F = Q⁄d2 = sy⁄v2/3
Now with all electrolytes, even with water, we have both positively and negatively charged ions, and y is consequently determined by the difference in the amounts adsorbed. Hence in the case of an electrolyte with an equal number of oppositely charged ions y = ma1c(1/n1)-ma2c(1/n2), where a1, a2, and n1, n2, are the appropriate constants for the particular ions concerned. Hence at constant temperature, pressure, etc., we may write
| F = | sm (a1c (1/n1) — a2c (1/n2)) |
| v2/3 |
The force tending to make a piece of gelatine swell is proportional to its mass, which is perhaps fairly obvious. The swelling force is also an inverse function of the volume of the gel, and as swelling proceeds therefore the force tending to swell further decreases. The force tending to swell is proportional to the specific surface of the disperse phase, other factors being constant. To illustrate this one has only to imagine that one particle of the disperse phase be split into two particles each carrying half the original charge. It is clear that a new repulsive force becomes operative, which did not before influence the swelling, and that the distance between the particles is halved. In the swelling of gelatine, however, we may consider the dispersity constant for constant temperature, and if we consider unit mass we see that the force causing swelling depends upon the operation of the adsorption law and upon the degree to which the gel is already swollen.
In the swelling of (say) one gram of gelatine to its maximum, both the contractile force of surface tension and the expanding force of electrical repulsion are in operation. At the commencement the latter is much the greater force—hence the rapid imbibition. Both these forces decrease in magnitude as the swelling proceeds, but the force tending to swell decreases at a more rapid rate, and the time comes when it has decreased to the precise value of the force tending to resist swelling. At this point equilibrium is established and the maximum swelling attained. Obviously this maximum will in many cases be determined largely by the value of a1c(1/n1)-a2c(1/n2). This factor, therefore, demands particular consideration.
Now, unfortunately, the adsorption law constants for the different ions have not yet been numerically determined, so that we are still somewhat in the dark as to the operation of ionic adsorptions. It is possible, however, to form conclusions of a qualitative or relative order, and these are such as to throw much light upon the question at issue. In the first place, we know that in general the various ions are not usually very widely different in the extent to which they are liable to be adsorbed. If this were otherwise, the valency rule would hardly operate so well in endosmosis, kataphoresis, and precipitation. In consequence we must expect the differences between the ions to appear in small rather than in large concentrations, the amounts adsorbed being under those conditions more affected by changes in the volume concentration. At the larger concentrations, therefore, the value of a1c(1/n1)-a2c(1/n2), is small, and the force causing swelling often tends to zero.
There are, however, noticeable differences at lower concentrations. Thus we know that if a substance be primarily a positive colloid, it will absorb kations more readily than anions. As gelatine falls into this class, we may therefore conclude that usually a1 > a2. Further, it often happens that very adsorbable substances are less affected by concentration changes, and in the case under consideration, therefore, we should expect that n1 > n2. Moreover, we know that the hydrion and hydroxyl ion are much more readily adsorbed than other ions, i.e. have a large value for a. Hence in the case of gelatine we expect that a1c(1/n1)-a2c(1/n2) will have a comparatively large value when one of the ions is H+ or OH-. Also we know that organic anions are usually much more strongly adsorbed than inorganic anions, and hence that in such cases a1 is more nearly approached by the value of a2. It should be emphasized perhaps, at this point, that these various considerations are not based upon any facts relating to the phenomena of imbibition in gels, or in gelatine in particular, but are based upon the behaviour of colloids in endosmosis, kataphoresis, electrolytic precipitation, adsorption, etc.
Fig. 1.
Now if we select a few simple figures which are in accord with the above considerations, we can examine the value of the factor a1c(1/n1)-a2c(1/n2) in a purely illustrative and typical way, and at any rate form some idea as to the manner in which it is likely to vary. The figures might be:—