with its resulting effect on pH and, consequently, on the potential of the hydrogen electrode, about all we can do is to select values which we assume to be most nearly representative.

Assuming that studies at higher and lower temperatures might alter the drifts in potential in a way that might throw light upon the situation, we made measurements at 15° C. and 46° C. The titration

EQUIVALENTS OF ALKALI

FIG. 1. Titration curves showing the pH values of solutions of alum at successive stages of treatment with alkali.

curves at these temperatures, like those at 30° C., were fairly satisfactory at pH values less than 7.0, but the drifts in the alkaline regions were erratic and can not be interpreted with confidence.

It may be mentioned in passing that titrations were also made with calcium hydroxide. The titration curves with calcium hydroxide as a base were, in the more acid regions, identical with those made

with sodium hydroxide, with the exception of very minor differences which are of no significance and which are of an order of magnitude attributable to experimental error. In the more alkaline regions the same difficulties mentioned above were encountered.

The points mentioned above do not affect in any very serious way the conclusion that the curves in Figure 1 represent the general trend of the titration process. In Figure 1 the curve marked by blackened circles represents the relation between pH (ordinates) and equivalents of added alkali (abscissa) in the case of a solution 0.0002 molecular with respect to Al when titrated with NaOH solution at 30°C. Upon the other curve, th circles represent the data obtained in the titration with NaOH of a solution 0.02 molecular with respect to Al. The crosses represent a duplicate series of measurements. If these curves are compared with those published by Hildebrand (1913) and by Blum (1913-14), there will be found a general agreement in the main features. However, the measurements by Hildebrand and by Blum were made with comparatively crude instruments, and for this reason the observers probably hesitated to call attention to detailed features in the titration curve which must have appeared to them. very peculiar. One notable feature is the distinct slope of the curve between pH 5 and 8. This certainly can not be due to the presence of a buffering impurity, such as bicarbonate, for our solutions were made with every precaution to exclude such impurities. The fact that the steeper part of the curve should occur so distinctly ahead of the three equivalents of alkali is also food for thought. The flatness of the curve at the start is, of course, accounted for by the throwing out of one or more constituents of the equilibrium as the titration proceeds.

In several papers published on the subject it has been assumed that the isoelectric point could be determined by mere inspection of the titration curve and the selection of the middle point in the steep part of the curve. This assumption was based on a procedure legitimate for the approximate determination of isoelectric points when there is symmetry in the number of acidic and basic dissociations and complete solubility of material concerned in the equilibrium equations. None of these conditions has been shown to apply to the case at hand.

In the titration of poly acids or bases it is usually found that the titration curve exhibits the several steps of neutralization. Even when the dissociation constants are close together, a distinct inflection. of the titration curve can usually be discerned. In the case at hand, there is but one inflection, and this may be considered to be between the region of acidic dissociation and the region of basic dissociation. It would, therefore, seem legitimate to neglect a distinction between

the first, second, and third basic dissociations and to consider them. together. Thus if we have

The use of equation (4) and the assumption of low solubility for Al(OH), would lead to a theoretical curve similar in its general form to the upper portions of the curves shown in Figure 1. Incidentally, the low solubility of Al(OH), will contribute very markedly to the position of the curve on the pH axis so that such a curve can not be directly compared with those curves used to depict equilibrium conditions when all components are soluble. By the same token the presence of any insoluble component would greatly modify the classic type curve and give it the general form of the experimental curve. Therefore we are not justified in using equation (4) as alone representative of actual conditions. Without committing ourselves at all to the assumption of the reality of particular modes of ionization, but merely to deal with convenient formulas, let us assume the ionizations which are involved in the following equilibrium equations:

If the solution is so dilute that the ionizations of all aluminium salts are complete, then the total aluminium will be the sum of the undissociated aluminium hydroxide, and each concentration of each ion. If, furthermore, the conditions are such that solid aluminium hydroxide is present in gram mols S, then the total aluminium, T, will be-

T=S+soluble [Al(OH),]+[Al(OH),O] + [AI+++] - - - - - - - - - - -- (7)

Substituting (5) and (6) and considering that at the equilibrium state [soluble Al(OH)] is a constant, C, we have

There will be a minimum variation of solid with variation of

But KC is the solubility product Kas, and KC is the solubility product Ks. Therefore

We know from the graphic expression of relations that

. (10)

d S

d [H+] passes from positive to negative in passing the isoelectric point. Therefore S is a maximum at the isoelectric point. To show that [H+] in (10) is the isoelectric point, that point at which there is electrical equivalence of positively and negatively charged aluminium ions, we proceed as follows: The condition to be fulfilled is

Substituting in (11) the relations of (5) and (6) we have

(11)

But K2 [A1 (OH),] and K1 [Al (OH),] are the acid and base solubility products Kas and Kbs, respectively.

Now, Heyrovsky (1920) working at 25° C. has given for Kas and Kbs the values 35 × 10- and 1× 10-33, respectively. Using the more favorable of the experimental data obtained from our titration. curves at 30° C., we find approximately Kas=1X 10-12 and Kbs=1×10-32. Heyrovsky's values introduced into equation (10) give an isoelectric point of pH 5.49. Our values give pH 5.6. According to the assumptions, then, the isoelectric point should lie between pH 5.5 and 5.6. We place little confidence in the calculation, however, because on returning to the titration curves themselves we find that we are unable to account for several features of the curves when using the solubility products mentioned above.

Furthermore, it is found that by specifying ionizations other than those used in the elementary treatment given, we obtain a variety of equations which, upon the assumption of one or several components of small solubility, will reduce to a form giving essentially the same picture as that presented. Thus the equations we have given furnish a correct description of principles but tell nothing whatever of the actual components entering into the problem. This has become evident in our attempts to formulate the very distinct slopes of the experimental curves found between the addition of two and the addition of three equivalents of alkali. As will appear later, the pH values found on these slopes are of the very greatest importance in the practical application of alum-coagulation, and no makeshift explanation will suffice for the problem at hand. The intersection of the curves shown in Figure 1 is of undoubted significance, but attempts to find a relation between intersections at various dilutions have failed so far.

In short, then, we are presented with a problem of very great complexity, owing, undoubtedly, to the very low and consequently variable solubility of one or more components of the equilibrium

state.

We have dwelt at some length upon this problem in order to indicate its general nature and to disarm those who, carrying over to the case at hand principles which furnish reliable data in other systems but which are inapplicable here, have assigned definite values to the isoelectric point. Inasmuch as a variation of 0.2 pH may have great practical significance, we shall need an accurate evaluation of the isoelectric point; and in the hope that future work will reveal it, we may now turn our attention to an indirect method of attack.