Molecule

MOLECULE

IN the conception of the atomic as opposed to the continuous and infinitely divisible constitution of matter, it is supposed that portions of matter called atoms exist, which are separated, or are capable of being separated, from each other by empty space. (See Atom). It may be the case that each atom has unchangeable shape and volume as well as unchangeable mass, but such a conception of an atom is not essential to the hypothesis. It is not even necessary, as explained in the article Atom (vol. iii., pp. 37, 38), to maintain that no part of space can be in two atoms at the same time. But one attribute of the atom upon which its permanence, or, so to speak, its personal identity, depends, is its constituent mass, and this remains the same, unchanged and unchangeable, through all time.

Boscovich, indeed, goes so far as to regard the atom as a mere centre of force, the result of whose existence is that no two atoms or centres can approach each other within a certain distance, while other physicists regard the atomic volume as a distinct portion of space occupied by that atom to the exclusion of every other, and comprising within it matter ideally infinitely divisible, but the parts of which in fact never have been, and never can be, separated from each other. In this latter mode of viewing the subject, all the conclusions of mechanics which are based on the conception of the continuity and infinite divisibility of matter may be applied to the equilibrium or motion of each individual atom, the atomic theory merely introducing the additional hypothesis that, in fact, these persistent entities called atoms do exist, and that out of them all substances which affect our senses are constructed. The theory of universal gravitation requires us to believe in the existence of forces or actions between every portion of matter and every other portion, determinate in magnitude and direction, and such that, when on the infinitely divisible hypothesis the volumes of these portions are indefinitely diminished, these mutual forces are inversely proportional to the square of the distance between the portions (the distance between any two points, one in the volume of each portion, being in this case taken as the distance between the portions), and directly proportional to the products of the masses, or quantities of the two portions of matter,—such forces being regarded provisionally as ultimate facts, while inviting further analysis and explanation. Chemical and chemico-physical investigations indicate the existence of other actions between portions of matter, following other and for the most part unknown laws, and rapidly becoming inappreciable as the distance between the reacting portions is increased. All these hypotheses are to be retained on the hypothesis of discrete atoms as above enunciated, the mutual actions between atoms being the resultant of the actions between the various portions of their constituent matter. The volumes of the atoms are so small that, for any sensible distances apart, the line of the resultant mutual action between them may be taken as coincident with the line joining any point in the volume of one to any point in the volume of the other, but, for distances or parts comparable with the linear dimensions of the atoms, the size and shape of their bounding surfaces must be taken into consideration, and perhaps also the law of distribution of their constituent matter within that surface. In all respects, unless we accept the Boscovichian hypothesis, we simply regard the atom as made up, so to speak, of infinitely divisible matter, while substances, as we know them, are built up of indestructible and unchangeable atoms.

With this conception of an atom, as thus explained, we might be content to rest, confessing our total ignorance of the mode in which such atoms are built up into actual substances, being satisfied to regard such substances as composed of these distinct portions of matter separated, or capable of being separated, by empty space from other portions. But the molecular hypothesis of the constitution of different kinds of substances aims at analysing this process by which such substances are built up out of their constituent atoms. The molecule of any substance is, by some chemists, defined as being the smallest portion of that substance to which can be attributed all the chemical properties of the substance; by others, as the smallest portion which, so long as the substance is chemically unchanged, keeps together without complete separation of its parts. In the language of Clausius’s theorem, if the parts of the molecule have internal motion, the kinetic energy of such internal motion is equal to the virial of the mutual attractive forces of the parts. Thus the formation of the molecule of each particular substance is viewed as an essential step in the process of building up that substance out of its constituent atoms. The molecule is first built up out of atoms arranged in its formation according to a definite type, and then the substance itself is constituted of these molecules. Of course molecules may be, and in fact in many particular substances are, supposed to be monatomic; that is to say, the intermediate step of building up the molecule out of the atoms has, in these particular substances, been omitted, the atoms and molecules becoming then identical. The particular arrangement of the formed molecules in the building up of the substance determines the physical state of that substance,—that is, its fluid, solid, gaseous, crystalline, or amorphous state; but the chemical properties of the substance depend upon the constitution of the molecule. As the investigations and theories of chemistry appear to indicate irresistibly the existence of permanent atoms, so do they also lead almost as necessarily to the conception of the molecule as an entity which bears the same relation to special substances that the atoms bear to matter generally. So long as the molecule endures, the substance of which it is the molecule retains its chemical properties; with the dissolution of the molecule, the substance, as that special substance, perishes; the atoms alone continue, and are free to enter into other combinations. The permanence of the molecule is relative, that of the atom absolute. This conception of the molecular constitution of substances suggests physical questions of great interest, such as the shape, volume, and mass of the constituent molecules, and the relative motions of which their parts are susceptible; and the answers to these questions cannot fail to be of great value in chemical and chemico-physical investigations, as well as in the theories of light and electricity.

Now, whatever differences may exist between the properties of different substances in the solid and liquid states, there are certain properties which, in the gaseous state, manifest themselves with no variation whatever in all substances alike. Hence the explanation of these common properties—or gaseous laws, as they are called—has long possessed a peculiar fascination for physicists. The tendency to expand or fill all accessible space, manifested by all gases, proves that on the molecular hypothesis their compound atoms or molecules must be continually tending to fly apart. We must conceive gases as constituted of molecules, not only separable but actually separated by space void of the matter of which these gases consist; and it may be most reasonably expected, therefore, that any general laws to which substances in this state conform may afford us a valuable insight into the constitution of these separate molecules.

Now the general laws to which all gases conform are: (1) Boyle’s law— that, in a given mass of any gas kept at constant temperature, the pressure per unit of area upon the containing surface increases in the same proportion as the volume occupied by the gas is diminished, or at least with very slight deviation from exact proportionality · (2) Charles’s law — that, if the temperature be varied while the pressure upon the gas remains the same, the gas increases by τrsd of its volume at zero centigrade for every degree of centigrade added to the temperature, or, which in combination with Boyle’s law is the same thing, that if the density be constant, the pressure is directly proportional to the temperature measured from the point - 273° centigrade, this point being called the zero of absolute temperature; (3) Avogadro’s law — which asserts that all gases at the same temperature and pressure contain the same number of molecules in the same volume; and (4) Dalton’s law— that in a mixture of different gases, when there is equilibrium, each gas behaves as a vacuum to all the rest.

It was at one time considered that these phenomena might be explained on the hypothesis of mutual repulsive forces between the parts of which the gas is composed, whether they were regarded as constituted of molecules or of infinitely divisible continuous matter, but it has been shown in the article Atom (vol. iii. p. 39 sq.) that there are at least two absolutely conclusive reasons why this explanation cannot be accepted. These objections, together with the experimental fact proved by Joule that gases, or at any rate atmospheric air, expand into vacuum with scarcely any appreciable change of temperature, must be considered fatal to any mutual-force theory of gaseous action, and, accordingly, physicists have been driven to seek for other methods of explaining these laws. The explanation which has been more developed than any other is that known as the kinetic theory of gases, which regards the intrinsic energy of a gaseous mass as residing, not in the potential energy of intermolecular forces, but mainly in the kinetic energy of the molecules themselves, which are assumed to be in a state of continual relative velocity, admitting at the same time a possible small intermolecular potential energy, and it may be also an interatomic energy, between the atoms of the individual molecules. That some such persistent relative motion does exist in every gaseous mass is evident from the rapidity with which odours penetrate the stillest air where no breath of wind—that is, of absolute motion of translation of the mass as a whole or any portion of finite size—is perceptible. It becomes an interesting question whether the laws of mechanics admit of a mass thus constituted ever arriving at a state of permanence; that is to say, whether, consistently with the hypothesis of infinite irregularities in the directions and magnitudes of velocities of individual molecules, there may be found any properties of the mass in the aggregate which remain constant, and in agreement with the accepted laws common to all gases. Now the physical theory of heat compels us to regard the intrinsic energy of any gaseous mass as dependent entirely or almost entirely upon the temperature. If, therefore, this intrinsic energy is to be sought for in the kinetic energy of the moving molecules, it follows that the average value of the kinetic energy of the molecules taken throughout the mass must be also a function of the temperature.

We will proceed to investigate the condition of permanence of a number of molecules moving about irregularly in any bounded space; and, for simplicity’s sake, we shall first of all restrict ourselves to the case of monatomic molecules.

We know nothing of the size or shape of these atoms, except that the volume of each one must be incomparably smaller than that of the containing region. In shape we shall, as the simplest hypothesis, regard them as spherical. We shall suppose that there are no intermolecular forces between any two such atoms, except of such a nature as to be practically insensible when the atoms are not geometrically in contact, and similarly as regards the forces between the atoms and the material bounding surface, such forces being of the nature called “conservative.” So that in point of fact we are investigating the mechanical properties of an infinitely large number of infinitely small and perfectly elastic spheres moving about in a given region and subject to frequent collisions.

Problem. — A very large number of smooth elastic spheres, equal in every respect, are in motion within a region of space of given volume, and therefore occasionally impinge upon each other with various degrees of relative velocity and in various relative directions; required to find the law of distribution of velocities in order that such distribution may be permanent.

Let Λ r be the total number of spheres, and let χ (w, V, w) du dv dw be the number of spheres whose component velocities, parallel to the axes, are intermediate between u and u + du, v and v + dv, w and w + dw respectively.

If c be the resultant velocity of any of these last-mentioned spheres, and if θ be the inclination of c fo the axis of z, and φ that of the plane cz to the plane xz, the last-mentioned expression will become, by changes of the independent variables from x, y, z to θ, φ, and c, χ (u, v, w) c? sin θ dθ dφ de.

Let a spherical surface of radius unity be described about any origin as centre, and let dσ be written for the element sin θ dθ dφ on this surface, then the last-written expression becomes

χ (it, v, w) c 2 de dσ.

Since for the same magnitude of the resultant velocities all directions of motion must be equally probable, it follows that the coefficient of de dσ in the last-written expression must be a function of c only, and therefore the number of spheres having component velocities between u and u + du, v and v + dv, w and w + dw, must be φ (c) du dv dw.

It is required to find the form of ψ in order that the value of this expression may be unaffected by collisions. The solution is, that the number of spheres with component velocities between the limits w and u + du, v and v + dv, w and w ÷ dw must be

Je^ ftc2 du dv dw; or Ae~ hc2 c 2 de dσ, employing the notation already used.

Integrating with respect to dσ from 0 to 4ττ, we find for the number of spheres with velocities between c and c + dc the expression

4τr Ae~^ ιc2 de.

Again, since the number with component velocities between u and u + du, v and v + dv, w and w + dw is Ac - h < u2 + v * + w2 > du dv dw, or (JJA~e- hu2 du} (l∕Ae~ hυ2 dυ) (ξ∕Λe~ hv,i dw), it follows that the number of spheres having velocities intermediate between u and u + du parallel to the x axis is

∕∙OO fCO

Ae~ fιu2 du i e ^ hυ2 dv ι e~ iu ° 2 dw; J oo √ oo that is, ^-Ae ~ hu2 du, h where A is to be determined by the equations -^A I e~ fιu2 du—N, J 00

. τri vr or A -= r = N; hi j Nhi therefore A = —v-; π i that is to say, the number of spheres having velocities between c and c + de is

4λi -7⅛c a a -=- Ne de.

√τr

Multiplying this expression by c, and integrating the product with regard to c from 0 to ∞, and dividing by N, the mean velocity for all the spheres becomes

2

V πh ’ and multiplying by c 2 instead of by c, we find the mean square of all the velocities to be

3_

2Λ'

In the preceding investigation no account has been taken of collisions between the spheres and the enclosing boundary of the region in which they are contained, because in every such collision the magnitude of the velocity of each sphere is unaltered and its direction is changed according to the ordinary law of reflexion, whence it is evident that the distribution is unaffected by such collisions. Also, the investigation has been confined to the cases of spheres colliding in pairs, but since there need be no limit to the smallness of the interval between any pair of collisions the result really embraces the cases of simultaneous collisions between three or more spheres; for if a sphere A collides with another B, and immediately afterwards with a third C, the resultant velocity of A after this second collision must be the same as if it had collided with B and C simultaneously.

The foregoing investigation has been given in some detail because the principles upon which it proceeds are essentially the same as those by which all questions of the distribution of energy among a great number of moving bodies are determined, although it may be found, as well as the detailed investigations of the results immediately following, in published memoirs and systematic treatises on the kinetic theory of gases.

If the spheres be not all of equal mass, but if there be within the region N spheres of mass m, N , of mass m', and so on, then it may be proved, by reasoning exactly similar to the foregoing, that when the permanent or stable state of motion has been attained the number of spheres of the N set with component velocities between u and u + du, v and v + dv, w and w + dw is

λmc 2 Ae 2 du dv dw, and the number of the N' set having component velocities between u' and u' + du', d and √ + dd, w and w + du∕, is

⅛m'c' 2 A'e 2 du dv' dw', where c 2 =m 2 + v 2 + w 2 , c' a =u' 2 + t∕ 2 + w 8 , h is a constant the same for both sets, and . N ∕ mh∖i N' ∕ ’ m'h∖i and so on if there be any other sets.

The mean velocity and mean square velocity of each sphere of the N set are 2 ∕^ 2 ~ j 3 .. 1

—7=. ∕ — τ and — r respectively, a v 7Γ z ∖∕ 77l∕i' ’ )TUl> and the mean kinetic energy of each of such spheres is 2h ’ the last result being common to all the sets.

If the spheres in the given region be acted on by any given forces tending to fixed centres, and functions of the distances of the centres of the spheres from the centres of force, we may not in such case assume, a priori, that the chances of velocities in all directions are the same; but we may assume that the number of spheres of any set (Λ 7 ) with coordinates of their centres intermediate between X and x + dx, y and y + dy, z and z + dz, and component velocities intermediate between u and u + du, v and υ + dv, w and w + dw, is ψ (x, y, z, u, v, w) dx dy dz du dv dw.

In the state of permanence the form of ψ must be independent of the time (t), so long as the sphere is moving free from collisions with any other.

From the last-mentioned condition it must follow that, if ≠ 1 =α 1 , ≠ 2 = α 2 , &c., be any equations among the variables determining the position and motion of any sphere obtained by the elimination of t from the equations of motion of that sphere, then ψ must be of the form ψ (≠j, φ. i , &c. ). With the assumption, then, that the number of spheres of the given set with variables between the above-mentioned limits is

≠ (≠ι, ≠ 2 ∙∙∙) dx...dw, we find for the form of φ, by reasoning like the foregoing, ⅛ ( I m ° 2 ∖

Ae × 2 ', where χ is the potential energy of the sphere in the position x, y, z, and cP=u 2 + v 2 + w 2 , and h is a constant, the same for all the sets.

j i ∕ mc 2 ∖

If we integrate the expression Ae × 2 'dx dy dz du dv dw for all values of x, y, z within the given region, we find for the number of spheres of any set with component velocities between u and u + du, v and v + dv, w and w + dw, hmc 2 Be 2 du dv dw, whence we easily see that the chances of velocities in all directions are the same, and that the mean velocity and mean square velocity of any sphere of this set are and respectively, and the mean

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kinetic energy of any such sphere is and therefore the same for all the sets.

Furthermore, if we integrate the expression

-A (χ+^)

Ae ' 2 'dx dy dz du dv dw for all values of u, v, and to from — co to + ∞ respectively, we obtain a result of the form Ce ~ dx dy dz, and therefore the number of spheres of the set in question with centres within the elementary volume dx dy dz, or, what is the same thing with the exception of a constant factor, the chance of the centre of any sphere of that set being within that elementary volume, is Ce ~ h %- dx dy dz, so that the density of the Wset of matter in the neighbourhood of the point x, y, z is

We are now in a position to compare the physical properties of a medium composed of monatomic molecules in motion, and free from any intermolecular or interatomic forces with those of ordinary gases, so long at least as the atoms are spherical.

Consider two contiguous portions of such a medium separated by any plane parallel to that of yz, and, since the distribution and motion of each set of spheres is independent of all the other sets, let us confine our attention to the spheres of the N set. Suppose that there are N such spheres per unit volume in the neighbourhood of the point x, y, z, whose component velocities parallel to the axis of x are between u and u + du. The number of these spheres which cross the elementary area dy dz in time dt will be the same as the number of the dN spheres whose centres are situated within the elementary parallelepiped dx dy dz, in which dx is equal to udt, and this number is Nu dy dz dt.

Each of these spheres carries across with it a momentum parallel to x equal to mu ; the total momentum parallel to x transferred across dy dz in time dt is therefore mNu 2 dy dz dt.

If u be positive, this is positive momentum transferred from the negative to the positive side of the plane y z ; and if u be negative, this is negative momentum similarly transferred from the positive to the negative side of that plane. In either case it follows that by the mere motion of these spheres across the area dy dz the positive momentum parallel to the axis of x is diminished by the quantity mNu i dy dz dt on the negative side of the plane y z, and increased by the same quantity on the positive side of that plane in the time dt; m being, as before, the mass of each sphere. Hence, on the whole, there is a transference of positive x momentum in the time dt across the area dy dz equal to mdy dzdf∑ “u 2 N ; that is, equal to - ∞

dy dz dt ptι 2 , where p is the density of the W matter at the point x, y, z, and u δ is the mean square of the x velocities.

But either by integration or general reasoning it is easily seen that u 2 ≈^, where v 2 is the mean square of the resultant velocities of the N spheres, and is equal, as we have proved, to

3 mh ’

Therefore, there is a transference of positive momentum from the negative to the positive side of the plane y z across the area dy dz in time dt equal to ρ dy dz dt mh

Each separate sphere whose component velocities are u, υ, and w carries across the same area y and z momenta equal to mv and mw respectively, so that in the time dt there are carried across the area dy dz y and z momenta equal to ∑muv dy dz dt and ∑muw dy dz dt, respectively. By symmetry it is clear that '∑muv and ^∑muw are separately zero. Therefore, the resultant mutual actions of the two portions of the medium under consideration in the time dt is the transference across the elementary area dy dz of a quantity of x momentum equal to pdy dz dt — from the negative to the positive side of the bounding plane. If this mutual action, or, as it is generally called, “pressure” when referred to unit of surface, be denoted by the symbol p, we get the equation p dy dz dt=p dy dz dt —, o v 2 p 0r

Since the momenta parallel to y and z remain unaltered, it follows that the mutual action or pressure between contiguous portions of the medium in the neighbourhood of any point is normal to the bounding surface at that point. Since also the expression for p or is independent of the direction of the x axis, it follows that the pressure at any point of the medium is the same in all directions.

If the contiguous portions of the medium be separated by a material instead of an ideal plane, it will be necessary for the maintenance of equilibrium that there should be an action between this plane and the adjacent medium, equivalent to the transference of momentum estimated above; but action measured by the rate per unit of time at which momentum is generated constitutes moving force or statical pressure. Hence the force or pressure between the plane and medium is normal to the plane, independent of the direction of the plane through the point, and equal to the value of — at the point.

mh

When several sets of spheres are present together in the region under consideration, the distribution of the centres and of the velocities of each set is, as we have seen, independent of the coexistence of the other sets. If therefore p 1 , ρ 2 , <fcc., be the densities of the matter of the different sets in the neighbourhood of the point x, y, z, and if p 1 , p i , &c., be the pressures at that point defined as above, and if m,, m 2 , &c., be the masses of the spheres of each of the sets, andj> the total pressure, we get P = Pi + Pi + &c·

= Ae + Ae + &c. m 1 h m 2 h

Hence we arrive at the following conclusions :—(1) there is one physical quantity having the same value for every set of spheres—

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namely, the mean kinetic energy of each sphere, or; let this quantity be called τ; (2) the distribution of the positions and velocities of the spheres of each set is independent of the coexistence of the remaining sets, and is in all respects the same as if that particular set existed alone in the region considered; (3) the pressure at any point referred to unit of surface at any point of

2 the medium arising from the action of any one of the sets is pτ, where p is the density of that particular set at the point in question, and τ is the physical quantity above referred to as common to all the sets.

This third inference may be expanded into the following three laws :—(a) if τ be kept constant, then the pressure arising from each set varies as the density of that set; (β) if p be kept constant, then the pressure from each set varies as τ; (γ) if the pressures for all the sets be the same, then — is also the same, or the num- m ber of spheres per unit volume is the same.

Now suppose there is a mixture of any number of gases in any < region; when there is equilibrium there is one physical quantity, namely, temperature, which is the same for all; the intrinsic energy of this mixture depends, as we know, upon its temperature, and the energy of these moving spheres is entirely kinetic, and may be conceived, therefore, to be a function of the mean vis viva. Let us then assume that in this medium of moving spheres we have a representation of a mass of gases, and that what is called the temperature of the gaseous mass is nothing else than the τ or mean kinetic energy of each moving sphere. Then, with this assumption, the three parts (a, β, y) of inference (3) above correspond to the gaseous laws connected with the names of Boyle, Charles, and Avogadro respectively, and inference (2) corresponds with the law of Dalton concerning gaseous mixtures.

We may also deduce the ordinary hydrostatical equations of equilibrium from the formulae which we have obtained.

For, since these equations give us p=-^~r, and p=mBc~^, r mh we get ⅛= -P d × = p‰ and‰ -τnX, dx dx dx and similarly ⅛I. = pY,⅛P = p Z, dy dz where X, Y, and Z are the component impressed forces, or the negatives of the space variations of χ along the coordinate axes.

So far, therefore, the physical properties of a perfect gas or mixture of such gases correspond, in all respects, with the physical properties of a medium consisting of a set of elastic spheres, or of a mixture of sets of such spheres, with the sole assumption that the physical property termed temperature, in the case of the gas, corresponds to, or is represented by, the mean kinetic energy of each of the spheres, and that each sphere represents the chemical atom.

There are, however, physical properties of gases which this theory fails to explain. The most important of these is the ratio of the specific heats at constant volume and constant temperature respectively. The specific heat of gas expanding while being heated under a constant pressure is greater than that of gas heated with a constant volume, as when it is contained in a rigid vessel, for the obvious physical reason that in the former case a portion of the heat is converted into mechanical work, namely, that performed by the expansion under the constant pressure. This ratio of the specific heat of gas under constant pressure to the specific heat with constant volume has been determined for many gases with great accuracy, chiefly from observations of the velocity of sound in such gases, in which velocity the value of this ratio bears a very important part.

Now, on the assumption of the gas being constituted of a number of elastic spheres in rapid but irregular motion among each other, and the physical property of temperature being represented or measured by the mean vis viva of each sphere, the ratio of these specific heats must be exactly 1 f.

For, if V be the volume occupied by a unit of mass of this moving sphere medium, and r the number of spheres to the unit mass, and if p be the density, it follows that rm=pυ=l.

Also we know that p, the pressure referred to unit surface, is given by the equation

2 pv = -g- rτ, where τ is the mean vis viva. If now τ increase from τ to τ + δτ, while v remains constant, the increase of intrinsic energy must be, from definition, rδτ. Also if there be a similar change in r without the restriction of v being constant, but supposing p to be constant, there is external mechanical work performed equal to pδv, where δv is the increase of volume. Also s 2 , pδv = — rδτ; and therefore the whole energy required to be supplied from without must be in this case τδτ + — rδτ.

3

Or the ratio of the energies to be supplied from without, in order that the mean vis viva of the moving sphere medium should be increased by the same amount in the two cases respectively, becomes

2 r + -q∙ r — , or 1⅞. r

If therefore the gaseous mass be adequately represented by the moving sphere medium, the ratio of the specific heats must be 1J.

Mercury vapour is the only gas for which the ratio has so large a value as this. Several of the more permanent gases have the ratio equal to 1 ‘408, while in others it falls as low as 1 ’ 26. The value for mercury vapour, as determined by Kundt and Warburg (Poggendorff, clvii. 353), is between 1 ’ 695 and 1 '631, the mean of all the observations being somewhat under 1 6. If any value above l - 6 be insisted on it will be impossible to retain the theory as above enunciated. In point of fact we may say, in anticipation of what has yet to come, that there is no modification of the kinetic theory as hitherto treated which could give a higher value for the ratio in question than If.

It follows from what has been proved that either all known gases and vapours, except the vapour of mercury, and perhaps cadmium, must be polyatomic, or else that the attempts to explain the constitution of gases by the kinetic theory must be abandoned. We must therefore proceed further to investigate the physical properties of a medium consisting of compound atoms or molecules built up of atoms in any definite arrangement, such molecules being in a condition of irregular motion among themselves, such as we have supposed in the cases of the spherical atoms hitherto considered.

It will be observed, on reference to the cases of the spheres hitherto investigated, that, whether there be forces to fixed centres in action on the medium or not, the chance of any sphere having the coordinates of its centre and its component velocities between x and x + dx, y and y-∖ r dy,z and z + dz, u and u + du, v and v + dv, w and w + dw, is proportional to e - dxdydz du dv dw, where E is the total energy, kinetic and potential, of the sphere in the state of position and motion defined by x, y, z, u, v, w.

We may generalize this proposition, and prove that when the sphere is replaced by a molecule of any shape and constitution, so as to be defined as to position and motion by r generalized coordinates q x .. .q r with their corresponding momenta p x .. .p r , the chances of the molecule having its defining variables between the limits q x and 5 , ι + t ⅛ι · · ∙P r Λτχβ, p r +dp r , or, what is the same thing, the number of such molecules at any time with variables thus limited, whether there be forces to fixed centres or not, and whether interatomic forces or intermolecular forces are or are not in action on the molecular aggregate, is proportional to ß ~ KE r dq x ...dp r , where h is a constant, the same for all molecules, and E r is the total energy, kinetic and potential, of the molecule in the free state as to position and motion, the potential energy being that of the fixed centre forces on the molecιile, together with that of its interatomic forces, in the given position.

The problem before us may be stated thus :—

A number of similar molecules possessing in the whole n degrees of freedom, where n is very large, are in motion in a region of space bounded by a material envelope, under the action either of forces to fixed centres (called external forces) or of forces between different molecules and different parts of the same molecule, as well as by forces between the fixed boundary and the contained molecules, all of them conservative, so that the total energy, kinetic and potential, of the aggregate remains always the same; it is required to find the chance of a group of any one or more molecules possessing in the whole r degrees of freedom, defined by the coordinates q x ...q r and momenta p x ...p r , where r is small compared with n, having its variables between the limits q l and q x + dq x .. .p r and p r + dp r ,

We might start with the assumption made above in the case of the spheres under central forces, that this chance must be of the form

≠ (0ι, 0 2 , &c ·) dq x ...dp r , where φ x = a x , φ. 2 = a. 2 , &c., are obtained by the elimination of t between the equations of motion of the r group under the fixed centre and boundary forces and those between its component atoms, because there is nothing in the conception of a molecule beyond that of a system with a number of degrees of freedom, and under internal forces; and in this case, considering the generality of the assumption as to the external forces, it would be impossible to conceive the existence of any general equation, independent of the time, between the variables, except that of the conservation of energy, so that the chance in question becomes

≠ (⅞) ⅛l-∙⅜w where E r is above defined, and it remains to determine the form of ψ.

If we considered a second group of one or more molecules containing s degrees of freedom (where s may or may not be equal to r, but, like r, is much smaller than n), and defined by the coordinates and momenta q r+x ...q r+f , p r+x ...p r+t , then the two groups together contain r + s degrees of freedom defined by the variables q x ...p r ±,, and since r + s is small compared with n, the chance of this group having its variables between q x and q x + dq x ...p r j rs au<∖ p r+t + dp r+t must be

0 (⅛) dq x ...dp r+e

But this chance must be equal to the chance of the r group being fixed in the state q x , q x + dq x ...p r , p r + dp r , multiplied by the chance of the remaining s group being in the state q r+x , q r+x + dq r ± x ... p , p r +t + dPr+v where the r group are so fixed.

Now to find this latter chance we observe that it is the chance of the s group being in their required limits of position and motion, when the internal forces between the r and s group become forces between the s group and fixed centres.

If the total kinetic energy of the r group in their given state be T r , and that of the r + s group be T r+t , the total kinetic energy of the s group must be 7 τ r+j , - T r .

Also if the total potential energy of the r + s group under the influence of all forces be χ r+r this is made up of—

(1) χ r , the potential energy of the r group to fixed centres, and of its internal forces;

(2) χ t , similarly taken for the s group; and

(3) r ‰> the potential energy of the r and s group forces.

And when the r group is fixed the potential energy of the s group is reduced to (2) and (3), or is χ r+ , - χ r ∙

Therefore the chance of the s group having its variables within the required limits when the r group is fixed must be ≠ ( E r+.~ E r) dq r ^ 1 ...dp r+r Therefore

≠(^r) ≠ ‰ 4 - E J dq 1 ...dp r j,,= ≠ (N r+1 ) ⅛...<⅛ r+ ,, or ≠ E r ) ≠ (E r ±,-E r ) = ψ (E r+t ).

Therefore ψ (x)=e Cx — e~" x suppose.

And the chances of the r group having its variables between the limits q 1 and q x + dq v .p r and p r + dp r must, in the state of permanent or stable motion, be proportional to e - hE r dq x .. .dp r , which was to be proved.

Supposing now that the aggregate of molecules under consideration consists of a number of sets of similar molecules, the number of molecules in one of these sets being N, where N is very large, and suppose that each of these A r molecules possesses σ degrees of freedom defined by the coordinates q x ...q σ with the momenta p x ...p σ an d th a t mass is m. Three of these coordinates may be taken as the rectangular coordinates of its centre of mass, in which case the corresponding momenta will be mu, mv, mw, where w, V, and w are the component velocities of translation of that centre of mass. Then in this case, if q i ...q σ p i -∙-P σ be the remaining coordinates and momenta of the molecule, the chance of the molecule’s variables being within the limits x and x + dx...p σ and Pσ + dp σ will be proportional to

-A(X+∕) , , , , , -A^(u2+v2+w2)

e dx dy dz dq i ...dp i e 2 du dv dw...(Y), where T, the kinetic energy of the molecule, is equal to

-0^(u 2 + υ 2 + w 2 ) +f, where f is a quadratic function of the p’s, having as coefficients known functions of the q’s.

If we integrate the expression (I) for all possible values of x, y, z, q i -..q σ Pt∙∙∙Pσ we °fitai∏ an expression of the form

-⅛21 c s Be 2 dudvdw (II), where B is independent of u, v, and w, and c 2 =w 2 +v 2 + u∙ 2 . From the form of (II) it follows, exactly as in thecasesofthe elastic spheres, that the chances of all directions of the velocity of translation of a molecule are equal, that the mean velocity and mean square velocity of translation of each molecule are 2 * and - 3 - ∖fnih, respectively, and that the mean kinetic energy of translation is —-, and the same for a molecule of any set.

Again, if T be the mean total kinetic energy of the molecule, then _ I ∕’/ ...T.e~ jl (χ+T) dx...dp σ τ = ∣∣∣ ... t -^ + τ> dsc ... ir ∕ σ∏) i and if we evaluate this expression, paying attention to the form of T as a quadratic function of the p’s mentioned above, we shall find for (III) the expression

It follows from this result that each additional degree of freedom of the molecule increases the mean total kinetic energy of the molecule by the quantity À, which is the mean kinetic energy of translation parallel to any one of the axes, and that the total kinetic energy is proportional to the number of such degrees of freedom.

If, again, we integrate the expression (I) for all values of the momenta, we obtain an expression of the form Ce~ h X dx dy dz dq i ...dq σ (IV), where χ is the potential energy of the molecule due to fixed centre and to interatomic forces in the position defined by x, y, z, q i ...q σ . The dimensions of the molecule are so small that we may regard forces from each fixed centre on different parts of the molecule as parallel and equal and functions of the distance of the centre of mass from that fixed centre, so that, if the part of χ arising from these fixed centre forces be called χ 1 , χ 1 will be a function of x, y, z, and of these variables only, the remaining part of χ (arising from interatomic forces), which may be called χ 2 , will be a function of the σ-3 variables q i ...q σ

If in (IV) we write χ 1 + χ 2 for χ, and then integrate for all values of q i ...q σ we obtain an expression of the form dx dy dz (V), where D is independent of x, y, z, and therefore p the density of the Λ r molecule matter in the neighbourhood of the point x, y, z, is mDe~ a Xj.

From these results all the propositions proved above with reference to the aggregate of elastic spheres or monatomic molecules, as to the correspondence of the physical properties of such an aggregate with those of gases as indicated by the gaseous laws, may be deduced also for this aggregate of polyatomic molecules. So that if T be equal to —, or the mean kinetic energy of agitation of any one of the aggregate of moving molecules, if v be the volume occupied by unit of mass, r the number of molecules in unit of volume, and m the mass of each molecule, we have, exactly as in the case referred to, mr=l, pv=l, o and pv= — τT.

3

We also get the ordinary hydrostatical equations ⅛=pX, ⅛=pY, d ½=f>Z dx dy dz from this expression for p combined with the equation ρ = mDe ~ remembering that

⅛1= - mX, ⅛ = - mY, ⅛⅛= - mZ, dx dy dz whence the coincidence of the physical properties of this aggregate of polyatomic moving molecules with those of a gas, on the assumption that the temperature represents the mean kinetic energy of agitation, is at once apparent.

It can be shown also that the aggregate of moving molecules, such as we conceive a gas to be, possesses another very important physical property which, by its analogy to the second law of thermodynamics, affords additional evidence of the relation between the phenomena of heat and those of aggregates in some kind of motion,— the property in question being that, if in any aggregate of moving molecules the mean kinetic energy of any one of them be called τ, and if δQ be an increment of energy imparted to the aggregate from without, then — is a perfect differential.

If to this aggregate we apply a certain small quantity δQ of heat or energy from without, and if δτ be the increase of the mean kinetic energy of agitation when the volume is unaltered, then this constancy of volume prevents any of the energy δQ from being absorbed in doing external work; but it is conceivable that the increase of r may cause such a change in the average state of the molecule as to produce a variation δχ in the mean potential energy of the molecule, δχ being proportional to δr.

Therefore δ <2=r-f d -l + ⅛I.∣ δτ. t dτ dτ J

But <r _ σ 3 _ cr r .

^ 2Λ^ 3^ ’ 2A~^3^ τ, therefore δQ = r i — + — ) δr.

13 dτ J If the volume vary by δυ, the pressure being constant, then we must add external work, or pδv, to the energy absorbed, so that if the whole external energy now applied be δ'Q, and the increase of temperature δτ be the same in both cases, we have r (— + δτ + pδυ δ'Q~ ∖ 3 dr) >Q ,(' t 4ft fc

∖ 3 dτ )

But if p be constant, then as before j>δr = 2 δτ, and therefore the ratio of δ'Q to δQ, or of the two specific heats at constant pressure and constant volume respectively, becomes ff + 3⅛ + 2 aτ

. + S⅜ dτ

~ is unknown in all respects except that it must be positive; also we know that σ must be integral and not less than 3; if we denote dy

3 —i by e we have for the ratio dτ j <τ + 2 + e σ + e or 1 +

σ + e which, with the necessary limitations of σ and e, cannot be greater than lj or 1 ∙6, and in this limiting case the gas must be monatomic.

If, therefore, any value above 1 '6 of the ratio for mercury vapour be insisted upon, the theory must be abandoned so far as present investigations are concerned. If, however, the difference between 1 '6 and any higher value given by the observations be regarded as within reasonable limits of experimental error, this value for mercury vapour, a gas which on chemical grounds is regarded as monatomic, may be viewed as confirming the theory, at least pro tanto.

If two spherical atoms were united by a rigid rod to form a molecule, such a molecule would have five degrees of freedom and the specific heat ratio would in this case be If, for e would then be zero. This value has a plausible approximation to the observed value l - 408 of the ratio in a great number of two-atom gases, such as hydrogen, nitrogen, oxygen, and others, but all observations agree so completely in the ratio 1 - 408, or from 1 ’ 405 to 1 - 408, that it hardly seems reasonable to regard the difference ’ 008 as within the limits of experimental error, unless, indeed, we had grounds for suspecting a tendency to excess in all the methods employed for the determination of the ratio. But there are other difficulties more formidable still, arising from the spectroscopic properties of heated gases. The light emitted by such gases, so long as they are of no great density, never presents a continuous spectrum, but a spectrum consisting of bright lines with intervening dark spaces. Thus the spectrum of hydrogen gives thirty-two bright lines, that of mercury vapour six lines, that of nitrogen eighteen, and so on. So long as light is regarded as an energy intercommunicable with heat, and light of definite refrangibility is refeιred to vibrations of given period, we must regard these discontinuous spectra as connected with, and arising from, vibrations of determinate periods in the molecule of the heated gas. And if a gas such as hydrogen or nitrogen be constituted, as we are supposing, of an indefinite repetition of similar molecules, it must follow that such molecules must be capable, at any rate when not too closely packed together, of as many independent vibrations as there are bright lines in the spectrum; that is to say, in addition to the three degrees of freedom arising from motion of translation in solid space, each molecule must possess as many additional degrees of freedom or possible relative motion of its parts as are indicated by the number of spectrum lines. The degrees of freedom corresponding to motion of translation cannot well contribute anything to these luminous vibrations owing to their assumed irregularity and independence of any law; but it is otherwise with the internal or relative degrees of freedom of each molecule, for, unless the gas be very dense, we may easily conceive a sufficient interval of time between one encounter and the next of any molecule with another for very many vibrations, each according to its own law, to take place in the relative positions of different parts of the molecule. At each encounter the whole molecule would be roughly shaken, and when the encounters increased in frequency the vibrations would become irregular and the spectrum would degenerate into a general diffused light of no definite refrangibility, just as music degenerates into ordinary noise. And this is exactly what occurs in the spectra of dense gases.

To bring the theory, therefore, into agreement with observed phenomena, we require very many more degrees of freedom in each molecule than could possibly be assigned to it in accordance with the observed value of the ratios of the specific heats,—mercury vapour, for example, admitting with difficulty the minimum number of three such degrees, as we have just now seen, while its spectrum would require at least nine. And the difficulty increases as we pass to hydrogen and other gases.

We might perhaps conceive, with the view of possibly explaining this difficulty, that there were in all gases a number of composite molecules with many degrees of freedom mixed up with the other molecules with three or five such degrees, but in so small a proportion to these molecules that their presence produces no appreciable effect upon the specific heats; or, since we have no experimental determination of the specific heats of gases at light-giving temperature, we might, at least until such experimental determination has been arrived at, conceive that our atoms may change their constitution under increased temperature, and become themselves capable of vibration. There is nothing in the conception of an atom as we are considering it which is really inconsistent with such an hypothesis.

Certain observed phenomena accompanying dissociation and combination give rise to other difficulties in the way of the acceptance of the kinetic theory in addition to those arising from the equal distribution of mean kinetic energy just now discussed. For when nitrogen and hydrogen, for example, are mixed in proportion to form ammonia it is observed (1) that at ordinary temperatures they do not exhibit the slightest tendency to combine directly with each other, while, on the other hand, (2) ammonia at ordinary temperatures does not exhibit the slightest tendency to decompose into nitrogen or hydrogen. But ammonia when subjected to certain very high temperatures becomes partially decomposed—that is, becomes a mixture of so many parts of ammonia and of so many other parts of nitrogen and hydrogen in the proportions to form ammonia; and if the temperature be high enough the decomposition may be complete. But, in accordance with the kinetic theory, the conditions, whatever they may be, which at high temperature cause the ammonia to decompose, must sometimes occur to individual molecules at ordinary temperature, because temperature, as we understand it, merely indicates a certain quantity ot kinetic energy, and therefore in a gas, however cold, there will be always some molecules in a state for dissociation; and this dissociation having taken place can by (1) never be compensated by recombination; therefore dissociation should be going on in ammonia at all temperatures, and this result is contrary to the observed phenomena (2). It might possibly be conceived, as away of meeting this last-mentioned difficulty, that the dissociation attendant upon high temperature—that is, upon an average large molecular velocity of translation—requires that there should be a fairly rapid repetition of encounters among molecules moving with dissociation velocity to ensure the production of dissociation, and that in the case of a gas at low temperature, or small average velocity, the chance of two molecules encountering one another at high velocities is small, and the chance of any molecule meeting with any rapid succession of such encounters is practically insensible, and therefore that the dissociation spoken of really never takes place.

As above stated, we conceive that in any gas at ordinary pressure and temperature the intermolecular forces are very small in the aggregate—that is, in Clausius’s language, have a very small virial,— by which is understood, not that the forces themselves, where acting, are small, but that, considering the whole aggregate of molecules at any instant, there are very few pairs near enough to each other to exert any appreciable force on each other. Or, if we could watch any individual molecule for any time, we should find that during by far the greater portion of the time it was sensibly free from any action by surrounding molecules. The distance traversed by the type molecule between the instant when it passes out of the sphere of action of one molecule and the instant when it passes into the sphere of action of the next—that is, from one encounter to another—is called its free path.

We may find the chance that a molecule starting from any point with velocity ω in a uniform gas shall have free path between x and x + dx from that point.

If a be the chance for such a molecule of free path at least unity, then α 2 is the chance of a free path at least 2. Hence the chance of free path at least x must be of the form a x .

Following the method employed by O. E. Meyer, let us write this in the form X e I > where therefore , _ 1 # log a ’ then the chance of free path x + dx is x + dx e I .

The chance that such a molecule shall have its first encounter between x and x + dx is the difference of these two expressions; that is, ≈ , e t —.

I

This is the chance of a free path between x and x + dx.

The mean free path for such a molecule must then be f 0 ° x C° o - x i

I e I -^-dx÷ I e I ~dx=l. Jθ l Jo l x

This is the meaning of the constant I in e I . But if we denote by B the number of encounters which a molecule moving through space with velocity ω experiences on the average per unit of time, τ > ω 12?

I I ω

Hence the chance for such a molecule of free path between x and x + dx is P -- — e ω dx, ω with the above definition of B.

The chance of a molecule whose velocity is ω having free path x is of course the same as the chance of its free path having the duration —. If t = —, the chance of duration between t and t + dt ω ω is thus E — e~ Bt ωdt ; or Be~ βt dt. ω

Meyer determines the value of B, if the molecules be spheres, in the form

* = λwω ∣ 1+ ∣jL 8 ^-jL 2 . s I. 5 ^ + (-) n - 1 γ 1 1 1— (ω 2 Λ) n ... I

[n 2n - 1 2n + 1 ' J ’

2 ~

where Q == —τ==∙, and s is the sum of the radii of two molecules. V τrh

It will be observed that the series converges very rapidly if ω 2 A is less than unity, the successive coefficients being 1 1 1 1 . 1 t

— · “ — « ^^F " « — — · 4 - — . &c.

3 30 ’ 210 ’ 1512 ’ 11880’

Having found B for the number of encounters experienced per unit of time by a molecule having velocity ω, we have for the average number of encounters experienced by any molecule per unit of time, which we denote by C,

4 s f 0 °

~ y Jh*J e~^ ιωi ω 2 Bdω.

From which Meyer deduces

C=2^.2W h

= Ω∖∕2. Λ τ τrs 2 .

Hence the mean value of the free path for all molecules, irrespective of velocity, is I.= 2- =— .

j, C τr√2Λ 7 s 2

Thus the kinetic theory of gases presents to us the conception of apparently perfect rest, as the result of motion irregular in detail but permanent and stable on the average. Whatever difficulty may be felt at first sight in the acceptance of this theory in the case of a medium at rest is greatly enhanced when we pass to the contemplation of a disturbed medium like a mass of gas through which a wave of sound is passing. In our ordinary investigations of such a disturbance the gas is treated as a continuous body, subjected to small relative motions of its parts, accompanied by corresponding variations of internal pressure. When a disturbance or a local condensation or rarefaction is set up in any portion of this gas we calculate the resulting effects by the well-known equations of sound motion. But on this kinetic theory the medium is supposed to consist of a number of discrete masses—elastic spheres or the like—which preserve the physical properties of the medium merely by the recurrence of their mutual collisions, such collisions obeying no law in individual cases, but preserving a certain average uniformity in the motion of the whole aggregate; and we need some further investigation to assure ourselves of the applicability of the ordinary treatment of wave motion to such a medium.

Now we observe that the physical properties of our medium, so far as the relation between pressure, density, and temperature is concerned, merely require that the temperature be measured by the mean total kinetic energy of translation, and that the mean kinetic energy of translation parallel to any fixed line be equal to one-third of the mean total energy of translation. If the molecules constituting any portion of this medium were animated by a common velocity or acceleration, the physical properties of this portion would be similarly determined by the velocities and kinetic energies relative to the common motion. When the distribution of such relative velocities is stable or permanent, the average relative kinetic energy in any fixed direction is one-third of the average relative total kinetic energy, such property constituting normal distribution.

Suppose that in any portion of a medium, consisting of equal elastic spheres, this distribution has been disturbed—that is, ∑mu 2 , ∑vnv 2 , and ∑znw 2 are unequal. If V were the relative velocity of any pair of spheres after such disturbance and before they collide, and 0 the angle between V and the common normal at the point of impact, then the normal and tangential relative velocities before impact are Vcosθ and Fsin 0, and after impact they become - Γeos θ and Fsin 0 respectively. The relative velocity after impact, resolved in the direction of relative velocity before impact, is therefore — V cos 2 θ + V sin 2 0, or - Pcos20; and the chance of 0 being between 0 and θ + dθ is sin 2θ dθ.

Therefore the average square relative velocity resolved in the original direction becomes after impact π _ V 2 f 7 cos 2 20 sin 20 dθ, or Jθ 3

The relative velocity after impact in the plane of V, and the normal perpendicular to the direction of V before impact is V sin 0 cos 0 + V sin 0 cos 0, or V sin 20.

And, if a fixed line be taken in the plane perpendicular to V, the average value of the square of the relative velocity after impact, resolved parallel to this line, is

_ _ IF

F 2 P 7r f 7 . F a w— · I I sin 3 20 cos 2 <∕> dθ dφ, or — as before.

z7r J 0 J 0 3

Hence we conclude that, in whatever manner the distribution is disturbed in any portion of the medium at any instant, it will, for all those pairs of spheres which within any given interval encounter each other, have assumed the normal distribution after that interval.

If τ denote the average time between two collisions for any given sphere, the chance that this sphere shall continue for any time t _ t_ free from collisions is, as we have seen, e τ .

If, therefore, Z>be the number of spheres within any region whose total relative velocity is between w and w + dw, but so distributed that the mean square of their relative velocities along any fixed line is not —, then after a time t considerably greater than τ, say o ten times τ, the number of the D spheres which have escaped collision will be utterly inconsiderable, and the distribution will have become normal throughout the region.

Suppose, for instance, that a sound wave is passing along a tube filled with air, the air in the tube is, at any instant, in a state of alternate compression and rarefaction, as at C, B, C, B above.

If the note sounded be (say) 500 vibrations per second, the length of the wave CB is about ⅛⅛% 0 feet, and the time taken by the wave in traversing that distance is about ∙^J^th of a second.

The air in any section of the tube near P has alternately a small positive momentum and an equal small negative momentum, the reversal taking place in every κ ⅞- σ th of a second; also the same cause which produces the average momentum in either case disturbs the distribution of energy among the x, y, and z directions, i.e. , it is always producing an excess or defect in mu 2 above or below that of mv 2 and mw 2 . By what has been proved above, this abnormal distribution of energy becomes inappreciable, owing to molecular collisions in a time considerably less than τ ∙ δ ⅛ 7j th of a second—in fact, in about τρ∙oVoτjth of a ∙ second, when the value of T for atmospheric air is considered. It is therefore legitimate, in calculating the velocity of sound in air (at least on the elastic sphere hypothesis), to regard the distribution as always normal in any section of the tube, the air in that section or in any elementary portion of it possessing, as a whole, any given velocity or acceleration, estimated as if we were dealing with a continuous mass.

Diffusion of Gases.

If any further light is to be thrown on the physical nature of a molecule from investigations, experimental or analytical, concerning gases, it will most probably be by means of experiments on the diffusion of gases, or else on the internal friction or viscosity of gases, and the comparison of these results with those obtained analytically by the methods of the kinetic theory. Such investigations have been undertaken experimentally by Graham, Loschmidt, Maxwell, O. E. Meyer, and others. An account of them will be found in O. E. Meyer’s work above referred to. The same problems have also been discussed analytically by Maxwell, and by Stefan, O. E. Meyer, and Boltzmann in the treatises referred to below. We proceed to give a short account of Meyer’s results.

The term “diffusion” has sometimes been applied to the process by which a gas passes through a porous diaphragm. This, however, is now generally denominated transpiration. It has also been applied to the expansion of a gas into vacuum, as on the removal of a diaphragm separating the gas from an exhausted receiver. This is now generally denominated free expansion. We shall understand, as is now usual, by the term diffusion the process by which, when two or more gases are mixed throughout any space in different proportions at different points, but so that if all molecules were of the same gas the whole would be in equilibrium, the different gases pass through each other and tend to equalize the proportions at all points in the space.

Suppose, for instance, a tube containing a mixture of two gases, A and B, at constant temperature and constant pressure of the combined gases throughout the tube and subject to no forces, but the density of gas A increasing and that of B diminishing from one end of the tube to the other. Let the axis of the tube be taken for the axis of x. If N a be the number of molecules of gas A, and N b the number of molecules of gas B in unit volume, we have, owing to the constant pressure and temperature at all points of the tube, N a + N b = N, a constant. But at a given instant N a and N b at any point are severally functions of x. It will be found that under these circumstances more molecules of gas A pass through any section of the tube, which may be in the plane of yz, in one direction, say from left to right, than in the opposite direction. On the other hand, more molecules of gas B pass from right to left than from left to right. And this will go on till the mixture becomes uniform throughout the tube.

The investigation of the rate at which the unequal distribution tends to equalize itself in this simple case—that is, the excess of the number of molecules of gas A which cross a section of the tube from left to right over the number crossing in the same time from right to left—is the problem of diffusion. We give the results obtained by O. E. Meyer as follows:—if the molecules of the two gases had the same mass and dimensions (to put an ideal case), then the excess of molecules of either gas passing through the section in one direction— that is, the stream velocity—would be -θ ωl, where I denotes the mean free path for a molecule having velocity ω, and ωl is the average value of that function for all molecules of the gas.

When we come to deal with two gases, the molecules of one not being of the same size and dimensions with those of the other, we shall find that, in the absence of any common velocity of the two gases at the plane of yz, more, or fewer, molecules of gas A would cross the plane per unit of time from left to right than of gas B from right to left, because, assuming constant pressure and temperature of the mixture at every point in the tube, the number of molecules of the two gases combined must be the same at every point—that is, N a + N b =N, where Wis constant. Hence dN a _ dN b dx dx ’

Now the excess of molecules of gas A coming from left to right per unit of time is -∣- and similarly the excess of mole cules of gas B crossing from right to left per unit of time is 1 dN b — j . . . , ,

*3^ f ∣ a , <A b , 11 we now distinguish by suffixes a and b quantities relating to the two gases respectively. Here l a and l b are mean free paths for velocity ω of the two kinds of molecules through the mixed gases, and ωl a is not generally equal to ωl b . Hence the total number of molecules crossing the plane from left to right exceeds the number coming from right to left by -y (ωl a - ωZ ft ).

Meyer here assumes that the combined gases have a common velocity - y ( t ->l a - ωl b ), and that such common velocity will not affect the relative motion of the molecules. On that hypothesis the rate of diffusion can be calculated as follows. The proportion of the stream of the combined gases which consists of molecules of gas A is

_ AΓ a 1 dN a f - , R r a + N b 3 dx < ωι a ~ ωi ι>i ·

Hence the total surplus number of molecules of gas A passing through unit area of the plane per unit of time is =l¾{.v⅛-, + ⅛}

1 dN a , _ _ 1

3 (ΛΓ β + N b ) dx ωl * + jv ≈ ωΖ 6> ’

or 1 dN * fκr ~1 , v - 7 1

3N dx ωla + 3 a ω i 1 ‘

The expression

3^-{W 4 U α +W α ω ∕ d }

is defined to be the “coefficient of diffusion” of gas A into gas B. It is evidently the same as that of gas B into gas A. The Relation of the Coefficient of Diffusion to Density.— It can be shown that l ω , the mean free path for a molecule having velocity ω, is for any single gas inversely proportional to the density, and for any mixture of gases inversely proportional to λ, the aggregate volume occupied by matter in unit space. Hence, in the expression -y ωl ωl is inversely proportional to the density, or to λ, as the case may be.

Now the rate of diffusion on this theory depends upon A-

3 dx

Hence, given the absolute increase of density of a gas per unit of dN length, that is, given the rate of diffusion ought to vary inversely as the density of the combined gases. On the other hand, given the proportional increase of the density, or the rate of diffusion ought to be independent of the density, because in that case dN — varies directly, and ωl inversely, as A r . The analytical result, dN that at given temperatures, and given the absolute value of , the rate of diffusion is inversely proportional to the density of the gases agrees with the experimental results obtained by Loschmidt for carbonic acid gas and air, carbonic acid gas and hydrogen, hydrogen and oxygen. Relation of the Coefficient of Diffusion to Temperature.— The coefficient of diffusion varies directly as the square root of the absolute temperature, for

- ω 2 , R=2Ns 2 ^ 7 ff e -bω2 + 2hω 2 + Ι i ω *d μe -μ 3 l. l "fh *∙ ω∖∕Λ Jo ‘ or, if ω∖∕h=y, τ> f -⅛ s . 2 3∕ 2 + l [ y -μ2j Ί

√Λ I y J o )

Hence 1 - , l ∖

B *JιrNs i ' f ' y ^ , where ≠ denotes a certain function, and

(?) “4><- λ "⅛

f° 0 v 4 1 1 ∕∙° 0 1

oc h dye~y 2 λ 2 s ≈ ~=~ dye~v 2 y 2 ∕y ∞ -=-. Jo λ x V h J 0 √ h

This analytical result also agrees fairly with Loschmidt’s experiments above referred to.

Friction or Viscosity of Gases.

Suppose two layers of gas separated by an imaginary plane, similar in all respects except that the molecules of one have a small common momentum in a certain direction parallel to the plane. We may take the imaginary plane for that of yz, and the average direction of motion of the molecules on one side of the plane, e.g., the left-hand side, for the axis of y, the molecules on the right-hand side of the plane having no average momentum. Then the molecules crossing from the left to the right side carry with them an average momentum in the direction y, and so tend to impress the right-hand stratum of gas with that momentum. On the other hand, the molecules of the righthand stratum crossing the plane into the left-hand one have, relatively to the molecules in the latter, an average momentum in the opposite direction, and therefore tend to diminish the average momentum of the left-hand stratum.

Hence, if we attempt to cause one stratum of gas to pass over another in parallel planes, we experience a resistance due to the interchange of molecules between the portions of gas separated by the plane. This is in some respects analogous to sliding friction between solid bodies, and is called by German writers the “friction” (Reibung), by Maxwell and others the “viscosity,” of the gas. Meyer investigates this effect of friction in a manner somewhat similar to that employed in case of diffusion, and obtains for the coefficient of viscosity ⅜ nιNωl. Relation of the Coefficient of Viscosity to Density and Temperature.— The viscosity of a gas is independent of the density, being, according to O. E. Meyer, 1 ~ ωl. Now, for any one gas, l ω is, as we have seen, inversely proportional to the density, and therefore <cl is inversely proportional to the density. On the other hand, N is directly proportional to the density. Hence the viscosity is independent of the density. This agrees with the result obtained by Maxwell from the kinetic theory in 1860, and with the results of experiments by Maxwell and Ο. E. Meyer. Also, experiments by 0. E. Meyer and Springmühl on the transpirât ion of gases show that the times in which two different gases under similar circumstances flow through a tube maintain the same constant ratio to one another. As in the case of the coefficient of diffusion, ωl is inversely proportional to the square root of the absolute temperature. As both the coefficient of diffusion and that of viscosity depend on the same function ωl, it should be possible from experiments on viscosity to determine the rate of diffusion. Experiments with this object have been conducted by Stefan ® with very satisfactory results, his calculated values for the coefficient of diffusion agreeing very closely with those determined by Loschmidt’s direct experiment.

We have given the above results for the coefficients of diffusion and viscosity from 0. E. Meyer’s work, because his method has met with very general acceptance. It has been shown, however, by Boltzmann / that the method is incomplete. Meyer’s results can only be obtained on the assumption that the molecules of a gas undergoing diffusion or internal friction, which have any given velocity, as w, are movingwith that velocity in all directions indifferently. We may calculate the number of molecules having velocity w that pass through a given plane during a short time dt, starting from encounters at any given distance from the plane. If we assume that the molecules, issuing from such encounters with velocity w, move indifferently in all directions, we obtain Meyer’s result. This assumption is true only of a gas at rest—that is, having no velocity of translation—so that our result so obtained would express, in case of diffusion, the rate at which two gases begin to diffuse, if given at any instant both at rest—that is, with no stream velocity —but mixed in unequal proportions in different parts of space. In any actual case of diffusion, either of the two diffusing gases acquires a small velocity of translation. If we take this velocity into account in calculating the number of molecules of the gas passing through a plane, according to Meyer’s method, we shall find that it introduces two new terms, one of which, when the motion becomes steady, is equal and opposite to the result obtained by Meyer. This is proved by Boltzmann in the case of viscosity in the treatise· above referred to. The same proof is easily applied in the case of diffusion.

Stefan's Method. — Stefan regards the two diffusing gases as having small velocities of translation, or stream velocities, u 1 and t⅛, in opposite directions, so that the molecules of one gas, of mass m 1 , have an average momentum wι 1 u 1 in direction from left to right, and those of the other gas, of mass m 2 , an average momentum τn 2 u 2 from right to left. By virtue of encounters between the two sets of molecules, each gas is always imparting to the other a portion of its own average momentum, and receiving from the other a corresponding momentum in the opposite direction. The momentum so transferred or interchanged is what Stefan calls the resistance which one gas offers to the other’s diffusion. In this investigation Stefan assumes that all classes of molecules of one gas, whatever their molecular velocity in space, have the same average velocity in the direction of diffusion—that is, the same stream velocity—so that the motion of the molecules of a diffusing gas would be exactly represented by considering the molecules of a gas at rest—that is, with only its molecular velocity—at the same temperature and pressure, and then giving to each molecule the additional common velocity u in the direction of diffusion. Boltzmann, however, shows that, in order correctly to represent the motion of the diffusing gas, we must impart to molecules having different molecular velocities independent of direction different common velocities in the direction of diffusion. And it will be found that the resistance of the gases is sensibly modified by this property.

The complete solution of the problem,—that is, the determination of u as a function of w, on the hypothesis that the molecules are elastic spheres,—is difficult.

If we assume molecules to be centres of force varying inversely as the nth power of the distance, so that the force at distance r is --, where μ. is constant, we obtain the following result. We assume the molecules of gas A whose absolute velocities are between w and w + dw to have an average stream velocity u in direction of the tube, where u is a function of w. Then, if the terminal condition at the ends of the tube be maintained constant, we obtain an equation of the form p dN j , _4 a-8 c , rnpn 2 τN t N t N dx 3n — 3 ' nι 1 + m 2 unit volume multiplied by the average value for all molecules of gas A of n-5

u V n ~ 1 , where V is the relative velocity of two molecules, one taken from each gas, and C is a constant, and wι 1 , m 2 the masses of the molecules of gas A and gas B respectively.

By making n infinite we obtain the result for elastic spheres: in n-5

that case V n ~ 1 = T r , and the problem is to find the average value of u V.

Since p varies as the absolute temperature, and the average value of V varies as the square root of the absolute temperature, we may infer that the average value of u— that is, the stream velocity—will vary approximately as the square root of the temperature, as it appears to do from experimental evidence. If, on the other hand, n=5, V disappears, and —¾ = I∙ I∏ ibis case the analytical determination of u presents no difficulty; but in the result the stream velocity varies as the absolute temperature, which accords less satisfactorily with experiments.

On Molecular Dimensions.

Many attempts have been made in recent years to form an estimate or conjecture, more or less accurate, of the numerical value of the dimensions of a molecule and the absolute force between molecules.

In accordance with the view of the subject considered in this article, we are here concerned with such speculations only in so far as they are founded upon the kinetic theory of gases, or supported by it. The phenomena of diffusion and viscosity especially have afforded grounds for estimates of molecular dimensions.

It is first necessary to define what is meant by the dimensions of a molecule. Regarded as an elastic sphere, it has dimensions with the conception of which we are familiar. It is not, of course, seriously contended by any physicists that the molecules of a gas are actually hard elastic spheres, exerting no force on each other at any distance greater than that of actual contact, and then an infinite force. It is necessary to conceive the forces as finite, although they may diminish so rapidly with the distance as that the motions of molecules in the aggregate differ little from what they would be if the molecules were ideal elastic spheres. Nevertheless, they must be finite forces; and, that being the case, it is difficult, if not impossible, to frame a definition of the boundary of a molecule, except as a certain surface at which the forces acting between the molecule in question and other molecules attain a certain value.

If, for instance, we were to regard a molecule as a centre of force, exerting an attractive force and a repulsive force r j, we might μ μ' define the molecule to be a sphere of radius a, such that ^ 4 =-^. In like manner, regarding a molecule as a centre of force, repelling according to the law of the inverse fifth power of the distance, we might define the magnitude of a molecule as a sphere of radius equal to the least distance to which two molecules, whose relative velocity is equal to the mean velocity of the centres of force, approach each other in a direct encounter.

If on any hypothesis concerning the nature of a molecule, or the law of force which acts during encounters, we can calculate the coefficient of viscosity or diffusion analytically, a comparison of the analytical result with results obtained by experiment may afford the means of determining the absolute numerical value of the constants used in the analysis. For example, if we consider the molecules as elastic spheres, and if we consider for a moment Meyer’s results as correct, or approximately correct, the coefficient of . Nm — viscosity for any single gas can be put in the form θ ωi ω , where N is the number of molecules in unit_of volume, m the mass of a molecule. Now, for every value of ω∕ ω , the mean free path of a molecule with that velocity is equal to √Λ∙ Nπ ip , w ^ ιere s * s twice the radius of a molecule multiplied by a numerical factor which can be determined to any required degree of accuracy. j Also at given temperature and pressure the numerical value of √Λ is known. It follows that we can calculate the numerical value of the coefficient of viscosity by analytical methods in terms of Nτrs 2 (j to any required degree of accuracy. Let it be · If by experiments on viscosity we can determine the numerical value of the same coefficient in the form C ’ 1 , when C ’ 1 is a mere numerical quan- C C tity, we have immediately the equation C l > or = ^q · This gives in absolute numerical measure the value of Nπs 2 , or four times the sum of the great circle areas of all the molecules in unit of volume, supposing them to be spherical. If we attempt to use the coefficient of diffusion instead of viscosity in this method, we are met by the difficulty that the analytical result contains now two unknown quantities instead of one—namely, the radii of the respective molecules of the two gases in question. If this difficulty be got over by a comparison of results obtained in different experiments, the greater certainty attending the observations on diffusion might perhaps compensate for the additional mathematical difficulty, and render diffusion at least equally trustworthy with viscosity as a method for estimating molecular dimensions. Again, on the hypothesis of repulsion between molecules according to the law of the inverse fifth power of the distance, we can calculate analytically the rate of diffusion between two reservoirs connected by a tube as above described, the result containing only one unknown constant, viz., μ, the constant of absolute force. Comparing the analytical result with the results of experiments on diffusion through such a tube as above described, if we find them capable of being harmonized by attributing any numerical value to μ, we should have good reason for concluding that the law of force assumed is to a certain extent at least the true law, and that the particular value of μ is that which harmonizes the analytical with the experimental results. And the determination of μ, the absolute force, corresponds to, or indeed is, the determination of the size of the molecule.

Until all the mathematical hypotheses have been fully developed, no very great reliance can be placed on the results of such comparisons, even assuming that the experimental results themselves are to be depended upon. However valuable the experiments may be for other purposes, they are not valuable for the purpose of determining molecular dimensions until our mathematical analysis is sufficiently advanced to enable us to interpret the experiment. At present it is perhaps impossible to deduce from the experiments any other result bearing on this question than that the coefficients of diffusion and viscosity increase with increasing temperature, and probably contain an important term proportional to the square root of the absolute temperature. If, indeed, it can be shown that that is the only term, and if it can be also shown that the density of one of . two diffusing gases in a tube through which steady diffusion is going on tends to vary in geometrical progression, then the analysis will lead us to the conclusion that molecules of gases behave in their physical relations to each other as if they were elastic spheres.

The following method has also been suggested for estimating the magnitude of molecules of mercury. Mercury is regarded by most chemists as monatomic. Let us assume that its molecules are conducting spheres; on that assumption we may calculate the specific inductive capacity of mercury vapour on Faraday’s hypothesis to be ^1 , where λ is the ratio which the aggregate volume of all the spherical molecules in unit volume bears to unit volume. If now K, the specific inductive capacity of mercury vapour, can be deter-

1 I 2λ

mined experimentally, the equation K~ _ χ affords a ground for estimating the value of λ,—that is, the aggregate volume of the molecules.

Another method, originally proposed by Van der Waals, is founded on the small deviations from Boyle’s law observed in all gases. Suppose a vessel of volume V containing a number ∆ r 1 of elastic spheres, each of mass m, moving with a certain average kinetic energy. Let 7> 1 be the pressure. Let a second class of elastic spheres, in number A 7 2 , each of the same mass m as the former class and having the same average kinetic energy, be introduced into the vessel. If the second class of spheres could freely penetrate the first, and vice versa, so that there should be no restrictions on a sphere of the first class and a sphere of the second being in the same place at the same time, then the pressure on the walls of the

JV +A r o vessel would be increased in the exact proportion ——'∙ Bθylθ ’ s law would be exactly fulfilled. But if the spheres cannot penetrate each other, the volume occupied by the second class of spueres 4

is not V, but V- -y A 7 1 τrr 3 , if r be the radius of a sphere of the first class. Consequently, the pressure due to the second class of spheres is rather greater than it should be, and there is a small deviation from Boyle’s law. Van der Waals treats the pressure as proportional to the number of encounters, and therefore inversely proportional to the mean free path, which is evidently diminished by any increase in the magnitude of the spheres, and diminished more than in proportion by any increase in the number.

(H. W. W.-S. H. B.)

Chemical Aspect.

The word Molecule is used by chemists to express the unit of a pure substance, that quantity of it which its formula ought to represent. What this quantity is, in any particular case, must be ascertained by studying the chemical actions by which the substance is produced and the chemical changes which it undergoes. We may give one or two illustrations to show how this can be done, as well as to indicate the limits within which these methods can be applied.

The formula usually assigned to acetic acid is C 2 H 4 O 2 ∙ This agrees with almost all the chemical actions in which it takes part. Thus, one quarter of the hydrogen is replaceable by other metals, as in C 2 H 3 KO 2 , &c.; and one, two, or three quarters of the hydrogen can be replaced by chlorine. There must, therefore, be four (or a multiple of four) atoms of hydrogen in the molecule. Similarly, half of the oxygen can be replaced by sulphur, and one-half of the oxygen along with one-quarter of the hydrogen can be replaced by chlorine. There must, therefore, be two (or a multiple of two) atoms of oxygen in the molecule. Again, the formation of marsh gas and carbonate of soda, when acetate of soda is heated with caustic soda, and the formation of aceto-nitrile from cyanide of potassium and iodide of methyl, show that the carbon in acetic acid is divisible by two, or that the molecule contains two (or a multiple of two) atoms of carbon. C 2 H 4 O 2 is the simplest formula which fulfils these conditions, but the existence of an acid acetate of potash and an acid acetate of ammonia, the formulae of which are usually written C 2 H 3 KO 2 , C 2 H 4 O 2 and C 2 H 3 (NH 4 )O 2 , C 2 H 4 O 2 , as if these were compounds derived from two molecules of acetic acid, might lead us to C 4 H 8 O 4 , as this shows that the hydrogen is divisible by eight. In the same way, we can easily satisfy ourselves that C 6 H 10 O 5 , or some multiple of it, is the formula of starch; that C 8 H 5 NO, or some multiple of it, is the formula of indigo blue, and so on. But it is not easy to determine by purely chemical methods whether these formulae themselves, or multiples of them, really represent the molecule. A simple formula may suffice for a great many of the reactions of a substance, and may enable us to represent a great many of its derivatives, and yet reactions and derivatives may be discovered which require a multiple of that simple formula. This has already been indicated in reference to acetic acid, and a very striking illustration is supplied by mellitic acid. For a long time the formula C 4 H 2 O 4 was used for this acid, and by means of it all the then known derivatives were represented. But later investigations by Baeyer proved that this formula must be multiplied by three, the new derivatives obtained by him not being capable of representation with any formula simpler than Ο 12 Η ϋ Ο 12 . Very many examples of the same kind might be adduced, but those given may serve to show the nature of the difficulty of settling the formula and with it the molecular weight of a substance. It need scarcely be said that the multiple formula represents everything which the simple formula represents and something more, and that chemists as a rule take the simplest formula which will answer the purpose. These chemical methods of determining the formula and molecular weight apply equally to all pure substances, but they do not give us absolute values, only numbers to which the molecular weights are proportional. And for purely chemical purposes these are all that we require. Thus, when a chemist speaks of acting on a molecule of succinic acid with two molecules of pentachloride of phosphorus, he means that he mixes them in the proportion of 118 parts of the former to 2 × 177*5 of the latter. For the sake of precision we sometimes speak of a molecule of water (or other substance) in grammes, or even of a gramme-molecule, a grain-molecule, <fcc. Thus, in the case just mentioned a gramme-molecule of succinic acid means 118 grammes· of succinic acid, Ac.

But, while for practical purposes these proportional numbers are quite sufficient, we cannot leave out of view their relation to the actual constitution of matter. There is good reason to believe that matter consists of discrete particles, and that every pure substance is made up of small portions of matter, all alike, so that one of them, if we could examine it, would give us a complete idea of the chemical composition, constitution, and character of the substance. These small portions, of which the smallest quantity of the substance which we can examine contains many millions, we may call molecules. From the character which we have supposed this molecule to possess—viz., that it fully represents all the chemical properties of the substance—it will be seen that these real, ultimate molecules must be proportional to the molecular weights ascertained by chemical means; so that, while for practical laboratory or manufacturing purposes we use the gramme, the pound, or the ton as our unit, and speak of 18 grammes, pounds, or tons, as the case may be, of water, as a molecule (or gramme-molecule, ton-molecule, <fcc.), in dealing with the actual constitution of matter we should use as our unit the mass of a single atom of hydrogen, and our gramme-molecule would then be a definite, very large, but not yet accurately ascertained, number of real molecules.

It has been already shown above that, on the kinetic theory of gas, a gas consists of a number of particles moving about in straight lines in all directions, and that in a homogeneous gas which follows Boyle’s and Charles’s laws these particles are all alike. The masses of the particles of different gases are therefore to one another in the same proportion as the densities of the gases, temperature and pressure being the same. Thus, in gases, the independently moving particles of the kinetic theory are the molecules of which the chemist is in search, and it becomes important that we should compare our chemically found molecular weights with the densities. Theoretically accurate results could be obtained only in the case of a perfect gas; but small deviations from Boyle’s and Charles’s laws do not interfere with the application of this method. Chemical methods, as we have already seen, lead us to a particular number, or a multiple of it, so that our choice is as a rule limited to two or three numbers widely differing from one another. We find that if we do not exceed the limits of chemical stability a gas approaches the state of a perfect gas as the temperature increases, or as the pressure diminishes. Now if one of the numbers rendered probable by chemical evidence nearly coincides with that given by comparison of gas densities, under conditions where the substance sensibly deviates from Boyle’s and Charles’s laws, we find that by diminishing the pressure or increasing the temperature within the limits of chemical stability, and thus bringing the substance nearer the state of a perfect gas, the correspondence between these two numbers becomes closer. This has already been pointed out and illustrated in the article Chemistry, vol. v. p. 469.

We can now compare the results, in the case of gases, of the chemical and of the physical determination of molecular weight, by giving some examples, placing side by side the formula and molecular weight adopted by chemists, and the mass, in grammes, of the gas occupying the volume of 22'33 × 760∕ j p × (273 + 0/273 litres. This volume is that which one gramme of an ideal gas having the molecular weight 1, and perfectly following Boyle’s and Charles’s laws, would occupy at pressure p millimetres of mercury and temperature t a C. If, then, w be the molecular weight of any gas, w grammes of it should occupy this volume, and slight deviation from this would indicate slight deviation from Boyle’s and Charles’s laws. In the annexed table w is the molecular weight and m the mass contained in 22'33 × 760/p × (273 + 0/273 litres. Where the temperature is not specially stated, the determinations were made under the usual atmospheric conditions. [table]

A comparison of the values of w and m leads to the following conclusions :—

( 1 ) In the case of a very great number of substances, of which only a few specimens are given in the table, the two determinations agree, the slight differences often observed being evidently due to deviation of the substance from the state of a perfect gas. (2) In a considerable number of substances, physical evidence leads to a multiple of the simplest number satisfying the chemical conditions. This cannot be looked upon as a disagreement between the methods, because, if a particular formula satisfies the chemical conditions, any multiple of it will necessarily do so; and indeed, in many of the cases we are now considering, it is possible from chemical considerations to justify the higher molecular weight after it has been suggested, although such chemical considerations might not in all cases have warranted its adoption without external support. Thus, we are not without chemical evidence in favour of the formulae H 2 , Cl 2 ,O 2 , or even Al 2 Br 6 and Fe 2 Cl 6 , although chemists would probably have contented themselves with H, Cl, O, A1Bγ 3 , and FeCl 3 , had it not been for the evidence of gas and vapour density, and certainly without the latter no one would have thought of P 4 , As 4 , or S 6 . (3) There are a number of substances in the case of which there is an apparent disagreement between the results of the two ways of determining molecular weight. Such substances are said to have an anomalous gas or vapour density. The expression anomalous vapour density is sometimes applied to the case of such substances as phosphorus and arsenic, but not very accurately. It would be better to say that these substances have an unexpected vapour density, because their complex molecular formulae, while not clearly indicated by their chemical character, are not at variance with any established law.

We shall therefore reserve the term “anomalous vapour density” for those substances the molecular weight of which as given by their vapour density is not reconcilable with any formula which is chemically admissible. In the case of some substances, such as the oxides of chlorine, it has been shown that the discrepancy was due to errors of observation, impure specimens having been used in the experiments; but there still remain many substances having, in the sense above indicated, an anomalous vapour density. These substances have therefore been examined with special care, with the result of completely vindicating the kinetic theory, and of disclosing a very interesting and theoretically important kind of chemical change. We shall take, as instances of such anomalous vapour densities, the substances in the last division of the table, and show how the anomaly has in these cases been explained.

Sal-ammoniac has the composition represented by the formula NH 4 C1. This formula agrees with all the chemical actions of the substance and of all the substances in any way related to it, but it does not agree with the results of vapour density determinations. When sal-ammoniac is heated it is converted into vapour or gas, and this vapour or gas is reconverted into solid sal-ammoniac when it is cooled. This looks exactly like the process of sublimation, and it was universally supposed that the vapour given off when sal-ammoniac is heated was really sal-ammoniac vapour. But its vapour density corresponds, not to the formula NH 4 C1 and the molecular weight 53'5, but to the half of this. Now this formula does not admit of division, and the explanation at once suggests itself, that the vapour examined was not really the vapour of salammoniac, but of hydrochloric acid and ammonia gases, the products of the decomposition of sal-ammoniac.

This would of course completely explain the apparent anomaly; each molecule NH 4 C1 dividing into two molecules NH 3 and HC1, the gas from a given weight of salammoniac would of course contain twice as many molecules and occupy twice the space which it would do if no such decomposition had occurred. On this supposition the mixed gases would remain uncombined as long as the temperature was above the decomposing point of salammoniac; if the temperature fell below this point they would unite and reproduce sal-ammoniac. It was necessary, however, to prove that this decomposition occurs.

As has been shown above (p. 618), the rate of diffusion of a gas depends upon its density. In this case the two gases into which the substance may be supposed to break up at the moment of volatilization differ considerably in density; we ought, therefore, to be able to effect partial separation by means of diffusion, and it has been shown that such partial separation actually does occur. Thus, if we have hydrogen gas on one side of a porous diaphragm and volatilized sal-ammoniac on the other side, we find after a time that, mixed with the hydrogen on the one side, we have what we may for shortness call salammoniac vapour—that is, a vapour which when cooled forms solid sal-ammoniac—with an excess of ammonia, which, being less dense than hydrochloric acid gas, has diffused faster; while on the other side, also mixed with hydrogen which has diffused through the diaphragm, we have sal-ammoniac vapour with excess of hydrochloric acid, the denser and more slowly diffusing gas. This of course proves that the decomposition has occurred, but it does not prove that the vapour of sal-ammoniac consists entirely of hydrochloric acid and ammonia mixed with one another. That this in fact is not the case has been shown by an ingenious experiment. The two gases were separately raised to a temperature higher than that at which sal-ammoniac volatilizes, and were then allowed to mix in a vessel kept at the same temperature as the two gases. In this vessel a delicate thermometer was placed, and it was found that the mixing of the two gases was accompanied by a small but very decided evolution of heat. This proves that some chemical combination takes place, and that the mixed gases must contain some vapour of NH 4 C1. Moreover, careful determinations of the vapour density of salammoniac prove that it is a little more than the mean of the densities of ammonia and hydrochloric acid (as compared with air at the same temperature and pressure, 1Ό1 instead of 0’9255 at 350° C.)} and this increase of density on mixing the hot gases is easily explained by supposing that a small proportion is in the condition of NH 4 C1, while the most of the gas consists of separate NH 3 and HC1 molecules.

In a similar way it has been shown that the vapour of oil of vitriol is a mixture of two vapours,—that of water, H 2 O, and that of sulphuric anhydride, SO 3 ; and that sulphide of ammonium when volatilized breaks up into two volumes of ammonia and one of sulphuretted hydrogen, (NH 4 ) 2 S = 2NH 3 + H 2 S. We find, therefore, that in the former case, as in that of sal-ammoniac, w = 2m, and in the latter, w = 3m.

This peculiar kind of decomposition is now known by the name “dissociation.” (See vol. v. pp. 475, 476.) In the cases we have mentioned the substances undergo nearly complete dissociation at the temperature at which they volatilize, and recombination takes place when they are cooled and again assume the solid, or, as in the case of oil of vitriol, the liquid state. These substances are therefore not suited for the illustration of the whole course of dissociation. This has been carefully studied in the case of some compounds, in which the dissociation is far from complete, at the boiling point of the substance, with the result that, if AB be the compound dissociating into the separate molecules A and B, we may represent the amount of dissociation as the ratio of the number of pairs of separate A and B molecules to the total number of pairs of A and B, both separate and combined. This ratio we may call B, so that when dissociation is complete 72=1.

(1) 72 increases as the temperature rises. (2) dR ∣ dt (where t is temperature) is a maximum when 72=⅛. (3) The presence of excess of either A or B diminishes the value of 72. For instance, PC1 5 is nearly completely dissociated into PC1 3 and Cl 2 at 300° C.; but if a large excess of PC1 3 is mixed with the vapour it is found to contain scarcely any Cl 2 , so that dissociation is greatly diminished by the presence of excess of PC1 3 . These experimental results are capable of explanation on the kinetic theory of gas, if we adopt Pfaundler’s hypothesis. This is, that for each case of dissociation there is a limiting value for the internal kinetic energy of a molecule of AB. If a molecule of AB, by encounters with other molecules or with the wall of the vessel containing the gas, acquires a greater amount of internal kinetic energy than this limit, it at once breaks up into A and B, so that in the gaseous mixture there are no molecules of AB having more internal kinetic energy than the limit. Further, if two molecules, one of A and one of B, meet one another with such a velocity and with such an amount of internal kinetic energy that together the internal kinetic energy is less than the limit, they will unite to form a molecule of AB. Thus the molecules with great internal kinetic energy will be separate molecules of A and B ; those with small internal kinetic energy will mostly be united as A B. This hypothesis has been to a considerable extent worked out and applied by Pfaundler and by Naumann, and the deductions from it agree fairly well with the results of experiment; but in some points the theory has not been fully developed, and in some it does not seem altogether to accord with observed facts. Some of these difficulties have been mentioned above. We know enough of the nature of dissociation to see that it belongs to the class of balanced chemical actions, in which a chemical change is reversible, and equilibrium is kept up, with constant external conditions, by the two opposite chemical changes taking place to an equal extent in a given time. We can see that all such cases are explicable by the statistical method, but we cannot apply this method mathematically until we know more of the intimate nature of the molecules and of the way in which they act upon one another. In this discussion of dissociation λve have looked specially at the cases in which A, B, and AB are all gaseous, because it was the question of anomalous vapour densities which led us to treat of the subject. Dissociation also occurs where one or two of the substances are solid or liquid.

We now see with what restrictions the method of vapour density is applicable to the determination of molecular weight, and we can understand more fully the example given in the article Chemistry, vol. V. p. 469. It is there shown that acetic acid vapour does not conform to the laws of Boyle and Charles until the temperature is raised to about 250°, at the ordinary barometric pressure. At and above that temperature the vapour density corresponds to the formula C 2 H 4 O 2 ∙ At lower temperatures the density corresponds to a higher molecular weight. Now Playfair and Wanklyn determined the vapour density at much lower temperatures than the ordinary boiling point of acetic acid, by greatly diminishing the pressure of the acetic acid vapour. This they accomplished by mixing it with a large quantity of hydrogen, so that the pressure due to acetic acid vapour formed only a small fraction of the total pressure. The vapour density of acetic acid at the low temperatures at which they worked was found to correspond very nearly with the formula C 4 H 8 O 4 , and, by comparing this result with what has been said (p. 620) of the chemical evidence as to the molecular weight of acetic acid, we may reasonably conclude that the molecule of acetic acid at low temperatures is C 4 II 8 O 4 , and that as the temperature is raised it undergoes dissociation, each molecule dividing into two of C 2 H 4 O 2 , This is then a case where A and B are equal, and A A divides into A + A. Another instance of the same kind is probably to be found in peroxide of nitrogen (Chemistry, p. 513), where N 2 O 4 divides into NO 2 + NO 2 ∙ Similarly, sulphur vapour has, at temperatures below 500° C., a density corresponding to the formula S β . This dissociates as the temperature rises until, about 1000° C., the density corresponds to the formula S 2 (Chemistry, p. 498).

We have now seen that chemistry receives great assistance in the determination of molecular weight from physics, but this assistance is almost entirely confined to the case of gases, or of substances which can be volatilized. The phenomena of the diffusion of liquids show us that there also there are independently moving particles; but the laws of liquid-diffusion have not been sufficiently generalized to give us much help in the determination of the relative masses of these particles. In liquids it is probable that the particles are very near each other, and that their shape and their mutual action, as well as their mass and the temperature, determine their rate of motion.

In solids we have no independently travelling particles, and it is perhaps scarcely correct to speak of a molecular structure of solids at all. Solids are no doubt composed of atoms, and those atoms are evidently arranged in what may be called a tactical order. When the solid is fused or dissolved or volatilized, it breaks into molecules, each repetition of the pattern, if we may use the expression, being ready to become an independent thing under favourable circumstances. But, while these potential molecules of solids cannot perhaps be properly called molecules in a physical sense, for chemical purposes we may call them so, for they are the smallest portions of the substance which fully represent it chemically, and, as we have seen, this is the chemical molecule, the quantity which should be represented by the formula. (A. C. B. )