DIFFUSION, in general, a spreading out, scattering or cir culation; in physics the term is applied to a special phenomenon, treated below. The word is from the Lat. di ff undere; dis-, asunder, and fundere, to pour out.
I. General Description.—When two different substances are placed in contact with each other they sometimes remain separate, but in many cases a gradual mixing takes place. This occurs when ever there is a difference of concentration or (in the case of gases) of partial pressure between the constituents of neigh bouring portions. This phenomenon is known as diffusion. Simple cases of diffusion are easily observed qualitatively. If a solution of a coloured salt is carefully introduced by a funnel into the bottom of a jar containing water, the two portions will at first be fairly well defined, but if the mixture can exist in all propor tions, the surface of separation will gradually disappear; and the rise of the colour into the upper part and its gradual weakening in the lower part, may be watched for days, weeks or even longer intervals. The diffusion of a strong aniline colouring matter into the interior of gelatine is easily observed, and is commonly seen in copying apparatus. Diffusion of gases may be shown to exist by taking glass jars containing vapours of hydrochloric acid and ammonia, and placing them in communication with the heavier gas downmost. The precipitation of ammonium chloride shows that diffusion exists, though the chemical action prevents this example from forming a typical case of diffusion. Again, when a film of Canada balsam is enclosed between glass plates, the disap pearance during a few weeks of small air bubbles enclosed in the balsam can be watched under the microscope.
In fluid media, whether liquids or gases, the process of mixing is greatly accelerated by stirring or agitating the fluids, and liquids which might take years to mix if left to themselves can thus be mixed in a few seconds. It is necessary carefully to dis tinguish the effects of agitation from those of diffusion proper. Agitation brings together portions of the fluid between which con siderable differences of concentration may exist. The interchange between such portions then proceeds much more rapidly. In many cases, especially in gases, the intermixing goes on until the con centration is uniform throughout. Thus a strong solution of will ultimately form a uniform weaker solution if brought into contact with water. In other cases, the material remains di vided into two or more regions (or phases) in each of which the relative concentration of the components is uniform, but changes in passing from one phase to the next. The passing of one or more components across the boundary between two phases is known as solution or evaporation or condensation; but the pro cess by which the uniform distribution in each phase is set up is diffusion.
Diffusion may take place in solids, that is, in regions occupied by matter which continues to exhibit the properties of the solid state. Thus, gold and lead brought into contact begin to diffuse into one another. An interesting series of examples is afforded by the passage of gases through partitions of metal, notably the passage of hydrogen through platinum and palladium and of he lium through silica ("fused" quartz) at high temperatures. When the process is considered with reference to a membrane or parti tion taken as a whole, the passage of a substance from one side to the other is commonly known as "osmosis" or "transpiration" (see SOLUTIONS) but what occurs in the material of the membrane itself is correctly described as diffusion. Agitation brings together portions of the fluid between which considerable differences of concentration may exist. The interchange between such por tions then proceeds much more rapidly.
To sum up, the ultimate process by which the individual mole cules of two different substances become mixed, producing finally a homogeneous mixture, is in every case diffusion.
In order to make accurate observations of diffusion in fluids it is necessary to guard against any cause which may set up cur rents; and in some cases this is exceedingly difficult. Thus, if gas is absorbed at the upper surface of a liquid, and if the gaseous solution is heavier than the pure liquid, currents may be set up, and a steady state of diffusion may cease to exist. This has been tested experimentally by C. G. von Hiif ner and W. E. Adney. The same thing may happen when a gas is evolved into a liquid at the surface of a solid even if no bubbles are formed; thus if pieces of aluminium are placed in caustic soda, the currents set up by the evolution of hydrogen are sufficient to set the aluminium pieces in motion, and it is probable that the motions of the Diatomaceae are similarly caused by the evolution of oxygen. In some pairs of substances diffusion may take place more rapidly than in others. Of course the progress of events in any experiment necessarily depends on various causes, such as the size of the containing ves sels, but it is easy to see that when experiments with different substances are carried out under similar conditions, however these "similar conditions" be defined, the rates of diffusion must be capable of numerical comparison, and the results must be express ible in terms of at least one physical quantity, which for any two substances can be called their coefficient of diffusion. How to select this quantity we shall see later.
2. Quantitative Methods of Observing Diffusion.—The simplest plan of determining the progress of diffusion between two liquids would be to draw off and examine portions from different strata at some stage in the process ; the disturbance produced would, however, interfere with the subsequent process of diffusion, and the observations could not be continued. By placing in the liquid column hollow glass beads of different average densities, and observing at what height they remain suspended, it is possible to trace the variations of density of the liquid column at differ ent depths, and different times. In this method, which was origi nally introduced by Lord Kelvin, difficulties were caused by the adherence of small air bubbles to the beads.
In general, optical methods are the most capable of giving exact results, and the following may be distinguished. (a) By refraction in a horizontal plane. If the containing vessel is in the form of a prism, the deviation of a horizontal ray of light in pass ing through the prism determines the index of refraction, and consequently the density of the stratum through which the ray passes. (b) By refraction in a vertical plane. Owing to the density varying with the depth, a horizontal ray entering the liquid also undergoes a small vertical deviation, being bent downwards towards the layers of greater density. The observation of this vertical deviation determines not the actual density, but its rate of variation with the depth, i.e., the "density gradient" at any point. A parallel-faced vessel is employed and the incident beam falls normally upon it. (c) By the saccharimeter. In the cases of solutions of sugar, which cause rotation of the plane of polarized light, the density of the sugar at any depth may be determined by observing the corresponding angle of rotation; this was done orig inally by W. Voigt.
3. Elementary Definitions of Coefficient of Diffusion.— The simplest case of diffusion is that of a substance, say a gas, diffusing in the interior of a homogeneous solid medium which re mains at rest, when no external forces act on the system. We may regard it as the result of experience that : if the density of the diffusing substance (i.e., the mass of that substance per unit volume) is everywhere the same, no diffusion takes place, and (2) if the density of the diffusing substance is different at different points, diffusion will take place from places of greater to those of lesser density, and will not cease until the density is everywhere the same. It follows that the rate of flow of the diffusing sub stance at any point in any direction must depend on the density gradient at that point in that direction, i.e., on the rate at which the density of the diffusing substance decreases as we move in that direction. We may define the coefficient jicient o f diffusion as the ratio of the total mass which flows per unit area across any small section, to the rate of decrease of the density with distance in a direction perpendicular to that section.
Further, the rate at which the quantity of substance is increas ing in an element between the distances x and x-i-dx is equal to the difference of the rates of flow in and out of the two faces, whence as in hydrodynamics, we have dp/dt= —dq/dx.
It follows that the equation of diffusion in this case assumes the form: which is identical with the equations representing conduction of heat, flow of electricity and other physical phenomena. For diffusion in three dimensions we have in like manner: and the corresponding equations in electricity and heat for aniso tropic substances would be available to account for any parallel phenomena, which may arise, or might be conceived, to exist in connection with diffusion through a crystalline solid. The solution of such an equation can usually be expressed in terms of an ex pansion in an infinite series (see FOURIER'S SERIES, SPHERICAL HARMONICS, etc.).
In the case of a very dilute solution, the coefficient of diffusion of the dissolved substance can be defined in the same way as when the diffusion takes place in a solid, because the effects of dif fusion will not have any perceptible influence on the solvent, and the latter may therefore be regarded as remaining practically at rest. But in most cases of diffusion between two fluids, both of the fluids are in motion, and hence there is far greater difficulty in determining the motion, and even in defining the coefficient of diffusion. It is important to notice in the first instance that it is only the relative motion of the two substances which constitutes diffusion. Thus when a current of air is blowing, under ordinary circumstances the changes which take place are purely mechanical, and do not depend on the separate diffusions of the oxygen and nitrogen of which the air is mainly composed. It is only when two gases are flowing with unequal velocity, that is, when they have a relative motion, that these changes of relative distribution, which are called diffusion, take place. The best way out of the difficulty is to investigate the separate motions of the two fluids, taking account of the mechanical actions exerted on them, and supposing that the mutual action of the fluids causes each fluid to resist the relative motion of the other.
This definition implies the following laws of resistance to dif fusion, which must be regarded as based on experience, and not as self-evident truths: (I) each fluid tends to assume, so far as diffusion is concerned, the same equilibrium distribution that it would assume if its motion were unresisted by the presence of the other fluid. (Of course, the rnutual attraction of gravitation of the two fluids might affect the final distribution, but this is prac tically negligible. Leaving such actions as this out of account the following statement is correct.) In a state of equilibrium, the density of each fluid at any point thus depends only on the partial pressure of that fluid alone, and it is the same as if the other fluids were absent. It does not depend on the partial pressures of the other fluids. If this were not the case, the resistance to diffusion would be analogous to the friction of solids, and would contain terms which were independent of the relative velocity u2—ui. (2) For slow motions the resistance to diffusion is (approximately at any rate) proportional to the relative velocity. (3) The coefficient of resistance C is not necessarily always constant ; it may, for example, and, in general, does, depend on the temperature.
If we form the equations of hydrodynamics for the different fluids occurring in any mixture, taking account of diffusion, but neglecting viscosity, and using suffixes r, 2 to denote the separate fluids, these assume the form given by James Clerk Maxwell ("Diffusion," in Ency. Brit., 9th ed.): and these equations imply that when diffusion and other motions cease, the fluids satisfy the separate conditions of equilibrium (Wax — The assumption made in the following ac count is that terms such as Dui/Dt may be negiected in the cases considered.
A further property based on experience is that the motions set up in a mixture by diffusion are very slow compared with those set up by mechanical actions, such as differences of pressure. Thus, if two gases at equal temperature and pressure be allowed to mix by diffusion, the heavier gas being below the lighter, the process will take a long time ; on the other hand, if two gases, or parts of the same gas, at different pressures be connected, equal ization of pressure will take place almost immediately. It follows from this property that the forces required to overcome the "inertia" of the fluids in the motions due to diffusion are minute in comparison. At any stage of the process, therefore, any one of the diffusing fluids may be regarded as in equilibrium under the action of its own partial pressure, the external forces to which it is subjected and the resistance to diffusion of the other fluids.
5. Slow Diffusion of Two Gases, Relation Between the Co efficients of Resistance and of Diffusion.—We now suppose the diffusing substances to be two gases which obey Boyle's law, and that diffusion takes place in a closed cylinder or tube of unit sec tional area at constant temperature, the surfaces of equal density being perpendicular to the axis of the cylinder, so that the direc tion of diffusion is along the length of the cylinder, and we sup pose no external forces, such as gravity, fo act on the system.
The densities of the gases are denoted by pi, p2, their velocities of diffusion by ui, u2, and if their partial pressures are pi, p2, we have by Boyle's law pi = kiPi, P2 = k2p2, where ki, k2 are constants for the two gases, the temperature being constant. The axis of the cylinder is taken as the axis of x.
From the considerations of the preceding section, the effects of inertia of the diffusing gases may be neglected, and at any instant of the process either of the gases is to be treated as kept in equilibrium by its partial pressure and the resistance to diffu sion produced by the other gas. Calling this resistance per unit volume R, and putting R=Cpip2(ui—u,), where C is the coeffi cient of resistance, the equations of equilibrium give where P is the total pressure of the mixture, and is everywhere constant, consistently with the conditions of mechanical equi librium.
Now is the pressure-gradient of the first gas, and is, by Boyle's law, equal to times the corresponding density gradient. Again is the mass of gas flowing across any section per unit time, and or can be regarded as representing the flux of partial pressure produced by the motion of the gas. Since the total pressure is everywhere constant, and the ends of the cylinder are supposed fixed, the fluxes of partial pressure due to the two gases are equal and opposite, so that or kipiui (3) From (2) (3) we find by elementary algebra = = = (ui and therefore p2 u1 = — p2 u2 = pi p2(ui — u")/P = ki pi p2(ui — u2)/P. Hence equations (I) (2) gives + C P ( = o, and -}- C P (p2u2) = ax ax whence also substituting pi = kepi, P2 = k2p2, and by transposing = k1k2 apt k1k2 apt phut C P a and p2u2 =— C P ax • We may now define the "coefficient of diffusion" of either gas as the ratio of the rate of flow of that gas to its density-gradient. With this definition, the coefficients of diffusion of both the gases in a mixture are equal, each being equal to The ratios of the fluxes of partial pressure to the corresponding pressure gradients are also equal to the same coefficient. Calling this coeffi cient K, we also observe that the equations of continuity for the two gases are exactly as in the case of diffusion through a solid.
If we attempt to treat diffusion in liquids by a similar method, it is, in the first place, necessary to define the "partial pressure" of the components occurring in a liquid mixture. This leads to the conception of "osmotic pressure," which is dealt with in the article SOLUTIONS. For dilute solutions at constant temperature, the assumption that the osmotic pressure is proportional to the density, leads to results agreeing fairly closely with experience, and this fact may be represented by the statement that a sub stance occurring in a dilute solution behaves like a perfect gas. (It is to be borne in mind that the partial pressures are no longer additive. For a solution containing I gm. molecule of sugar per litre the osmotic pressure is about 3o atmospheres even though the total pressure is only I Atm.) 6. Relation of the Coefficient of Diffusion to the Units of Length and Time.—We may write the equation defining K in the form dp u=—KX I - p dx Here—dp/pdx represents the proportional rate at which the density decreases with the distance x; and we thus see that the coefficient of diffusion represents the ratio of the velocity of flow to the proportional rate at which the density decreases with the distance measured in the direction of flow. This proportional rate being of the nature of a number divided by a length, and the velocity being of the nature of a length divided by a time, we may state that K is of two dimensions in length and — I in time, i.e., of dimensions Example I. Taking K=o•1423 for carbon dioxide and air (at temperature o° C and pressure cm. of mercury) referred to a centimetre and a second as units, we may interpret the result as follows :—Supposing in a mixture of carbon dioxide and air, the density of the carbon dioxide decreases by, say, I, 2, or 3% of itself in a distance of I cm., then the corresponding velocities of the diffusing carbon dioxide will be respectively o•01, 0•02 and 0•03 times 0.1423, that is, 0•001423, 0.002846 and 0.004269 cm. per second in the three cases.
Example 2. If we wished to take a foot and a second as our units, we should have to divide the value of the coefficient of diffusion in Example 1 by the square of the number of centimetres in I ft., that is, roughly speaking, by 900, giving the new value of K = 0•00016 roughly.
7. Numerical Values of the Coefficient of Diffusion.—The table with this article gives the values of the coefficient of diffusion of several principal pairs of gases at a pressure of 76 cm. of mercury, and also of a number of other substances. In the values for gases the centimetre and second are taken as fundamental units, in other cases the centimetre and day. The numbers given must be taken as indicating the order of magnitude only since considerably different values are obtained by different observers. Thus Obermayer obtained the value 0.67 for hydrogen-oxygen.
8. Diffusion Through a Membrane or Partition. Theory of the Semi-permeable Membrane.—It has been pointed out that diffusion of gases frequently takes place in the interior of solids; moreover, different gases behave differently with respect to the same solid at the same temperature. A membrane or par tition formed of such a solid can therefore be used to effect a more or less complete separation of gases from a mixture. This method is employed commercially for extracting oxygen from the atmosphere, in particular for use in projection lanterns where a high degree of purity is not required. A similar method is of ten applied to liquids and solutions and is known as "dialysis." In such cases as can be tested experimentally it has been found that a gas always tends to pass through a membrane from the side where its density, and therefore its partial pressure, is greater to the side where it is less ; so that for equilibrium the partial pressures on the two sides must be equal. This result is unaffected by the presence of other gases on one or both sides of the membrane. For example, if different gases at the same pressure are separated by a partition through which one gas can pass more rapidly than the other, the diffusion will give rise to a difference of pressure on the two sides, which is capable of doing mechanical work in moving the partition. In evidence of this conclusion Max Planck quotes a test experiment made by him in the Physical Institute of the University of Munich in 1883, depending on the fact that platinum foil at white heat is permeable to hydrogen but impermeable to air, so that if a platinum tube filled with hydrogen be heated the hydrogen will diffuse out, leaving a vacuum.
The details of the experiment may be quoted here :—"A glass tube of about 5 mm. internal diameter, blown out to a bulb at the middle, was provided with a stop-cock at one end. To the other a platinum tube To cm. long was fastened, and closed at the end. The whole tube was exhausted by a mercury pump, filled with hydrogen at ordinary atmospheric pressure, and then closed. The closed end of the platinum portion was then heated in a horizontal position by a Bunsen burner. The connection be tween the glass and platinum tubes, having been made by means of sealing-wax, had to be kept cool by a continuous current of water to prevent the softening of the wax. After four hours the tube was taken from the flame, cooled to the temperature of the room, and the stop-cock opened under mercury. The mercury rose rapidly, almost completely filling the tube, proving that the tube had been very nearly exhausted." In order that diffusion through a membrane may be reversible so far as a particular gas is concerned, the process must take place so slowly that equilibrium is set up at every stage. In order to separate one gas from another consistently with this condition it is necessary that no diffusion of the latter gas should accom pany the process. The name "semi-permeable" is applied to an ideal membrane or partition through which one gas can pass, and which offers an insuperable barrier to any diffusion whatever of a second gas. By means of two semi-permeable partitions acting oppositely with respect to two different gases A and B these gases could be mixed or separated by reversible methods.
Most physicists admit, as Planck does, that it is impossible to obtain an ideal semi-permeable substance; indeed such a sub stance would necessarily have to possess an infinitely great resist ance to diffusion for such gases as could not penetrate it. But in an experiment performed under actual conditions the losses of available energy arising from this cause would be attributable to the imperfect efficiency of the partitions and not to the gases themselves ; moreover, these losses are, in every case, found to be completely in accordance with the laws of irreversible thermo dynamics. The reasoning in this article being somewhat con densed, the reader must necessarily be referred to treatises on thermodynamics for further information on points of detail connected with the argument.
9. Work That Can Be Gained on Mixing Perfect Gases Reversibly.—In the case of perfect gases the partial pressures of the respective gases are independent of the presence of other gases. Take two separate gases, each at the same pressure, pi-Fp2, and at the same temperature; their volumes being and V2. These can be mixed in a reversible way by means of suitable cylinders and pistons.
I. II.
Ord. Piston Semi-permeable Ord. Piston Place the gases in cylinders I and 2. The central pistons initially in contact must be semi-permeable, No. I to gas No. I and No. 2 to gas No. 2. By adjusting the ordinary pistons the gases may be brought to pressures pi and p2 respectively. The isothermal work done by the system is By suitably moving all four pistons the gases can be trans ferred reversibly to the space between the two inner pistons the pressure remaining at the values pi and p2 while the intervening region remains throughout at constant pressure, pi+p2. The work done in this stage is — V'1 p1— V'2 p2+ (pi+p2) (Vi+ V 2) which, on inserting the values of and is seen to be zero. Hence the total work obtained is that done in the first stage and on putting and (pi+p2) V2 = R2T it becomes (p1+p2) Vilog p1+pi +(p1+p2) V2log pi P2 When the two gases are mixed by diffusion in an enclosed space, the total pressure remaining constant at pi + throughout, all this work is lost for good since the gases can only be separated again by having work performed on them at least equal to that which might have been gained.
I o. Kinetic Models of Diffusion.—Imagine in the first in stance that a very large number of red balls are distributed over one half of a billiard table, and an equal number of white balls over the other half. If the balls are set in motion with different velocities in various directions, diffusion will take place, the red balls finding their way among the white ones, and vice versa; and the process will be retarded by collisions between the balls. The simplest model of a perfect gas studied in the kinetic theory of gases (see KINETIC THEORY OF MATTER) differs from the above illustration in that the bodies representing the molecules move in space instead of in a plane, and, unlike billiard balls, their motion is unresisted, and they are perfectly elastic, so that no kinetic energy is lost either during their free motions, or at a collision.
The mathematical analysis connected with the application of the kinetic theory to diffusion is very long and cumbersome. We shall therefore confine our attention to regarding a medium formed of elastic spheres as a mechanical model, by which the most im portant features of diffusion can be illustrated. We shall assume the results of the kinetic theory, according to which :—(I) In a dynamical model of a perfect gas the mean kinetic energy of translation of the molecules represents the absolute temperature of the gas. (2) The pressure at any point is proportional to the product of the number of molecules in unit volume about that point into the mean square of the velocity. (The mean square of velocity is different from but proportional to the square of the mean velocity, and in the subsequent arguments either of these two quantities can generally be taken.) (3) In a gas mixture represented by a mixture of molecules of unequal masses, the mean kinetic energies of the different kinds are equal.
Consider now the problem of diffusion in a region containing two kinds of molecules A and B of unequal mass. The molecules of A in the neighbourhood of any point will, by their motion, spread out in every direction until they come into collision with other molecules of either kind, and this spreading out from every point of the medium will give rise to diffusion. If we imagine the velocities of the A molecules to be equally distributed in all directions, as they would be in a homogeneous mixture, it is ob vious that the process of diffusion will be greater, ceteris paribus, the greater the velocity of the molecules, and the greater the length of the free path before a collision takes place. If we assume consistently with this, that the coefficient of diffusion of the gas A is proportional to the mean value of Iva/a, where wa is the velocity and /a is the length of the path of a molecule of A, this expression f or the coefficient of diffusion is of the right dimensions in length and time. If, moreover, we observe that when diffusion takes place in a fixed direction, say that of the axis of x, it depends only on the resolved part of the velocity and length of path in that direction: this hypothesis readily leads to our taking the mean value of iwa/a as the coefficient of diffusion for the gas A. This value was obtained by O. E. Meyer and others.
Unfortunately, however, it makes the coefficients of diffusion unequal for the two gases, a result inconsistent with that obtained above from considerations of the coefficient of resistance, and leading to the consequence that differences of pressure would be set up in different parts of the gas. To equalize these differences of pressure, Meyer assumed that a counter current is set up, this current being, of course, very slow in practice ; and J. Stefan as sumed that the diffusion of one gas was not affected by collisions between molecules of the same gas. When the molecules are mixed in equal proportions both hypotheses lead to the value e( [wa/al [wb/b] ), (square brackets denoting mean values). When one gas preponderates largely over the other, the phenom ena of diffusion are too difficult of observation to allow of ac curate experimental tests being made. Moreover, in this case no difference exists unless the molecules are different in size or mass.
Instead of supposing a velocity of translation added after the mathematical calculations have been performed, a better plan is to assume from the outset that the molecules of the two gases have small velocities of translation in opposite directions, superposed on the distribution of velocity, which would occur in a medium representing a gas at rest. When a collision occurs between mole cules of different gases a transference of momentum takes place between them, and the quantity of momentum so transferred in one second in a unit of volume gives a dynamical measure of the resistance to diffusion. It is to be observed that, however small the relative velocity of the gases A and B, it plays an all-important part in determining the coefficient of resistance; f or without such relative motion, and with the velocities evenly distributed in all directions, no transference of momentum could take place. The coefficient of resistance being found, the motion of each of the two gases may be discussed separately.
One of the most important consequences of the kinetic theory is that if the volume be kept constant the coefficient of diffusion varies as the square root of the absolute temperature. To prove this, we merely have to imagine the velocity of each molecule to be suddenly increased n fold; the subsequent processes, including diffusion, will then go on n times as fast ; and the temperature T, being proportional to the kinetic energy, and therefore to the square of the velocity, will be increased n2 fold. Thus K, the co efficient of diffusion, varies as VT.
The relation of K to the density when the temperature re mains constant is more difficult to discuss, but it may be sufficient to notice that if the number of molecules is increased n fold, the chances of a collision are n times as great, and the distance tra versed between collisions is (not therefore but as the result of more detailed reasoning) on the average i/n of what it was before. Thus the free path, and therefore the coefficient of diffusion, varies inversely as the density, or directly as the volume. If the pressure p and temperature T be taken as variables, K varies inversely as p and directly as VT3.
Now according to the experiments first made by J. C. Maxwell and J. Loschmidt, it appeared that with constant density K was proportional to T more nearly than to VT. The inference is that in this respect a medium formed of colliding spheres fails to give a correct mechanical model of gases. It has been found by L. Boltzmann, Maxwell and others that a system of particles whose mutual actions vary according to the inverse fifth power of the distance between them represents more correctly the relation between the coefficient of diffusion and temperature in actual gases. Other recent theories of diffusion have been advanced by M. Thiesen, P. Langevin and W. Sutherland. On the other hand, J. Thovert finds experimental evidence that the coefficient of diffusion is proportional to molecular velocity in the cases exam ined of non-electrolytes dissolved in water at 8° at 2.5 grams per litre.
. Applications of Diffusion to Separating Gases.—Lord Rayleigh has applied the different rates at which diffusion takes place through porous partitions to the partial separation of mixed gases. Let x and y denote the quantities of the respective gases remaining at any moment in the chamber, so that —dx and —dy can stand for the quantities diffusing out in time dt. The values of dx/dt and dy/dt will depend upon the character of the porous partition and upon the actual pressure. Calling the relative rates of diffusion and v, we have dy/ dx = (I) The integral of (I) is Y log = x A ± constant. If the values, at any moment which we take as the initial time, y X are Y and X the constant can be eliminated. If we write — — = r x Y the value of r represents the enrichment of the residue in regard to the second constituent (y) and we can write x y X A>+ These equations show that the residue becomes purer without limit, and this is so whatever may be the original proportions. This is an outline of the theory that might be expected to apply to Graham's atmolyser in which the gaseous mixture is caused to travel along a tobacco pipe on the outside of which a vacuum is maintained. The third Lord Rayleigh applied this method to the separation of argon from air from which the oxygen had pre viously been removed. If an enrichment in the ratio 2 tO I is desired the diffusion must continue until the total quantity of gas is reduced to less than 2%. In his experiments even more than this reduction was required (Rayleigh, Phil. Mag., xiii. 493, 1896).