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molecular science as it now exists. In the earliest times / pied by matter, for extension cannot be an extension of the most ancient philosophers whose speculations are nothing. known to us seem to have discussed the ideas of number

Ac proinde si quæratur quid fiet, si Deus auferat omne corpus and of continuous magnitude, of space and time, of matter quod in aliquo vase continetur, et nullum aliud in ablati locumi and motion, with a native power of thought which has venire permittat ? respondendum est, vasis latera sibi invicem hoc probably never been surpassed. Their actual knowledge, ipso fore contigua. Cum enim inter duo corpora nihil interjacet, however, and their scientific experience were necessarily sive ut inter ipsa sit distantia, et tamen ut ista distantia sit nihil ; limited, because in their days the records of human thought quia omnis distantia est modus extensionis, et ideo sine substantia were only beginning to accumulate. It is probable that extensa esse non potest.”—Principia, ii. 18. the first exact notions of quantity were founded on the This identification of extension with substance runs consideration of number. It is by the help of numbers through the whole of Descartes's works, and it forms one of that concrete quantities are practically measured and the ultimate foundations of the system of Spinoza. Descalculated. Now, number is discontinuous. We pass cartes, consistently with this doctrine, denied the existence from one number to the next per saltum. The magnitudes, of atoms as parts of matter, which by their own nature are on the other hand, which we meet with in geometry, are indivisible. He seems to admit, however, that the Deity essentially continuous. The attempt to apply numerical might make certain particles of matter indivisible in this methods to the comparison of geometrical quantities led to sense, that no creature should be able to divide them. the doctrine of incommensurables, and to that of the infinite These particles, however, would be still divisible by their divisibility of space. Meanwhile, the same considerations own nature, because the Deity cannot diminish his own had not been applied to time, so that in the days of Zeno power, and therefore must retain his power of dividing of Elea time was still regarded as made up of a finite them. Leibnitz, on the other hand, regarded his monad number of “moments," while space was confessed to be as the ultimate element of everything. divisible without limit. This was the state of opinion There are thus two modes of thinking about the constiwhen the celebrated arguments against the possibility of tution of bodies, which have had their adherents both in motion, of which that of Achilles and the tortoise is a ancient and in modern times. They correspond to the specimen, were propounded by Zeno, and such, apparently, two methods of regarding quantity-the arithmetical and continued to be the state of opinion till Aristotle pointed the geometrical. To the atomist the true method of out that time is divisible without limit, in precisely the estimating the quantity of matter in a body is to count the same sense that space is. And the slowness of the develop- atoms in it. The void spaces between the atoms count ment of scientific ideas may be estimated from the fact that for nothing. To those who identify matter with extension, Bayle does not see any force in this statement of Aristotle, the volume of space occupied by a body is the only measure but continues to admire the paradox of Zeno. (Bayle's of the quantity of matter in it. Dictionary, art. “Zeno"). Thus the direction of true Of the different forms of the atomic theory, that of scientific progress was for many ages towards the recogni- Boscovich may be taken as an example of the purest tion of the infinite divisibility of space and time.

monadism. According to Boscovich matter is made up It was easy to attempt to apply similar arguments to of atoms. Each atom is an indivisible point, having matter. If matter is extended and fills space, the same position in space, capable of motion in a continuous path, mental operation by which we recognise the divisibility of and possessing a certain mass, whereby a certain amount of space may be applied, in imagination at least, to the matter force is required to produce a given change of motion. which occupies space. From this point of view the atomic Besides this the atom is endowed with potential force, doctrine might be regarded as a relic of the old numerical that is to say, that any two atoms attract or repel each way of conceiving magnitude, and the opposite doctrine of other with a force depending on their distance apart. The the infinite divisibility of matter might appear for a time law of this force, for all distances greater than say the the most scientific. The atomists, on the other hand, thousandth of an inch, is an attraction varying as the asserted very strongly the distinction between matter and inverse square of the distance. For smaller distances the space. The atoms, they said, do not fill up the universe ; force is an attraction for one distance and a repulsion for there are void spaces between them. If it were not so, another, according to some law not yet discovered. BosLucretius tells us, there could be no motion, for the atom covich himself, in order to obviate the possibility of two which gives way first must have some empty place to atoms ever being in the same place, asserts that the ultimove into.

mate force is a repulsion which increases without limit as

the distance diminishes without limit, so that two atoms “Quapropter locus est intactus, inane, vacansque Quod si non esset, nulla ratione moveri

can never coincide. But this seems an unwarrantable Res possent; namque, officium quod corporis exstat,

concession to the vulgar opinion that two bodies cannot Officere atque obstare, id in omni tempore adesset

co-exist in the same place. This opinion is deduced from
Omnibus : haud igitur quicquam procedere posset, our experience of the behaviour of bodies of sensible size,
Principium quoniam cedendi nulla daret res.
-De Rerum Natura, i. 335.

but we have no experimental evidence that two atoms may

not sometimes coincide. For instance, if oxygen and The opposite school maintained then, as they have always hydrogen combine to form water, we have no experimental done, that there is no vacuum—that every part of space

is evidence that the molecule of oxygen is not in the very full of matter, that there is a universal plenum, and that same place with the two molecules of hydrogen. Many all motion is like that of a fish in the water, which yields persons cannot get rid of the opinion that all matter is in front of the fish because the fish leaves room for it extended in length, breadih, and depth. This is a prebehind.

judice of the same kind with the last, arising from our “Cedere squamigeris latices nitentibus aiunt

experience of bodies consisting of immense multitudes of Et liquidas aperire vias, quia post loca pisces

atoms. The system of atoms, according to Boscovich, Linquant, quo possint cedentes confluere undæ.”

occupies a certain region of space in virtue of the forces --i. 373.

acting between the component atoms of the system and In modern times Descartes held that, as it is of the any other atoms when brought near them. No other essence of matter to be extended in length, breadth, and system of atoms can occupy the same region of space at thickness, so it is of the essence of extension to be occu- the same time, because, before it could do so, the mutual

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action of the atoms would have caused a repulsion between one of the intervals between the pieces ; to him, therefore, the two systems insuperable by any force which we can the gravel is by no means a homogeneous and continuous command. Thus, a number of soldiers with firearms may substance. occupy an extensive region to the exclusion of the enemy's In the same way, a theory that some particular substance, armies, though the space filled by their bodies is but small. say water, is homogeneous and continuous may be a good In this way Boscovich explained the apparent extension of working theory up to a certain point, but may fail when bodies consisting of atoms, each of which is devoid of we come to deal with quantities so minute or so attenuated extension. According to Boscovich's theory, all action that their heterogeneity of structure comes into prominence. between bodies is action at a distance. There is no such Whether this heterogeneity of structure is or is not conthing in nature as actual contact between two bodies. sistent with homogeneity and continuity of substance is When two bodies are said in ordinary language to be in another question. contact, all that is meant is that they are so near together The extreme form of the doctrine of continuity is that that the repulsion between the nearest pairs of atoms stated by Descartes, who maintains that the whole universe belonging to the two bodies is very great.

is equally full of matter, and that this matter is all of one Thus, in Boscovich's theory, the atom has continuity of kind, having no essential property besides that of extension. existence in time and space. At

any instant of time it is All the properties which we perceive in matter he reduces to at some point of space, and it is never in more than one its parts being movable among one another, and so capable place at a time. It passes from one place to another along of all the varieties which we can perceive to follow from a continuous path. It has a definite mass which cannot be the motion of its parts (Principia, ii. 23). Descartes's own increased or diminished. Atoms are endowed with the attempts to deduce the different qualities and actions of power of acting on one another by attraction or repulsion, bodies in this way are not of much value. More than a the amount of the force depending on the distance between century was required to invent methods of investigating them. On the other hand, the atom itself has no parts or the conditions of the motion of systems of bodies such as dimensions. In its geometrical aspect it is a mere geo- Descartes imagined. But the hydrodynamical discovery of metrical point. It has no extension in space. It has not Helmholtz that a vortex in a perfect liquid possesses certain the so-called property of Impenetrability, for two atoms may permanent characteristics, has been applied by Sir W. exist in the same place. This we may regard as one Thomson to form a theory of vortex atoms in a homoextreme of the various opinions about the constitution of geneous, incompressible, and frictionless liquid, to which bodies.

we shall return at the proper time. The opposite extreme, that of Anaxagoras -- the theory that bodies apparently homogeneous and continuous are so

OUTLINE OF MODERN MOLECULAR SCIENCE, AND IN PARin reality-is, in its extreme form, a theory incapable of

TICULAR OF THE MOLECULAR THEORY OF GASES. development. To explain the properties of any substance by this theory is impossible. We can only admit the We begin by assuming that bodies are made up of parts, observed properties of such substance as ultimate facts. each of which is capable of motion, and that these parts There is a certain stage, however, of scientific progress in act on each other in a manner consistent with the principle which a method corresponding to this theory is of service. of the conservation of energy. In making these assumpIn hydrostatics, for instance, we define a fluid by means of tions, we are justified by the facts that bodies may be one of its known properties, and from this definition we divided into smaller parts, and that all bodies with which make the system of deductions which constitutes the science we are acquainted are conservative systems, which would of hydrostatics. In this way the science of hydrostatics not be the case unless their parts were also conservative may be built upon an experimental basis, without any con- systems. sideration of the constitution of a fluid as to whether it is We may also assume that these small parts are in motion. molecular or continuous. In like manner, after the French This is the most general assumption we can make, for it mathematicians had attempted, with more or less ingenuity, includes, as a particular case, the theory that the small to construct a theory of elastic solids from the hypothesis parts are at rest. The phenomena of the diffusion of gases that they consist of atoms in equilibrium under the action and liquids through each other show that there may be a of their mutual forces, Stokes and others showed that all motion of the small parts of a body which is not perceptible the results of this hypothesis, so far at least as they agreed

to us. with facts, might be deduced from the postulate that elastic We make no assumption with respect to the nature of bodies exist, and from the hypothesis that the smallest the small parts—whether they are all of one magnitude. portions into which we can divide them are sensibly homo- We do not even assume them to have extension and figure. geneous. In this way the principle of continuity, which Each of them must be measured by its mass, and any two is the basis of the method of Fluxions and the whole of of them must, like visible bodies, have the power of acting modern mathematics, may be applied to the analysis of on one another when they come near enough to do so. The problems connected with material bodies by assuming them, properties of the body, or medium, are determined by the for the purpose of this analysis, to be homogeneous. All configuration and motion of its small parts. that is required to make the results applicable to the real The first step in the investigation is to determine the case is that the smallest portions of the substance of which amount of motion which exists among the small parts, we take any notice shall be sensibly of the same kind. independent of the visible motion of the medium as a Thus, if a railway contractor has to make a tunnel through whole. For this purpose it is convenient to make use of a a hill of gravel, and if one cubic yard of the gravel is so general theorem in dynamics due to Clausius. like another cubic yard that for the purposes of the contract When the motion of a material system is such that the they may be taken as equivalent, then, in estimating the time average of the quantity (mx) remains constant, the work required to remove the gravel from the tunnel, he state of the system is said to be that of stationary motion. may, without fear of error, make his calculations as if the When the motion of a material system is such that the gravel were a continuous substance. But if a worm has to sum of the moments of inertia of the system, about three make his way through the gravel, it makes the greatest axes at right angles through its centre of mass, never varies possible difference to him whether he tries to push right by more than small quantities from a constant value, against a piece of gravel, or directs his course through the system is said to be in a state of stationary motion.

The kinetic energy of a particle is half the product of where @ is the temperature reckoned from absolute zero, its mass into the square of its velocity, and the kinetic and R is a constant.

The fact that this equation expresses energy of a system is the sum of the kinetic energy of all with considerable accuracy the relation between the volume, its parts.

pressure, and temperature of a gas when in an extremely When an attraction or repulsion exists between two rarified state, and that as the gas is more and more compoints, half the product of this stress into the distance pressed the deviation from this equation becomes more between the two points is called the virial of the stress, apparent, shows that the pressure of a gas is due almost and is reckoned positive when the stress is an attraction, entirely to the motion of its molecules when the gas is rare, and negative when it is a repulsion. The virial of a system and that it is only when the density of the gas is consideris the sum of the virials of the stresses which exist in it. ably increased that the effect of direct action between the If the system is subjected to the external stress of the molecules becomes apparent. pressure of the sides of a vessel in which it is contained, The effect of the direct action of the molecules on each this stress will introduce an amount of virial ŽPV, where other depends on the number of pairs of molecules which p is the pressure on unit of area and V is the volume of at a given instant are near enough to act on one another. the vessel.

The number of such pairs is proportional to the square of The theorem of Clausius may now be stated as follows : the number of molecules in unit of volume, that is, to the In a material system in a state of stationary motion the square of the density of the gas. Hence, as long as the time-average of the kinetic energy is equal to the time- medium is so rare that the encounter between two molecules average of the virial. In the case of a fluid enclosed in a is not affected by the presence of others, the deviation from vessel

Boyle's law will be proportional to the square of the 13(mv2) = {pV+ $E(Rr),

density. If the action between the molecules is on the where the first term denotes the kinetic energy, and is half whole repulsive, the pressure will be greater than that given the sum of the product of each mass into the mean square by Boyle's law. If it is, on the whole, attractive, the

It of its velocity. In the second term, p is the pressure on

pressure will be less than that given by Boyle's law. unit of surface of the vessel, whose volume is V, and the appears, by the experiments of Regnault and others, that third term expresses the virial due to the internal actions the pressure does deviate from Boyle's law when the

In the case of carbonic between the parts of the system. A double symbol of density of the gas is increased. summation is used, because every pair of parts between acid and other gases which are easily liquefied, this deviawhich action exists must be taken into account. any

tion is very great. In all cases, however, except that of

We have next to show that in gases the principal part of the hydrogen, the pressure is less than that given by Boyle's pressure arises from the motion of the small parts of the law, showing that the virial is on the whole due to medium, and not from a repulsion between them.

attractive forces between the molecules.

Another kind of evidence as to the nature of the action In the first place, if the pressure of a gas arises from the repulsion of its parts, the law of repulsion must be inversely by Dr Joule. Of two vessels, one was exhausted and the

between the molecules is furnished by an experiment made as the distance. For, consider a cube filled with the gas

other filled with a gas at a pressure of 20 atmospheres ; at pressure p, and let the cube expand till each side is n times its former length. The pressure on unit of surface

and both were placed side by side in a vessel of water,

which was constantly stirred. The temperature of the according to Boyle's law is now

and since the

whole was observed. Then a communication was opened of a face of the cube is n2 times what it was, the whole between the vessels, the compressed gas expanded to

twice its volume, and the work of expansion, which at pressure on the face of the cube is of its original value. first produced a strong current in the gas, was soon con

When But since everything has been expanded symmetrically, the verted into heat by the internal friction of the gas. distance between corresponding parts of the air is now n

all was again at rest, and the temperature uniform, the times what it was, and the force is n times less than it was.

temperature was again observed. In Dr Joule’s original Hence the force must vary inversely as the distance.

experiments the observed temperature was the same as But Newton has shown (Principia, bk. i. prop. 93) that before. In a series of experiments, conducted by Dr Joule this law is inadmissible, as it makes the effect of the dis- and Sir W. Thomson on a different plan, by which the tant parts of the medium on a particle greater than that of thermal effect of free expansion can be more accurately the neighbouring parts. Indeed, we should arrive at the measured, a slight cooling effect was observed in all the conclusion that the pressure depends not only on the density gases examined except hydrogen. Since the temperature of the air but on the form and dimensions of the vessel depends on the velocity of agitation of the molecules, it which contains it, which we know not to be the case.

appears that when a gas expands without doing external If, on the other hand, we suppose the pressure to arise work the velocity of agitation is not much affected, but entirely from the motion of the molecules of the gas, the

that in most cases it is slightly diminished. Now, if the interpretation of Boyle's law becomes very simple. For, molecules during their mutual separation act on each other, in this case

their velocity will increase or diminish according as the pV= {}(mv2).

force is repulsive or attractive. It appears, therefore, from

the experiments on the free expansion of gases, that the The first term is the product of the pressure and the volume, force between the molecules is small but, on the whole, which according to Boyle's law is constant for the same attractive. quantity of gas at the same temperature. The second term

Having thus justified the hypothesis that a gas consists is two-thirds of the kinetic energy of the system, and we of molecules in motion, which act on each other only have every reason to believe that in gases when the when they come very close together during an encounter, temperature is constant the kinetic energy of unit of mass

but which, during the intervals between their encounters is also constant. If we admit that the kinetic energy of which constitute the greater part of their existence, are unit of mass is in a given gas proportional to the absolute describing free paths, and are not acted on by any moletemperature, this equation is the expression of the law of cular force, we proceed to investigate the motion of such a Charles as well as of that of Boyle, and may

be written

system. pᏙ = ᎡᎾ ,

The mathematical investigation of the properties of such


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a system of molecules in motion is the foundation of mole cogency. But it is purely chemical reasoning; it is not cular science. Clausius was the first to express the dynamical reasoning. It is founded on chemical experirelation between the density of the gas, the length of the ence, not on the laws of motion. free paths of its molecules, and the distance at which Our definition of a molecule is purely dynamical. A they encounter each other. He assumed, however, at least molecule is that minute portion of a substance which moves in his earlier investigations, that the velocities of all the about as a whole, so that its parts, if it has any, do not part molecules are equal. The mode in which the velocities are company during the motion of agitation of the gas. The distributed was first investigated by the present writer, result of the kinetic theory, therefore, is to give us informawho showed that in the moving system the velocities of tion about the relative masses of molecules considered as the molecules range from zero to infinity, but that the moving bodies. The consistency of this information with number of molecules whose velocities lie within given the deductions of chemists from the phenomena of comlimits can be expressed by a formula identical with that bination, greatly strengthens the evidence in favour of the which expresses in the theory of errors the number of actual existence and motion of gaseous molecules. errors of observation lying within corresponding limits. Another confirmation of the theory of molecules is The proof of this theorem has been carefully investigated derived from the experiments of Dulong and Petit on the by Boltzmann, who has strengthened it where it appeared specific heat of gases, from which they deduced the law weak, and to whom the method of taking into account the which bears their name, and which asserts that the specific action of external forces is entirely due.

heats of equal weights of gases are inversely as their comThe mean kinetic energy of a molecule, however, has a bining weights, or, in other words, that the capacities for definite value, which is easily expressed in terms of the heat of the chemical equivalents of different gases are quantities which enter into the expression for the distribu- equal. We have seen that the temperature is determined tion of velocities. The most important result of this investi- by the kinetic energy of agitation of each molecule. The gation is that when several kinds of molecules are in motion molecule has also a certain amount of energy of internal moand acting on one another, the mean kinetic energy of a mole- tion, whether of rotation or of vibration, but the hypothesis cule is the same whatever be its mass, the molecules of of Clausius, that the mean value of the internal energy greater mass having smaller mean velocities. Now, when always bears a proportion fixed for each gas to the energy gases are mixed their temperatures become equal. Hence of agitation, seems highly probable and consistent with we conclude that the physical condition which determines experiment. The whole kinetic energy is therefore equal that the temperature of two gases shall be the same is that to the energy of agitation multiplied by a certain factor. the mean kinetic energies of agitation of the individual mole. Thus the energy communicated to a gas by heating it is cules of the two gases are equal. This result is of great divided in a certain proportion between the energy of agitaimportance in the theory of heat, though we are not yet tion and that of the internal motion of each molecule. For able to establish any similar result for bodies in the liquid a given rise of temperature the energy of agitation, say of a or solid state.

million molecules, is increased by the same amount whatIn the next place, we know that in the case in which the ever be the gas. The heat spent in raising the temperature whole pressure of the medium is due to the motion of its ( is measured by the increase of the whole kinetic energy. molecules, the pressure on unit of area is numerically the thermal capacities, therefore, of equal numbers of equal to two-thirds of the kinetic energy in unit of volume. molecules of different gases are in the ratio of the factors Hence, if equal volumes of two gases are at equal pressures by which the energy of agitation must be multiplied to the kinetic energy is the same in each. If they are also obtain the whole energy. As this factor appears to be at equal temperatures the mean kinetic energy of each nearly the same for all gases of the same degree of atomicity, molecule is the same in each. If, therefore, equal volumes Dulong and Petit's law is true for such gases. of two gases are at equal temperatures and pressures, the Another result of this investigation is of considerable number of molecules in each is the same, and therefore, importance in relation to certain theories,” which assume the the masses of the two kinds of molecules are in the same existence of æthers or rare media consisting of molecules ratio as the densities of the gases to which they belong. very much smaller than those of ordinary gases. According

This statement has been believed by chemists since the to our result, such a medium would be neither more nor time of Gay-Lussac, who first established that the weights less than a gas. Supposing its molecules so small that of the chemical equivalents of different substances are they can penetrate between the molecules of solid substances proportional to the densities of these substances when in such as glass, a so-called vacuum would be full of this rare the form of gas. The definition of the word molecule, gas at the observed temperature, and at the pressure, whathowever, as employed in the statement of Gay-Lussac's law ever it may be, of the ætherial medium in space. The is by no means identical with the definition of the same specific heat, therefore, of the medium in the so-called word as in the kinetic theory of gases. The chemists vacuum will be equal to that of the same volume of any ascertain by experiment the ratios of the masses of the other gas at the same temperature and pressure. Now, the different substances in a compound. From these they purpose for which this inolecular æther is assumed in these deduce the chemical equivalents of the different substances, theories is to act on bodies by its pressure, and for this that of a particular substance, say hydrogen, being taken purpose the pressure is generally assumed to be very great. as unity. The only evidence made use of is that furnished Hence, according to these theories, we should find the by chemical combinations. It is also assumed, in order to specific heat of a so-called vacuum very

considerable comaccount for the facts of coinbination, that the reason why pared with that of a quantity of air filling the same space. substances combine in definite ratios is that the molecules We have now made a certain definite amount of progress of the substances are in the ratio of their chemical equiva- towards a complete molecular theory of gases.

We know lents, and that what we call combination is an action the mean velocity of the molecules of each gas in metres which takes place by a union of a molecule of one substance per second, and we know the relative masses of the molecules to a molecule of the other.

of different gases.

We also know that the molecules of This kind of reasoning, when presented in a proper form one and the same gas are all equal in mass.

For if they and sustained by proper evidence, has a high degree of

? See Gustav Hansemann, Die Atorne und ihre Bewegungen. 1871. Sitzungsberichte der K. K. Akad., Wien, 8th Oct. 1868. (H. G. Mayer.)

are not, the method of dialysis, as employed by Graham, | temperature of different parts of the medium, and constitutes would enable us to separate the molecules of smaller mass the phenomenon of the conduction of heat in gases. from those of greater, as they would stream through porous These three phenomena—the diffusion of matter, of substances with greater velocity. We should thus be able motion, and of heat in gases-have been experimentally to separate a gas, say hydrogen, into two portions, having investigated,—the diffusion of matter by Graham and different densities and other physical properties, different Loschmidt, the diffusion of motion by Oscar Meyer and combining weights, and probably different chemical pro- | Clerk Maxwell, and that of heat by Stefan. perties of other kinds. As no chemist has yet obtained These three kinds of experiments give results which in specimens of hydrogen differing in this way from other the present imperfect state of the theory and the extreme specimens, we conclude that all the molecules of hydrogen difficulty of the experiments, especially those on the conare of sensibly the same mass, and not merely that their duction of heat, may be regarded as tolerably consistent mean mass is a statistical constant of great stability. with each other. At the pressure of our atmosphere, and

But as yet we have not considered the phenomena which at the temperature of melting ice, the mean path of a enable us to form an estimate of the actual mass and molecule of hydrogen is about the 10,000th of a millidimensions of a molecule. It is to Clausius that we owe metre, or about the fifth part of a wave-length of green light. the first definite conception of the free path of a molecule The mean path of the molecules of other gases is shorter and of the mean distance travelled by a molecule between than that of hydrogen. successive encounters. He showed that the number of The determination of the molecular volume of a gas is encounters of a molecule in a given time is proportional to subject as yet to considerable uncertainty. The most the velocity, to the number of molecules in unit of volume, obvious method is that of compressing the gas till it and to the square of the distance between the centres of assumes the liquid form. It seems probable, from the great two molecules when they act on one another so as to have resistance of liquids to compression, that their molecules an encounter. From this it appears that if we call this are at about the same distance from each other as that at distance of the centres the diameter of a molecule, and the which two molecules of the same substance in the gaseous volume of a sphere having this diameter the volume of a form act on each other during an encounter. If this is the molecule, and the sum of the volumes of all the molecules case, the molecular volume of a gas is somewhat less than the molecular volume of the gas, then the diameter of a the volume of the liquid into which it would be condensed molecule is a certain multiple of the quantity obtained by by pressure, or, in other words, the density of the molecules diminishing the free path in the ratio of the molecular is somewhat greater than that of the liquid. volume of the gas to the whole volume of the gas. The Now, we know the relative weights of different molecules numerical value of this multiple differs slightly, according with great accuracy, and, from a knowledge of the mean to the hypothesis we assume about the law of distribution path, we can calculate their relative diameters approxiof velocities. It also depends on the definition of an mately. From these we can deduce the relative densities encounter. When the molecules are regarded as elastic of different kinds of molecules. The relative densities so spheres we know what is meant by an encounter, but if calculated have been compared by Lorenz Meyer with the they act on each other at a distance by attractive or repul observed densities of the liquids into which the gases may sive forces of finite magnitude, the distance of their be condensed, and he finds a remarkable correspondence centres varies during an encounter, and is not a definite between them. There is considerable doubt, however, as quantity. Nevertheless, the above statement of Clausius to the relation between the molecules of a liquid and those enables us, if we know the length of the mean path and of its vapour, so that till a larger number of comparisons the molecular volume of a gas, to form a tolerably near have been made, we must not place too much reliance on estimate of the diameter of the sphere of the intense action the calculated densities of molecules. Another, and perhaps of a molecule, and thence of the number of molecules in a more refined, method is that adopted by M. Van der unit of volume and the actual mass of each molecule. To Waals, whò deduces the molecular volume from the deviacomplete the investigation we have, therefore, to determine tions of the pressure from Boyle's law as the gas is comthe mean path and the molecular volume. The first pressed. numerical estimate of the mean path of a gaseous molecule The first numerical estimate of the diameter of a molecule was made by the present writer from data derived from the was that made by Loschmidt in 1865 from the mean path internal friction of air. There are three phenomena which and the molecular volume. Independently of him and of depend on the length of the free path of the molecules of a each other, Mr Stoney, in 1868, and Sir W. Thomson, in gas. It is evident that the greater the free path the more 1870, published results of a similar kind--those of Thomson rapidly will the molecules travel from one part of the being deduced not only in this way, but from considerations medium to another, because their direction will not be so derived from the thickness of soap bubbles, and from the often altered by encounters with other molecules. If the electric action between zinc and copper. molecules in different parts of the medium are of different The diameter and the mass of a molecule, as estimated kinds, their progress from one part of the medium to by these methods, are, of course, very small, but by no another can be easily traced by analysing portions of the means infinitely so. About two millions of molecules of medium taken from different places. The rate of diffu- hydrogen in a row would occupy a millimetre, and about sion thus found furnishes one method of estimating the two hundred million million million of them would weigh length of the free path of a molecule. This kind of a milligramme. These numbers must be considered as diffusion goes on not only between the molecules of exceedingly rough guesses ; they must be corrected by more different gases,

the molecules of the same gas, extensive and accurate experiments as science advances ; only in the latter case the results of the diffusion cannot but the main result, which appears to be well established, be traced by analysis. But the diffusing molecules carry is that the determination of the mass of a molecule is a with them in their free paths the momentum and the energy legitimate object of scientific research, and that this mass which they happen at a given instant to have. The

The is by no means immeasurably small. diffusion of momentum tends to equalise the apparent Loschmidt illustrates these molecular measurements by motion of different parts of the medium, and constitutes a comparison with the smallest magnitudes visible by means the phenomenon called the internal friction or viscosity of of a microscope. Nobert, he tells us, can draw 4000 lines gases. The diffusion of energy tends to equalise the in the breadth of a millimetre. The intervals between

III. - 6

but among

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