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 proportions, 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 disappearance 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 to carefully distinguish the effects of agitation from those of diffusion proper. By shaking up two liquids which do not mix we split them up into a large number of different portions, and so greatly increase the area of the surface of separation, besides decreasing the thicknesses of the various portions. But even when we produce the appearance of a uniform turbid mixture, the small portions remain quite distinct. If however the fluids can really mix, the final process must in every case depend on diffusion, and all we do by shaking is to increase the sectional area, and decrease the thickness of the diffusing portions, thus rendering the completion of the operation more rapid. If a gas is shaken up in a liquid the process of absorption of the bubbles is also accelerated by capillary action, as occurs in an ordinary sparklet bottle. To state the matter precisely, however finely two fluids have been subdivided by agitation, the molecular constitution of the different portions remains unchanged. The ultimate process by which the individual molecules of two different substances become mixed, producing finally a homogeneous mixture, is in every case diffusion. In other words, diffusion is that relative motion of the molecules of two different substances by which the proportions of the molecules in any region containing a finite number of molecules are changed.

In order, therefore, to make accurate observations of diffusion in fluids it is necessary to guard against any cause which may set up currents; 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 Hüfner 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 vessels, 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 expressible 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 different depths, and different times. In this method, which was originally 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 passing 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, (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 originally 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 remains at rest, when no external forces act on the system. We may regard it as the result of experience that: (1) if the density of the diffusing substance 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 substance 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 of diffusion as the ratio of the total mass per unit area which flows across any small section, to the rate of decrease of the density per unit distance in a direction perpendicular to that section.

In the case of steady diffusion parallel to the axis of x, if ρ be the density of the diffusing substance, and q the mass which flows across a unit of area in a plane perpendicular to the axis of x, then the density gradient is -dρ/dx and the ratio of q to this is called the “coefficient of diffusion.” By what has been said this ratio remains finite, however small the actual gradient and flow may be., and it is natural to assume, at any rate as a first approximation, that it is constant as far as the quantities in question are concerned. Thus if the coefficient of diffusion be denoted by K we have q= -K(dρ/dx).

Further, the rate at which the quantity of substance is increasing in an element between the distances x and x+dx is equal to the difference of the rates of flow in and out of the two faces, whence as in hydrodynamics, we have dρ/dt =-dq/dx.

It follows that the equation of diffusion in this case assumes the form

dρ = d( K dρ),
dt dx dx