SCIENTIFIC WORK--MOLECULAR THEORY.
Maxwell in his article “Atom,” in the ninth edition of the
_Encyclopædia Britannica_, has given some account of Modern Molecular
Science, and in particular of the molecular theory of gases. Of this
science, Clausius and Maxwell are the founders, though to their names
it has recently been shown that a third, that of Waterston, must be
added. In the present chapter it is intended to give an outline of
Maxwell’s contributions to molecular science, and to explain the
advances due to him.
The doctrine that bodies are composed of small particles in rapid
motion is very ancient. Democritus was its founder, Lucretius--de Rerum
Naturâ--explained its principles. The atoms do not fill space; there is
void between.
“Quapropter locus est intactus inane vacansque,
Quod si non esset, nullâ ratione moveri
Res possent; namque officium quod corporis extat
Officere atque obstare, id in omni tempore adesset
Omnibus. Haud igitur quicquam procedere posset
Principium quoniam cedendi nulla daret res.”
According to Boscovitch an atom is an indivisible point, having
position in space, capable of motion, and possessing mass. It is also
endowed with the power of exerting force, so that two atoms attract
or repel each other with a force depending on their distance apart.
It has no parts or dimensions: it is a mere geometrical point without
extension in space; it has not the property of impenetrability, for two
atoms can, it is supposed, exist at the same point.
In modern molecular science according to Maxwell, “we begin by assuming
that bodies are made up of parts each of which is capable of motion,
and that these parts act on each other in a manner consistent with the
principle of the conservation of energy. In making these assumptions
we are justified by the facts that bodies may be divided into
smaller parts, and that all bodies with which we are acquainted are
conservative systems, which would not be the case unless their parts
were also conservative systems.
“We may also assume that these small parts are in motion. This is the
most general assumption we can make, for it includes as a particular
case the theory that the small parts are at rest. The phenomena of the
diffusion of gases and liquids through each other show that there may
be a motion of the small parts of a body which is not perceptible to us.
“We make no assumption with respect to the nature of the small
parts--whether they are all of one magnitude. We do not even assume
them to have extension and figure. Each of them must be measured by its
mass, and any two of them must, like visible bodies, have the power
of acting on one another when they come near enough to do so. The
properties of the body or medium are determined by the configuration of
its parts.”
These small particles are called molecules, and a molecule in its
physical aspect was defined by Maxwell in the following terms:--
“A molecule of a substance is a small body, such that if, on
the one hand, a number of similar molecules were assembled
together, they would form a mass of that substance; while on
the other hand, if any portion of this molecule were removed,
it would no longer be able, along with an assemblage of other
molecules similarly treated, to make up a mass of the original
substance.”
We are to look upon a gas as an assemblage of molecules flying about
in all directions. The path of any molecule is a straight line, except
during the time when it is under the action of a neighbouring molecule;
this time is usually small compared with that during which it is free.
The simplest theory we could formulate would be that the molecules
behaved like elastic spheres, and that the action between any two was
a collision following the laws which we know apply to the collision of
elastic bodies. If the average distance between two molecules be great
compared with their dimensions, the time during which any molecule
is in collision will be small compared with the interval between the
collisions, and this is in accordance with the fundamental assumption
just mentioned. It is not, however, necessary to suppose an encounter
between two molecules to be a collision. One molecule may act on
another with a force, which depends on the distance between them, of
such a character that the force is insensible except when the molecules
are extremely close together.
It is not difficult to see how the pressure exerted by a gas on the
sides of a vessel which contains it may be accounted for on this
assumption. Each molecule as it strikes the side has its momentum
reversed--the molecules are here assumed to be perfectly elastic.
Thus each molecule of the gas is continually gaining momentum from
the sides of the vessel, while it gives up to the vessel the momentum
which it possessed before the impact. The rate at which this change of
momentum proceeds across a given area measures the force exerted on
that area; the pressure of the gas is the rate of change of momentum
per unit of area of the surface.
Again, it can be shown that this pressure is proportional to the
product of the mass of each molecule, the number of molecules in a unit
of volume, and the square of the velocity of the molecules.
Let us consider in the first instance the case of a jet of sand or
water of unit cross section which is playing against a surface. Suppose
for the present that all the molecules which strike the surface have
the same velocity.
Then the number of molecules which strike the surface per second, will
be proportional to this velocity. If the particles are moving quickly
they can reach the surface in one second from a greater distance than
is possible if they be moving slowly. Again, the number reaching the
surface will be proportional to the number of molecules per unit of
volume. Hence, if we call _v_ the velocity of each particle, and N
the number of particles per unit of volume, the number which strike
the surface in one second will be N _v_; if _m_ be the mass of each
molecule, the mass which strikes the surface per second is N _m_ _v_;
the velocity of each particle of this mass is _v_, therefore the
momentum destroyed per second by the impact is N _m_ _v_ × _v_, or N
_m_ _v_², and this measures the pressure.
Hence in this case if _p_ be the pressure
_p_ = N _m_ _v_².
In the above we assume that _all_ the molecules in the jet are moving
with velocity _v_ perpendicular to the surface. In the case of a crowd
of molecules flying about in a closed space this is clearly not true.
The molecules may strike the surface in any direction; they will not
all be moving normal to the surface. To simplify the case, consider a
cubical box filled with gas. The box has three pairs of equal faces at
right angles. We may suppose one-third of the particles to be moving at
right angles to each face, and in this case the number per unit volume
which we have to consider is not N, but ⅓ N. Hence the formula becomes
_p_ = ⅓ N _m_ _v_².
Moreover, if _ρ_ be the density of the gas--that is, the mass of
unit volume--then N_m_ is equal to _ρ_, for _m_ is the mass of each
particle, and there are N particles in a unit of volume.
Hence, finally, _p_ = ⅓ _ρ_ _v_².
Or, again, if V be the volume of unit mass of the gas, then _ρ_ V is
unity, or ρ is equal to 1/V.
Hence _p_V = ⅓_v_².
Formulæ equivalent to these appear first to have been obtained by
Herapath about the year 1816 (Thomson’s “Annals of Philosophy,” 1816).
The results only, however, were stated in that year. A paper which
attempted to establish them was presented to the Royal Society in 1820.
It gave rise to very considerable correspondence, and was withdrawn
by the author before being read. It is printed in full in Thomson’s
“Annals of Philosophy” for 1821, vol. i., pp. 273, 340, 401. The
arguments of the author are no doubt open to criticism, and are in many
points far from sound. Still, by considering the problem of the impact
of a large number of hard bodies, he arrived at a formula connecting
the pressure and volume of a given mass of gas equivalent to that just
given. These results are contained in Propositions viii. and ix. of
Herapath’s paper.
In his next step, however, Herapath, as we know now, was wrong. One
of his fundamental assumptions is that the temperature of a gas is
measured by the momentum of each of its particles. Hence, assuming
this, we have T = _m_ _v_, if T represents the temperature: and
_p_ = ⅓ N _m_ _v_² = ⅓ (N/_m_) (_m_ _v_)².
Or, again--
_p_ = ⅓ N·T·_v_ = ⅓·(N/_m_)·T².
These results are practically given in Proposition viii., Corr. (1)
and (2), and Proposition ix.[49] The temperature as thus defined by
Herapath is an absolute temperature, and he calculates the absolute
zero of temperature at which the gas would have no volume from the
above results. The actual calculation is of course wrong, for, as
we know now by experiment, the pressure is proportional to the
temperature, and not to its square, as Herapath supposed. It will be
seen, however, that Herapath’s formula gives Boyle’s law; for if the
temperature is constant, the formula is equivalent to
_p_ V = a constant.
Herapath somewhat extended his work in his “Mathematical Physics”
published in 1847, and applied his principles to explain diffusion, the
relation between specific heat and atomic weight, and other properties
of bodies. He still, however, retained his erroneous supposition
that temperature is to be measured by the momentum of the individual
particles.
The next step in the theory was made by Waterston. His paper was read
to the Royal Society on March 5th, 1846. It was most unfortunately
committed to the Archives of the Society, and was only disinterred by
Lord Rayleigh in 1892 and printed in the Transactions for that year.
In the account just given of the theory, it has been supposed that all
the particles move with the same velocity. This is clearly not the
case in a gas. If at starting all the particles had the same velocity,
the collisions would change this state of affairs. Some particles will
be moving quickly, some slowly. We may, however, still apply the
theory by splitting up the particles into groups, and, supposing that
each group has a constant velocity, the particles in this group will
contribute to the pressure an amount--_p_₁--equal to ⅓ N₁ _m_ _v_₁²,
where _v_₁ is the velocity of the group and N₁ the number of particles
having that velocity. The whole pressure will be found by adding that
due to the various groups, and will be given as before by _p_ = ⅓ N _m_
_v_², where _v_ is not now the actual velocity of the particles, but a
mean velocity given by the equation
N _v_² = N₁ _v_₁² + N₂ _v_₂² + .....,
which will produce the same pressure as arises from the actual impacts.
This quantity v² is known as the _mean square_ of the molecular
velocity, and is so used by Waterston.
In a paper in the _Philosophical Magazine_ for 1858 Waterston gives an
account of his own paper of 1846 in the following terms:--“Mr. Herapath
unfortunately assumed heat or temperature to be represented by the
simple ratio of the velocity instead of the square of the velocity,
being in this apparently led astray by the definition of motion
generally received, and thus was baffled in his attempts to reconcile
his theory with observation. If we make this change in Mr. Herapath’s
definition of heat or temperature--viz., that it is proportional
to the vis-viva or square velocity of the moving particle, not to
the momentum or simple ratio of the velocity--we can without much
difficulty deduce not only the primary laws of elastic fluids, but also
the other physical properties of gases enumerated above in the third
objection to Newton’s hypothesis. [The paper from which the quotation
is taken is on ‘The Theory of Sound.’] In the Archives of the Royal
Society for 1845–46 there is a paper on ‘The Physics of Media that
consist of perfectly “Elastic Molecules in a State of Motion,”’ which
contains the synthetical reasoning on which the demonstration of these
matters rests.... This theory does not take account of the size of the
molecules. It assumes that no time is lost at the impact, and that if
the impacts produce rotatory motion, the vis viva thus invested bears
a constant ratio to the rectilineal vis viva, so as not to require
separate consideration. It does, also, not take account of the probable
internal motion of composite molecules; yet the results so closely
accord with observation in every part of the subject as to leave no
doubt that Mr. Herapath’s idea of the physical constitution of gases
approximates closely to the truth.”
In his introduction to Waterston’s paper (Phil. Trans., 1892) Lord
Rayleigh writes:--“Impressed with the above passage, and with the
general ingenuity and soundness of Waterston’s views, I took the first
opportunity of consulting the Archives, and saw at once that the memoir
justified the large claims made for it, and that it marks an immense
advance in the direction of the now generally received theory.”
In the first section of the paper Waterston’s great advance consisted
in the statement that the mean square of the kinetic energy of each
molecule measures the temperature.
According to this we are thus to put in the pressure equation--½ _m_
_v_² = T, the temperature, and we have at once--_p_ V = ⅔ N · T.
Now this equation expresses, as we know, the laws of Boyle and Gay
Lussac.
The second section discusses the properties of media, consisting of
two or more gases, and arrives at the result that “in mixed media
the mean square molecular velocity is inversely proportional to the
specific weights of the molecules.” This was the great law rediscovered
by Maxwell fifteen years later. With modern notation it may be put
thus:--If _m_₁, _m_₂ be the masses of each molecule of two different
sets of molecules mixed together, then, when a steady state has been
reached, since the temperature is the same throughout, _m_₁ _v_₁² is
equal to _m_₂ _v_₂². The average kinetic energy of each molecule is the
same.
From this Avogadros’ law follows at once--for if _p_₁, _p_₂ be the
pressures, N₁, N₂ the numbers of molecules per unit volume--
_p_₁ = ⅓ N₁ _m_₁ _v_₁²,
_p_₂ = ⅓ N₂ _m_₂ _v_₂².
Hence, if _p_₁, is equal to _p_₂, since _m_₁ _v_₁² is equal to _m_₂
_v_₂², we must have N₁ equal to N₂, or the number of molecules in equal
volumes of two gases at the same pressure and temperature is the same.
The proof of this proposition given by Waterston is not satisfactory.
On this point, however, we shall have more to say. The third section of
the paper deals with adiabatic expansion, and in it there is an error
in calculation which prevented correct results from being attained.
At the meeting of the British Association at Ipswich, in 1851, a paper
by J. J. Waterston of Bombay, on “The General Theory of Gases,” was
read. The following is an extract from the Proceedings:--
The author “conceives that the atoms of a gas, being perfectly elastic,
are in continual motion in all directions, being constrained within
a limited space by their collisions with each other, and with the
particles of surrounding bodies.
“The vis viva of these motions in a given portion of a gas constitutes
the quantity of heat contained in it.
“He shows that the result of this state of motion must be to give the
gas an elasticity proportional to the mean square of the velocity of
the molecular motions, and to the total mass of the atoms contained in
unity of bulk” (unit of volume)--that is to say, to the density of the
medium.
“The elasticity in a given gas is the measure of temperature.
Equilibrium of pressure and heat between two gases takes place when the
number of atoms in unit of volume is equal and the vis viva of each
atom equal. Temperature, therefore, in all gases is proportional to the
mass of one atom multiplied by the mean square of the velocity of the
molecular motions, being measured from an absolute zero 491° below the
zero of Fahrenheit’s thermometer.”
It appears, therefore, from these extracts that the discovery of the
laws that temperature is measured by the mean kinetic energy of a
single molecule, and that in a mixture of gases the mean kinetic energy
of each molecule is the same for each gas, is due to Waterston. They
were contained in his paper of 1846, and published by him in 1851. Both
these papers, however, appear to have been unnoticed by all subsequent
writers until 1892.
Meanwhile, in 1848, Joule’s attention was called by his experiments
to the question, and he saw that Herapath’s result gave a means of
calculating the mean velocity of the molecules of a gas. For according
to the result given above, _p_ = ⅓ _ρ v_²; thus _v_² = 3 _p/ρ_, and _p_
and _ρ_ being known, we find _v_². Thus for hydrogen at freezing-point
and atmospheric pressure Joule obtains for _v_ the value 6,055 feet per
second, or, roughly, six times the velocity of sound in air.
Clausius was the next writer of importance on the subject. His first
paper is in “Poggendorff’s Annalen,” vol. c., 1857, “On the Kind
of Motion we call Heat.” It gives an exposition of the theory, and
establishes the fact that the kinetic energy of the translatory motion
of a molecule does not represent the whole of the heat it contains. If
we look upon a molecule as a small solid we must consider the energy it
possesses in consequence of its rotation about its centre of gravity,
as well as the energy due to the motion of translation of the whole.
Clausius’ second paper appeared in 1859. In it he considers the average
length of the path of a molecule during the interval between two
collisions. He determines this path in terms of the average distance
between the molecules and the distance between the centres of two
molecules at the time when a collision is taking place.
These two papers appear to have attracted Maxwell’s attention to the
matter, and his first paper, entitled “Illustrations of the Dynamical
Theory of Gases,” was read to the British Association at Aberdeen and
Oxford in 1859 and 1860, and appeared in the _Philosophical Magazine_,
January and July, 1860.
In the introduction to this paper Maxwell points out, while there was
then no means of measuring the quantities which occurred in Clausius’
expression for the mean free path, “the phenomena of the internal
friction of gases, the conduction of heat through a gas, and the
diffusion of one gas through another, seem to indicate the possibility
of determining accurately the mean length of path which a particle
describes between two collisions. In order, therefore, to lay the
foundation of such investigations on strict mechanical principles,” he
continues, “I shall demonstrate the laws of motion of an indefinite
number of small, hard and perfectly elastic spheres acting on one
another only during impact.”
Maxwell then proceeds to consider in the first case the impact of two
spheres.
But a gas consists of an indefinite number of molecules. Now it is
impossible to deal with each molecule individually, to trace its
history and follow its path. In order, therefore, to avoid this
difficulty Maxwell introduced the statistical method of dealing with
such problems, and this introduction is the first great step in
molecular theory with which his name is connected.
He was led to this method by his investigation into the theory of
Saturn’s rings, which had been completed in 1856, and in which he
had shown that the conditions of stability required the supposition
that the rings are composed of an indefinite number of free particles
revolving round the planet, with velocities depending on their
distances from the centre. These particles may either be arranged in
separate rings, or their motion may be such that they are continually
coming into collision with each other.
As an example of the statistical method, let us consider a crowd
of people moving along a street. Taken as a whole the crowd moves
steadily forwards. Any individual in the crowd, however, is jostled
backwards and forwards and from side to side; if a line were drawn
across the street we should find people crossing it in both directions.
In a considerable interval more people would cross it, going in the
direction in which the crowd is moving, than in the other, and the
velocity of the crowd might be estimated by counting the number which
crossed the line in a given interval. This velocity so found would
differ greatly from the velocity of any individual, which might have
any value within limits, and which is continually changing. If we knew
the velocity of each individual and the number of individuals we could
calculate the average velocity, and this would agree with the value
found by counting the resultant number of people who cross the line in
a given interval.
Again, the people in the crowd will naturally fall into groups
according to their velocities. At any moment there will be a certain
number of people whose velocities are all practically equal, or, to be
more accurate, do not differ among themselves by more than some small
quantity. The number of people at any moment in each of these groups
will be very different. The number in any group, which has a velocity
not differing greatly from the mean velocity of the whole, will be
large; comparatively few will have either a very large or a very small
velocity.
Again, at any moment, individuals are changing from one group to
another; a man is brought to a stop by some obstruction, and his
velocity is considerably altered--he passes from one group to a
different one; but while this is so, if the mean velocity remains
constant, and the size of the crowd be very great, the number of people
at any moment in a given group remains unchanged. People pass from that
group into others, but during any interval the same number pass back
again into that group.
It is clear that if this condition is satisfied the distribution is
a steady one, and the crowd will continue to move on with the same
uniform mean velocity.
Now, Maxwell applies these considerations to a crowd of perfectly
elastic spheres, moving anyhow in a closed space, acting upon each
other only when in contact. He shows that they may be divided into
groups according to their velocities, and that, when the steady state
is reached, the number in each group will remain the same, although the
individuals change. Moreover, it is shown that, if A and B represent
any two groups, the state will only be steady when the numbers which
pass from the group A to the group B are equal to the numbers which
pass back from the group B to the group A. This condition, combined
with the fact that the total kinetic energy of the motion remains
unchanged, enables him to calculate the number of particles in any
group in terms of the whole number of particles, the mean velocity, and
the actual velocity of the group.
From this an accurate expression can be found for the pressure of the
gas, and it is proved that the value found by others, on the assumption
that all the particles were moving with a common velocity, is correct.
Previous to this paper of Maxwell’s it had been realised that the
velocities could not be uniform throughout. There had been no attempt
to determine the distribution of velocity, or to submit the problem to
calculation, making allowance for the variations in velocity.
Maxwell’s mathematical methods are, in their generality and elegance,
far in advance of anything previously attempted in the subject.
So far it has been assumed that the particles in the vessel are all
alike. Maxwell next takes the case of a mixture of two kinds of
particles, and inquires what relation must exist between the average
velocities of these different particles, in order that the state may be
steady.
Now, it can be shown that when two elastic spheres impinge the effect
of the impact is always such as to reduce the difference between their
kinetic energies.
Hence, after a very large number of impacts the kinetic energies of the
two balls must be the same; the steady state, then, will be reached
when each ball has the same kinetic energy.
Thus if _m_₁, _m_₂ be the masses of the particles in the two sets
respectively, _v_₁, _v_₂ their mean velocities we must have finally--
½ _m_₁ _v_₁² = ½ _m_₂ _v_₂²
This is the second of the two great laws enunciated by Waterston in
1845 and 1851, but which, as we have seen, had remained unknown until
1859, when it was again given by Maxwell.
Now, when gases are mixed their temperatures become equal. Hence we
conclude, in Maxwell’s words, “that the physical condition which
determines that the temperature of two gases shall be the same, is that
the mean kinetic energy of agitation of the individual molecules of the
two gases are equal.”
Thus, as the result of Maxwell’s more exact researches on the motion of
a system of spherical particles, we find that we again can obtain the
equations--
T = ½ _mv_²
_p_ = ⅓ N _mv_² = ⅔ NT = ⅔ _ρ_ T/_m_
From these results we obtain as before the laws of Boyle, Charles and
Avrogadro.
Again if _σ_ be the specific heat of the gas at constant volume, the
quantity of heat required to raise a single molecule of mass _m_ one
degree will be _σ_ _m_.
Thus, when a molecule is heated, the kinetic energy must increase by
this amount. But the increase of temperature, which in this case is 1°,
is measured by the increase of kinetic energy of the single molecule.
Hence the amount of heat required to raise the temperature of a single
molecule of all gases 1° is the same. Thus the quantity _σ_ _m_ is the
same for all gases; or, in other words, the specific heat of a gas is
inversely proportional to the mass of its individual molecules. The
density of a gas--since the number of molecules per unit volume at
a given pressure and temperature is the same for all gases--is also
proportional to the mass of each individual molecule. Thus the specific
heats of all gases are inversely proportional to their densities.
This is the law discovered experimentally by Dulong and Petit to be
approximately true for a large number of substances.
* * * * *
In the next part of the paper Maxwell proceeded to determine the
average number of collisions in a given time, and hence, knowing the
velocities, to determine, in terms of the size of the particles and
their numbers, the mean free path of a particle; the result so found
differed somewhat from that already obtained by Clausius.
Having done this he showed how, by means of experiments on the
viscosity of gases, the length of the mean free path could be
determined.
An illustration due to Professor Balfour Stewart will perhaps make this
clear. Let us suppose we have two trains running with uniform speed in
opposite directions on parallel lines, and, further, that the engines
continue to work at the same rate, developing just sufficient energy to
overcome the resistance of the line, etc., and to maintain the speed
constant. Now suppose passengers commence to jump across from one train
to the other. Each man carries with him his own momentum, which is in
the opposite direction to that of the train into which he jumps; the
result is that the momentum of each train is reduced by the process;
the velocities of the two decrease; it appears as though a frictional
force were acting between the two. Maxwell suggests that a similar
process will account for the apparent viscosity of gases.
Consider two streams of gas, moving in opposite directions one over
the other; it is found that in each case the layers of gas near the
separating surface move more slowly than those in the interior of
the streams; there is apparently a frictional force between the two
streams along this surface, tending to reduce their relative velocity.
Maxwell’s explanation of this is that at the common surface particles
from the one stream enter the other, and carry with them their own
momentum; thus near this surface the momentum of each stream is
reduced, just as the momentum of the trains is reduced by the people
jumping across. Internal friction or viscosity is due to the diffusion
of momentum across this common surface. The effect does not penetrate
far into the gas, for the particles soon acquire the velocity of the
stream to which they have come.
Now, the rate at which the momentum is diffused will measure the
frictional force, and will depend on the mean free path of the
particles. If this is considerable, so that on the average a particle
can penetrate a considerable distance into the second gas before a
collision takes place and its motion is changed, the viscosity will be
considerable; if, on the other hand, the mean free path is small, the
reverse will be true. Thus it is possible to obtain a relation between
the mean free path and the coefficient of viscosity, and from this, if
the coefficient of viscosity be known, a value for the mean free path
can be found.
Maxwell, in the paper under discussion, was the first to do this,
and, using a value found by Professor Stokes for the coefficient of
viscosity, obtained as the length of the mean free path of molecules
of air 1/447000 of an inch, while the number of collisions per second
experienced by each molecule is found to be about 8,077,200,000.
Moreover, it appeared from his theory that the coefficient of viscosity
should be independent of the number of molecules of gas present, so
that it is not altered by varying the density. This result Maxwell
characterises as startling, and he instituted an elaborate series of
experiments a few years later with a view of testing it. The reason
for this result will appear if we remember that, when the density is
decreased, the mean free path is increased; relatively, then, to the
total number of molecules present, the number which cross the surface
in a given time is increased. And it appears from Maxwell’s result that
this relative increase is such that the total number crossing remains
unchanged. Hence the momentum conveyed across each unit area per second
remains the same, in spite of the decrease in density.
Another consequence of the same investigation is that the coefficient
of viscosity is proportional to the mean velocity of the molecules.
Since the absolute temperature is proportional to the square of the
velocity, it follows that the coefficient of viscosity is proportional
to the square root of the absolute temperature.
The second part of the paper deals with the process of diffusion of two
or more kinds of moving particles among one another.
If two different gases are placed in two vessels separated by a porous
diaphragm such as a piece of unglazed earthenware, or connected by
means of a narrow tube, Graham had shewn that, after sufficient time
has elapsed, the two are mixed together. The same process takes place
when two gases of different density are placed together in the same
vessel. At first the denser gas may be at the bottom, the less dense
above, but after a time the two are found to be uniformly distributed
throughout.
Maxwell attempted to calculate from his theory the rate at which
the diffusion takes place in these cases. The conditions of most of
Graham’s experiments were too complicated to admit of direct comparison
with the theory, from which it appeared that there is a relation
between the mean free path and the rate of diffusion. One experiment,
however, was found, the conditions of which could be made the subject
of calculation, and from it Maxwell obtained as the value of the mean
free path in air 1/389000 of an inch.
The number was close enough to that found from the viscosity to afford
some confirmation of his theory.
However, a few years later Clausius criticised the details of this
part of the paper, and Maxwell, in his memoir of 1866, admits the
calculation to have been erroneous. The main principles remained
unaffected, the molecules pass from one gas to the other, and this
constitutes diffusion.
Now, suppose we have two sets of particles in contact of such a nature
that the mean kinetic energy of the one set is different from that of
the other; the temperatures of the two will then be different. These
two sets will diffuse into each other, and the diffusing particles will
carry with them their kinetic energy, which will gradually pass from
those which have the greater energy to those which have the less, until
the average kinetic energy is equalised throughout. But the kinetic
energy of translation is the heat of the particles. This diffusion of
kinetic energy is a diffusion of heat by conduction, and we have here
the mechanical theory of the conduction of heat in a gas.
Maxwell obtained an expression, which, however, he afterwards modified,
for the conductivity of a gas in terms of the mean free path. It
followed from this that the conductivity of air was only about 1/7000
of that of copper.
Thus the diffusion of gases, the viscosity of gases, and the conduction
of heat in gases, are all connected with the diffusion of the particles
carrying with them their momenta and their energy; while values of the
mean free path can be obtained from observations on any one of these
properties.
In the third part of his paper Maxwell considers the consequences
of supposing the particles not to be spherical. In this case the
impacts would tend to set up a motion of rotation in the particles.
The direction of the force acting on any particle at impact would not
necessarily pass through its centre; thus by impact the velocity of its
centre would be changed, and in addition the particles would be made to
spin. Some part, therefore, of the energy of the particles will appear
in the form of the translational energy of their centres, while the
rest will take the form of rotational energy of each particle about its
centre.
It follows from Maxwell’s work that for each particle the average value
of these two portions of energy would be equal. The total energy will
be half translational and half rotational.
This theorem, in a more general form which was afterwards given to
it, has led to much discussion, and will be again considered later.
For the present we will assume it to be true. Clausius had already
called attention to the fact that some of the energy must be rotational
unless the molecules be smooth spheres, and had given some reasons
for supposing that the ratio of the whole energy to the energy of
translation is in a steady state a constant. Maxwell shows that for
rigid bodies this constant is 2. Let us denote it for the present by
the symbol β. Thus, if the translational energy of a molecule is ½ _m_
_v_², its whole energy is ½ β _m_ _v_².
The temperature is still measured by the translational energy, or ½ _m_
_v_²; the heat depends on the whole energy. Hence if H represent the
amount of heat--measured as energy--contained by a single molecule,
and T its temperature, we have--
H = βT
From this it can be shewn[50] that if γ represent the ratio of the
specific heat of a gas at constant pressure to the specific heat at
constant volume, then--
β = ⅔ 1/(γ-1)
For air and some other gases the value of γ has been shown to be 1·408.
From this it follows that β = 1·634. Now, Maxwell’s theory required
that for smooth hard particles, approximately spherical in shape, β
should be 2, and hence he concludes “we have shown that a system of
such particles could not possibly satisfy the known relation between
the two specific heats of all gases.”
Since this statement was made many more experiments on the value of γ
have been undertaken; it is not equal to 1·408 for _all_ gases. Hence
the value of β is different for various gases.
It is of some importance to notice that the value of β just found for
air is very approximately 1·66 or 5/3.
For mercury vapour the value of γ has been shown by Kundt to be 1·33
or 1⅓, and hence β is equal to 1. Thus all the energy of a particle of
mercury vapour is translational, and its behaviour in this respect is
consistent with the assumption that a particle of mercury vapour is a
smooth sphere.
The two results of this theory which seemed to lend themselves most
readily to experimental verification were (1) that the viscosity of
a gas is independent of its density, and (2) that it is proportional
to the square root of the absolute temperature. The next piece
of work connected with the theory was an attempt to test these
consequences, and a description of the experiments was published in the
“Philosophical Transactions” for 1865, in a paper on the “Viscosity
or Internal Friction of Air and other Gases,” and forms the Bakerian
lecture for that year.
The first result was completely proved. It is shewn that the value of
the coefficient[51] of viscosity “is the same for air at 0·5 inch and
at 30 inches pressure, provided that the temperature remains the same.”
It was clear also that the viscosity depended on the temperature,
and the results of the experiments seemed to show that it was nearly
proportional to the absolute temperature. Thus for two temperatures,
185° Fah. and 51° Fah., the ratio of the two coefficients found was
1·2624; the ratio of the two temperatures, each measured from absolute
zero, is 1·2605.
This result, then, does not agree with the hypothesis that a gas
consists of spherical molecules acting only on each other by a kind of
impact, for, if this were so, the coefficient would, as we have seen,
depend on the square root of the absolute temperature. But Maxwell’s
result, connecting viscosity with the first power of the absolute
temperature, has not been confirmed by other investigators. According
to it we should have as the relation between μ, the coefficient of
viscosity at t° and μ₀, that at zero the equation--
μ = μ₀ (1 + .00365 t).
The most recent results of Professor Holman (_Philosophical Magazine_,
Vol. xxi., p. 212) give--
μ = μ₀ (1 + .00275 t - .00000034 t²).
And results similar to this are given by O. E. Meyer, Puluj, and
Obermeyer. Maxwell’s coefficient ·00365 is too large, but ·00182, the
coefficient obtained by supposing the viscosity proportional to the
square root of the temperature, would be too small.
It still remains true, therefore, that the laws of the viscosity of
gases cannot be explained by the hypothesis of the impact of hard
spheres; but some deductions drawn by Maxwell in his next paper from
his supposed law of proportionality to the first power of the absolute
temperature require modification.
It was clear from his experiments just described that the simple
hypothesis of the impact of elastic bodies would not account for all
the phenomena observed. Accordingly, in 1866, Maxwell took up the
problem in a more general form in his paper on the “Dynamical Theory of
Gases,” Phil. Trans., 1866.
In it he considered the molecules of the gas not as elastic spheres
of definite radius, but as small bodies, or groups of smaller
molecules, repelling one another with a force whose direction always
passes very nearly through the centre of gravity of the molecules,
and whose magnitude is represented very nearly by some function of
the distance of the centres of gravity. “I have made,” he continues,
“this modification of the theory in consequence of the results of my
experiments on the viscosity of air at different temperatures, and I
have deduced from these experiments that the repulsion is inversely as
the fifth power of the distance.”
Since more recent observation has shown that the numerical results of
Maxwell’s work connecting viscosity and temperature are erroneous, this
last deduction does not hold; the inverse fifth power law of force
will not give the correct relation between viscosity and temperature.
Maxwell himself at a later date, “On the Stresses in Rarefied Gases,”
Phil. Trans., 1879, realised this; but even in this last paper he
adhered to the fifth power law because it leads to an important
simplification in the equations to be dealt with.
The paper of 1866 is chiefly important because it contains for the
first time the application of general dynamical methods to molecular
problems. The law of the distribution of velocities among the molecules
is again investigated, and a result practically identical with that
found for the elastic spheres is arrived at. In obtaining this
conclusion, however, it is assumed that the distribution of velocities
is uniform in all directions about any point, whatever actions may be
taking place in the gas. If, for example, the temperature is different
at different points, then, for a given velocity, all directions are not
equally probable. Maxwell’s expression, therefore, for the number of
molecules which at any moment have a given velocity only applies to the
permanent state in which the distribution of temperature is uniform.
When dealing, for example, with the conduction of heat, a modification
of the expression is necessary. This was pointed out by Boltzmann.[52]
In the paper of 1866, Maxwell applies his generalised results to the
final distribution of two gases under the action of gravity, the
equilibrium of temperature between two gases, and the distribution of
temperature in a vertical column. These results are, as he states,
independent of the law of force between the molecules. The dynamical
causes of diffusion viscosity and conduction of heat are dealt with,
and these involve the law of force.
It follows also from the investigation that, on the hypotheses assumed
as its basis, if two kinds of gases be mixed, the difference between
the average kinetic energies of translation of the gases of each kind
diminishes rapidly in consequence of the action between the two. The
average kinetic energy of translation, therefore, tends to become the
same for each kind of gas, and as before, it is this average energy of
translation which measures the temperature.
A molecule in the theory is a portion of a gas which moves about as a
single body. It may be a mere point, a centre of force having inertia,
capable of doing work while losing velocity. There may be also in each
molecule systems of several such centres of force bound together by
their mutual actions. Again, a molecule may be a small solid body of
determinate form; but in this case we must, as Maxwell points out,
introduce a new set of forces binding together the parts of each
molecule: we must have a molecular theory of the second order. In any
case, the most general supposition made is that a molecule consists of
a series of parts which stick together, but are capable of relative
motion among each other.
In this case the kinetic energy of the molecule consists of the energy
of its centre of gravity, together with the energy of its component
parts, relative to its centre of gravity.[53]
Now Clausius had, as we have seen, given reasons for believing that the
ratio of the whole energy of a molecule to the energy of translation of
its centre of gravity tends to become constant. We have already used β
to denote this constant. Thus, while the temperature is measured by the
average kinetic energy of translation of the centre of gravity of each
molecule, the heat contained in a molecule is its whole energy, and is
β times this quantity. Thus the conclusions as to specific heat, etc.,
already given on page 130, apply in this case, and in particular we
have the result that if γ be the ratio of the specific heat at constant
pressure to that at constant volume, then--
β = ⅔ 1/(γ-1)
Maxwell’s theorem of the distribution of kinetic energy among a system
of molecules applied, as he gave it in 1866, to the kinetic energy of
translation of the centre of gravity of each molecule. Two years later
Dr. Boltzmann, in the paper we have already referred to, extended
it (under certain limitations) to the parts of which a molecule is
composed. According to Maxwell the average kinetic energy of the centre
of gravity of each molecule tends to become the same. According to
Boltzmann the average kinetic energy of each part of the molecule tends
to become the same.
Maxwell, in the last paper he wrote on the subject (“On Boltzmann’s
Theorem on the Average Distribution of Energy in a System of
Material Points,” Camb. Phil. Trans., XII.), took up this problem.
Watson had given a proof of it in 1876 differing from Boltzmann’s,
but still limited by the stipulation that the time, during which a
particle is encountering other particles, is very small compared with
the time during which there is no sensible action between it and
other particles, and also that the time during which a particle is
simultaneously within the distance of more than one other particle may
be neglected.
Maxwell claims that his proof is free from any such limitation. The
material points may act on each other at all distances, and according
to any law which is consistent with the conservation of energy; they
may also be acted on by forces external to the system, provided these
are consistent with that law.
The only assumption which is necessary for the direct proof is that
the system, if left to itself in its actual state of motion, will
sooner or later pass through every phase which is consistent with the
conservation of energy.
In this paper Maxwell finds in a very general manner an expression for
the number of molecules which at any time have a given velocity, and
this, when simplified by the assumptions of the former papers, reduces
to the form already found. He also shows that the average kinetic
energy corresponding to any one of the variables which define his
system is the same for every one of the variables of his system.
Thus, according to this theorem, if each molecule be a single small
solid body, six variables will be required to determine the position
of each, three variables will give us the position of the centre of
gravity of the molecule, while three others will determine the position
of the body relative to its centre of gravity. If the six variables
be properly chosen, the kinetic energy can be expressed as a sum of
six squares, one square corresponding to each variable. According to
the theorem the part of the kinetic energy depending on each square is
the same. Thus, the whole energy is six times as great as that which
arises from any one of the variables. The kinetic energy of translation
is three times as great as that arising from each variable, for it
involves the three variables which determine the position of the centre
of gravity. Hence, if we denote by K the kinetic energy due to one
variable, the whole energy is 6 K, and the translational energy is 3 K;
thus, for this case--
β = 6K/3K = 2
Or, again, if we suppose that the molecule is such that _m_ variables
are required to determine its position relatively to its centre of
gravity, since 3 are needed to fix the centre of gravity, the total
number of variables defining the position of the molecule is _m_ + 3,
and it is said to have _m_ + 3 degrees of freedom. Hence, in this case,
its total energy is (_m_ + 3) K and its energy of translation is 3 K,
thus we find--
β = (_m_ + 3)/3
Hence γ = 1 + 2/(_m_ + 3) = 1 + 2/_n_
if _n_ be the number of degrees of freedom of the molecule.
Thus, if this Boltzmann-Maxwell theorem be true, the specific heat of a
gas will depend solely on the number of degrees of freedom of each of
its molecules. For hard rigid bodies we should have _n_ equal to 6, and
hence γ = 1·333. Now the fact that this is not the value of γ for any
of the known gases is a fundamental difficulty in the way of accepting
the complete theory.
Boltzmann has called attention to the fact that if _n_ be equal to
five, then γ has the value 1·40. And this agrees fairly with the value
found by experiment for air, oxygen, nitrogen, and various other gases.
We will, however, return to this point shortly.
There is, perhaps, no result in the domain of physical science in
recent years which has been more discussed than the two fundamental
theorems of the molecular theory which we owe to Maxwell and to
Boltzmann.
The two results in question are (1) the expression for the number of
molecules which at any moment will have a given velocity, and (2) the
proposition that the kinetic energy is ultimately equally divided
among all the variables which determine the system.
With regard to (1) Maxwell showed that his error law was one possible
condition of permanence. If at any moment the velocities are
distributed according to the error law, that distribution will be a
permanent one. He did not prove that such a distribution is the only
one which can satisfy all the conditions of the problem.
The proof that this law is a necessary, as well as a sufficient,
condition of permanence was first given by Boltzmann, for a single
monatomic gas in 1872, for a mixture of such gases in 1886, and for a
polyatomic gas in 1887. Other proofs have been given since by Watson
and Burbury. It would be quite beyond the limits of this book to go
into the question of the completeness or sufficiency of the proofs. The
discussion of the question is still in progress.
The British Association Report for 1894 contains an important
contribution to the question, in the shape of a report by Mr. G. H.
Bryan, and the discussion he started at Oxford by reading this report
has been continued in the pages of _Nature_ and elsewhere since that
time.
Mr. Bryan shows in the first place what may be the nature of the
systems of molecules to which the results will apply, and discusses
various points of difficulty in the proof.
The theorem in question, from which the result (1) follows as a simple
deduction, has been thus stated by Dr. Larmor.[54]
“There exists a positive function belonging to a group of molecules
which, as they settle themselves into a steady state--on the average
derived from a great number of configurations--maintains a steady
downward trend. The Maxwell-Boltzmann steady state is the one in which
this function has finally attained its minimum value, and is thus a
unique steady state, it still being borne in mind that this is only a
proposition of averages derived from a great number of instances in
which nothing is conserved in encounters, except the energy, and that
exceptional circumstances may exist, comparatively very few in number,
in which the trend is, at any rate, temporarily the other way.”
This theorem, when applied to cases of motion, such as that of a gas at
constant temperature enclosed in a rigid envelope impermeable to heat,
appears to be proved. For such a case, therefore, the Maxwell-Boltzmann
law is the only one possible.
But whether this be so or not, the law first introduced by Maxwell is
one of those possible, and the advance in molecular science due to its
introduction is enormous.
We come now to the second result, the equal partition of the energy
among all the degrees of freedom of each molecule. Lord Kelvin
has pointed out a flaw in Maxwell’s proof, but Boltzmann showed
(_Philosophical Magazine_, March, 1893) how this flaw can easily
be corrected, and it may be said that in all cases in which the
Boltzmann-Maxwell law of the distribution of velocities holds,
Maxwell’s law of the equal partition of energy holds also.
Three cases are considered by Mr. Bryan, in which the law of
distribution fails for rigid molecules: the first is when the molecules
have all, in addition to their velocities of agitation, a common
velocity of translation in a fixed direction; the second is when the
gas has a motion of uniform rotation about a fixed axis; while the
third is when each molecule has an axis of symmetry. In this last case
the forces acting during a collision necessarily pass through the
axis of symmetry, the angular velocity, therefore, of any molecule
about this axis remains constant, the number of molecules having a
given angular velocity will remain the same throughout the motion,
and the part of the kinetic energy which depends on this component of
the motion will remain fixed, and will not come into consideration
when dealing with the equal partition of the energy among the various
degrees of freedom.
Such a molecule has five, and not six, degrees of freedom; three
quantities are needed to determine the position of its centre of
gravity, and two to fix the position of the axis of symmetry.
In this case, then, as Boltzmann points out, in the expression for the
ratio of the specific heats, we must have _n_ equal to 5, and hence
γ = 1 + 2/_n_ = 1 + 2/5 = 1·4
agreeing fairly with the value found for air and various other
permanent gases.
For cases, then, in which we consider each atom as a single rigid body,
the Boltzmann-Maxwell theorem appears to give a unique solution,
and the Maxwell law of the distribution of the energy to be in fair
accordance with the results of observation.[55]
If we can never go further--and it must be admitted that the
difficulties in the way of further advance are enormous--it may,
I think, be claimed for Maxwell that the progress already made is
greatly due to him. Both these laws, for the case of elastic spheres,
are contained in his first paper of 1860; and while it is to the
genius of Boltzmann that we owe their earliest generalisation, and in
particular the proof of the uniqueness of the solution under proper
restrictions, Maxwell’s last paper contributed in no small degree to
the security of the position. Not merely the foundations, but much of
the superstructure of molecular science is his work.
The difficulties in the way of advance are, as we have said, enormous.
Boltzmann, in one of his papers, has considered the properties of a
complex molecule of a gas, consisting maybe of a number of atoms and
possibly of ether atoms bound with them, and he concludes that such a
molecule will behave in its progressive motion, and in its collisions
with other molecules, nearly like a rigid body. But to quote from Mr.
Bryan: “The case of a polyatomic molecule, whose atoms are capable of
vibrating relative to one another, affords an interesting field for
investigation and speculation. Is the Boltzmann distribution still
unique, or do other permanent distributions exist in which the kinetic
energy is unequally divided?”
Again, the spectroscope reveals to us vibrations of the ether, which
are connected in some way with the vibrations of the molecules of
gas, whose spectrum we are observing. It seems clear that the law of
equal partition does not apply to these, and yet, if we are to suppose
that the ether vibrations are due to actual vibrations of the atoms
which constitute a molecule, why does it not apply? Where does the
condition come in which leads to failure in the proof? Or, again,
is it, as has been suggested, the fact that the complex spectrum
of a gas represents the terms of a Fourier Series, into which some
elaborate vibration of the atoms is resolved by the ether? or is the
spectrum due simply to electro-magnetic vibrations on the surface of
the molecules--vibrations whose period is determined chiefly by the
size and shape of the molecule, but in which the atoms of which it is
composed take part? There are grave difficulties in the way of either
of these explanations, but we must not let our dread of the task which
remains to be done blind our eyes to the greatness of Maxwell’s work.
One other important paper, and a number of shorter articles, remain to
be mentioned.
The Boltzmann-Maxwell law applies only to cases in which the
temperature is uniform throughout. In a paper published in the
Philosophical Transactions for 1879, on “Stresses in Rarefied Gases
Arising from Inequalities of Temperature,” Maxwell deals, among other
matters, with the theory of the radiometer. He shows that the observed
motions will not take place unless gas, in contact with a solid, can
slide along the surface of the solid with a finite velocity between
places where the temperature is different; and in an appendix he proves
that, on certain assumptions regarding the nature of the contact of the
solid and the gas, there will be, even when the pressure is constant, a
flow of gas along the surface from the colder to the hotter parts.
Among his less important papers bearing on molecular theory must be
mentioned a lecture on “Molecules” to the British Association at its
Bradford meeting; “Scientific Papers of Clerk Maxwell,” vol. ii., p.
361; and another on “The Molecular Constitution of Bodies,” Scientific
Papers, vol. ii., p. 418.
In this latter, and also in a review in _Nature_ of Van der Waals’
book on “The Continuity of the Gaseous and Liquid States,”[56] he
explains and discusses Clausius’ virial equation, by means of which the
variations of the permanent gases from Boyle’s law are explained. The
lecture gives a clear account, in Maxwell’s own inimitable style, of
the advances made in the kinetic theory up to the date at which it was
delivered, and puts clearly the difficulties it has to meet. Maxwell
thought that those arising from the known values of the ratio of the
specific heats were the most serious.
In the articles, “Atomic Constitution of Bodies” and “Diffusion,” in
the ninth edition of the _Encyclopædia Britannica_, we have Maxwell’s
later views on the fundamental assumptions of the molecular theory.
The text-book on “Heat” contains some further developments of the
theory. In particular he shows how the conclusions of the second law
of thermo-dynamics are connected with the fact that the coarseness of
our faculties will not allow us to grapple with individual molecules.
The work described in the foregoing chapters would have been sufficient
to secure to Maxwell a distinguished place among those who have
advanced our knowledge; it remains still to describe his greatest work,
his theory of Electricity and Magnetism.