been convinced and have
abandoned one after the other the sceptical position that

208 ATOMS

was for a long time legitimate and no doubt useful. Equili-
brium between the instincts towards caution and towards
boldness is necessary to the slow progress of human science ;
the conflict between them will henceforth be waged in other
realms. of thought.

But in achieving this victory we see that all the definite-
ness and finality of the original theory has vanished. Atoms
are no longer eternal indivisible entities, setting a limit to
the possible by their irreducible simplicity ; inconceivably
minute though they be, we are beginning to see in them a
vast host of new worlds. In the same way the astronomer is
discovering, beyond the familiar skies, dark abysses that the
light from dim star clouds lost in space takes eons to span.
The feeble light from Milky Ways immeasurably distant tells
of the fiery life of a million giant stars. Nature reveals
the same wide grandeur in the atom and the nebula, and
each new aid to knowledge shows her vaster and more
diverse, more fruitful and more unexpected, and, above all,
unfathomably immense.

INDEX

A.

AGITATION,
molecular and diffusion, 4.
and expansion of fluids, 6.
Arrhenius’s hypothesis, 40.
Atom, :
gramme, 21.
material part concentrated
at centre, 157. —
of electricity, 164.
Atomic,
co-efficients, 19.
disintegration, 188.
hypothesis, 10.
projectiles, visibility of
paths, 204.
weights, 21 et seq.
Atoms, 7.
ageing of, 192.
counting of, 199, 200.
dimensions of, 49—52.
relative weights of, 11, 12.
Avogadro’s hypothesis, 17—19.
proof of, 59.
Number, 26, 49.

B.

BLACK bodies, 144, 146.

composition of light from,
148.

Boltzmann, 60 et seq.

Brownian movement, 83—89.
and Carnot’s principle, 86.
definition of activity of,

— 109—111.
in emulsions, 99.
irregularity of, 116.
rotational, 113, 124.

C.

CARNOT’S principle, 86.
A.

Centrifuging, fractional, 94.
Charge,
minimum elementary,
48.
on gramme ion, 44.
- on aprojectile, 199.
Chemical,
discontinuity, 9.
formule, 13.
Constant, Planck’s, 152.
Constitutional formule, 33.
Corpuscles, 177.
Cosmogeny, 195.
Crystals, liquid, 143.

43,

ay

DECOMPOSITION, 7.
Definite proportions, law of, 9.
Density, fluctuations of, 134.
Diffusion,

of emulsions, 111.

of large molecules, 127.

of visible granules, 129.
Discontinuity,.

chemical, 9.

of energy, 69—70.
Dissociation, —

electrolytic, 42.

molecular, light as cause of,

161.
Distribution of grains in emul-
sions, 102.

Divisibility of matter, 48.
Dulong and Petit’s law, 21.

E.

EFFUSION, 61.
Einstein’s theory, 109.
verification of, 114.
Electricity, atom of, 164.
Electrolytes, dissociation of, 42.
Emanations, 189.
P

210

Emulsions, 89—106.

gas laws, and, 89.

preparation of, 94, 95.
Energy, discontinuity of,

70, 73.

quantum of, 70.
Equilibrium, in gas column, 90.
Equipartition of energy, 60.
Equivalents, 16.

F.

Fits, thin, 49.

Fluctuations, 134—142.

Formule,
constitutional, 33.
molecular, 27.

G.

GASES,
monatomic, 65.
specific heat of, 69 et seq.
viscosity of, 74.

Gas laws, extension to emulsions,

89.
Gay-Lussac, 18.
Gramme atom, 21.
molecule, 26.

H.

HELIUM,
number of atoms in known
volume of, 201.
radioactive production of,
191.
Hofi’s, van’t, law, 39.
Hypothesis,
atomic, 10.
Arrhenius’s, 40.
Avogadro’s, 17, 19.
Prout’s, 24.

I.

Impact, molecular, 77—81.
Ions, 40.
gaseous, charge on, 166—
169, 175.

69,

S
|

|

|
|

|
|

INDEX

K.

KATHODE rays, 164.

i,

Law,
of definite proportions, 9.
Dulong and Petit’s, 21.
van’t Hoff’s, 39.
of multiple proportions, 10.
Proust’s, 9—10.
Raoult’s, 36.
Stefan’s, 146.
Stokes’, 97—99.

| Light and quanta, 148.

| Liquid crystals, 1438.

M.

MAGNETONS, 183.

Matter, divisibility of, 48.
Masked transformations, 199..

_ Mean free path, 74.
| Membranes, semi-permeable, 38.

Mendélejeff’s rule, 24.

| Mixtures, persistence of compo-

nents in, 1.
Moleeular
agitation, 4.
formule, 27.
magnitudes, from Brownian
movement, 122.

determination of, 81,
5 Ue aes
from black body radia
tion, 153.
orientation, fluctations in,
143.
size, upper limit of, 48.
structure, 28.
Molecule,
dislocation of, during re
action, 31.
distribution of matter in,
160.
Molecules,

diameter of impact of, 77.
free paths of, 74.

in constant impact, 71.
rotation of, 67.

size of, 48.

INDEX 211

Molecules—continued. Raoult’s laws, 36.
velocities of, 53, 56. a Rays, 191.
vibration of, 64. Rays,
Multiple proportions, law of, 9. kathode, 164.
positive, 180.
X-, 165.
N. Rotation, unstable, 156.
NumBER, Avogadro’s, 49.
Numbers, proportional, 13. 5.
| ScINTILLATIONS, 199.
O. _Semi-permeable membranes, 38.
pa Similar compounds, 15.
OPALESCENCE. critical, 135. | Simple substances, i
Csemotic pressure, 38. Sky, blueness of, 139—140.

| Solution, 36.
| Specific heat,

E. gases, 69.
solids, 77, 154.

a PARTICLES, see a Projectiles. | Spectral lines, width of, 62.
Planck’s constant, 152. Sphere of protection, 66.
Positive rays, 180. Stefan’s law, 146.

’ a Projectiles, 192—203. Stereochemistry, 35.
Projectiles, atomic, 196—204. Stokes’ law, 97, 99.
Protection, sphere of, 66. Substitution, 28.

Proust’s law, 9—11.
Prout’s hypothesis, 24.
Pure substances, 2—3. T

Tun films, 49.
Q. Transmutation, 186.

QUANTA, 150, 154. casi
and light, 147. E
and rotation, 159. ;
and specific heats of solids, V.

154.
VALENCY, 31.

bond, strength of, 161.
R electrical, 44.
: Van der Waal’s equation, 79.

RADIOACTIVE Equilibrium, 198.
series, 194.
Radioactivity, 186 et seq. X.
Radium, number of atoms in
known mass of, 203. X-rays, 165. —

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