THE COLLIERY GUARDIAN

AND JOURNAL OF THE COAL AND IRON TRADES.

Vol. CV.

FRIDAY, MAY 9, 1913.

No. 2732.

ON THE KINETIC THEORY OF CASES.

[Rights of Reproduction Reserved.']

By Sir Henry Cunynghame, Chairman of the Royal
Commission upon Accidents in Mines.

This lecture was originally given to an audience of
mining students in Birmingham in 1911, and was an
attempt to give an account of the theory, not in too rigid
a mathematical dress, but in a form more suited to those
about to engage in practical work.

At the request of the Editor of the Colliery Guardian,
the lecture has been written down with such additions
as appeared necessary.

It may perhaps be considered that the sections
devoted to an explanation of the law of distribution of
velocities are disproportionately long. It seems, how-
ever, impossible to give even a sketch of this theory in
a shorter form.

It was omitted in the lecture, as it would have taken
up too much time, but no excuse need be offered for its
insertion here, inasmuch as the reader can pass it over
if he pleases.

Introduction.

In spite of the fact that most of our modern methods
of industry are based on scientific principles, there still
exists a very unreasonable antagonism between practice
and theory. The practical man, with a pitying smile,
often treats the theorist as a dreamer. The theorist
retaliates with disdainful air by treating the practical
man as a boor. But there is no need for this con-
temptuous attitude. Theory is only practice methodised,
and based on experiment. No practical man, I suppose,
objects to the use of method. What he objects to is that
a man should come from a university with a bundle of
mathematical books, and try to dictate to him how to
work a mine. And he is right. Untried theory is as
bad as unmethodised practice, and the true position of
theory as applied to the practical arts is that of a sign-
post pointing out the way to practical results, and
telling us where to expect and how to look for them.
No one in a matter like mining should dare to use a
theory until it has stood the test of the most practical
trial which is obtainable.

It is in this spirit that I have endeavoured, in as
simple language as I can command, to fulfil my promise—
rashly given perhaps—to explain the main outlines of
the kinetic theory of gases.

This theory has been placed upon a basis so firm that
its truth is now universally admitted,—but parts of it
are still very unsettled. It is like a huge building seen
in a fog. Its main outlines can be recognised, but its
details are still shadowy and uncertain.

And yet it has its uses. We cannot employ it to
calculate the size of the cap on the flame of a lamp or the
dimensions of an airway. But when we are experi-
mentally groping after some method of preventing
explosions in mines, or endeavouring to explain the
different caprices of explosives, of gases, or of coaldust,
how useful is the conception of gases as consisting of
dying molecules whose speed gives rise to what we call
heat, and whose chemical action is produced by the
shattering effects of collisions.

The kinetic theory of gases presents peculiar
difficulties. It deals with bodies so small that no
microscope can see them—indeed, so much smaller than
waves of light, that we may never perhaps be able to
see them.

They move so fast that the eye could not follow their
flight, even if they were visible. We know not their
shape nor appearance, and hence can only speculate
vaguely as to their modes of action.

And yet on examination, we become certain that they
exist. We are like observers placed in a balloon watching
the operations of huge armies. We see their general
motions, the effects of their collisions, then* progress as
a whole, and their power of surmounting obstacles, and
in certain cases we can even see the effects produced
by the action of individual molecules. From these
observations we have to guess the characteristics of the
soldiers of which these armies of molecules are
composed.

When we consider the difficulties of the problem, the
wonder is, not that man knows so little of this subject,
but that he has ever succeeded in learning anything at
all.

The present lecture does not pretend to give a full
mathematical proof of the various formulae. For this
the student must have recourse to some work like Jean’s
Kinetic Theory. All that is attempted here is what may
be termed a mathematical sketch, correct in main
outlines, but with details and refinements omitted.

Section I.

In former times, air and other gases were treated as
elastic fluids. Gas was described as if it were a sort of
liquid indiarubber. If you squeezed it, on release it
recovered its volume. If you pulled it out, it shrank
back again. It was thus supposed to resemble an
elastic tangled mass of very fine steel wire or hair
cushion which resisted either compression or extension,
and, like a spring, the extension or compression varied
with the force which produced it.

Torricelli’s experiment with the barometer was the
first which somewhat shook this theory. For he showed
that when air is sucked out of a tube water will rise into
it, not because of an elastic drawing together of the air,
but by the pressure of other air which drives the water
in. The notion of “ suction ” was thus abolished.

But this explanation only creates a fresh difficulty.
For if we are to regard air as a continuous elastic fluid
how can a finite mass of air fill infinite space ? And yet
if it cannot, there. must be a degree of attenuity after
which the air will refuse to expand, and thus have
boundaries.

The difficulties attendant upon the conception of air as
an elastic fluid induced Bernouilli and others to propose
a new explanation, which, however, at first sight seemed
so startling that it was slow in obtaining credence.

This new idea is founded on the ancient atomic
theory, which treated matter as composed of an
enormous number of atoms, spherical, small in size,
and of perfect elasticity. In the solid state these
molecules cohere, in the liquid state they glide about
over one another like marbles in a box.

But in a gaseous state they are isolated from one
another, and in rapid motion. As their mutual attrac-
tion when at normal distances from one another is so
small as to be negligible, they fly about in straight
lines with repeated collisions. Thus they can separate
indefinitely, and thus in a sense they can appear to fill
infinite space.

They are subject to gravitational attraction, and
thus congregate round the worlds in space, but if by
some chance one or more of them acquires exceptional
speed it may fly off, out of the attraction of the world in
which it happens to be, and find a new home in some
other planetary body.

The repeated collisions of the molecules against the
walls of the containing vessel cause a pressure to be
exerted against it, and this is the reason of the continual
effort of a confined gas to escape. Thus if it were not for
gravitation, all the free molecules of gas on the surface
of the earth would leave it.

But the question may occur, How upon this theory can
these motions continue ? Balls which repeatedly collide
lose their motion by friction, and come to rest. But mole-
cules of gas remain in perpetual motion, as is evidenced by
the fact that the pressure of a given quantity of confined
air at a uniform temperature remains unchanged.

To answer this question, we may ask what “ friction ”
Js. Friction is the operation whereby the motion of a
mass of molecules is shared with other masses having
less motion ; it is a method of communicating heat, or
in other words motion, by attrition.

But in the case of gases we are dealing with the
motion of individual molecules, and this motion cannot
be destroyed, only converted into other motions of
molecules—so that no degradation of motion really takes
place. For the molecular motion is itself that which we
call heat, and to talk of changing molecular motion into
heat is only to talk of changing molecular motion into
molecular motion.

Hence molecular motion must be frictionless, except

so far as it may be converted into the rotation of the
molecules on an axis, or into internal molecular motion.
But such conversion could not go on indefinitely. As a
result the molecules of gases and of liquids (and solids)
are in perpetual motion, and in the case of liquids this
perpetual motion can even be seen in what are known as
the Brownian movements.

It might seem an almost impossible task to measure
experimentally the velocity of motion of the molecules
of a gas, but in reality it is a very simple and easy
operation. We can measure the mass of a given volume
of gas. We know, for instance, that a cubic centimetre
of oxygen at a pressure, say 29 barometric inches, and
60 degs. Fahr., has a certain weight, and must there-
fore be composed of a certain “ mass ” of material. At
760 mm. of mercury and 0 degs. Cent., it has a mass of
0*00143 gramme. If then the static pressure of one atmo-
sphere, or 14*7 lb. to the square inch, or 1,013,200 dynes
to the square centimetre, is known, then the bombarding
effect of the gas due to the movement of its molecules
ought to be proportional to that pressure.

It is here necessary to observe that bodies have both
“ mass ” and “ weight,” and these must not be confused
together, as is done in the Act of Parliament which
prescribes standard weights and measures. The mass is
estimated in grammes, that is, the mass of a cubic centi-
metre of water. The gramme is 15*43 English grains.
The weight of the gramme is found by multiplying its
mass by the number of units of force which will be
equivalent to the attraction of gravitation upon a
gramme. The metric unit of force is the dyne, or
that force which, acting on a gramme for one second, will
give it a velocity of 1 cm. per second. The attraction of
gravitation on a mass of a. gramme is 981 dynes.

Before, however, we proceed to determine the velocity
which the molecules of a gas must have in order to exert
the pressure which it is known to have on the sides of
the vessel which contains it, we must arrive at a method
of computing the statical pressure produced by a rapid
series of blows.

According to the laws of motion, if m be a mass and v
its velocity, its momentum or quantity of motion = m v.
But again, according to the theory of motion put into
its final form by Newton, a force or pressure acting
upon a mass m will produce a quantity of motion in it
directly proportional to the force and to the time
during which that force acts, whence then, if a
pressure P acting on a mass m for a time t produces
a velocity v, then P t = m v, that is to say, the
velocity produced is directly as the pressure, and
directly as the time during which it acts, and inversely
as the mass of the body moved. This law rests on
experiment. Thus, then, a pressure P, acting for a
time t, on a mass wt, will produce a momentum m v in a
mass m. From this we may infer, though it is not a
rigid proof, that if a momentum mv, or a series of
momenta amounting to mv, act for a time t, it will
produce a pressure P. It is hardly a proof, for it does
not follow if a cause A produces an effect B, that a
cause B will produce an effect A. But this is true
here. For suppose that a series of small taps of a light
hammer of mass m strike a surface with velocity v.
Then, if there are n of such hammer taps per second,
the pressure P produced thereby will = n m v.

We may experimentally verify this among other ways
as follows (fig. 1):—

A tank of mercury G consisting of a 12 oz. glass
bottle with a hole in its bottom is fitted with a cork in
the hole and a small glass jet made by drawingout a piece
of tubing in the gas flame. The internal bore of the jet
may be, say, 0’3 mm. diameter. This will discharge a
jet of mercury at such a rate that, say, 4 grammes will
issue in a second, as may be found by collecting what
runs through in a minute, ^weighing it and dividing it
by 60. Here we have our hammer. The mercury stream
is broken into tiny drops, which patter down pretty
regularly. We do not know the mass m of any one drop
nor their number n, but we know n x w, the mass which
falls in a second, which in this case is 4 grammes. Now
let this stream impinge on the scale of a balance, so
that we can weigh the pressure which it exerts. A small