ciation is influenced so little by the presence of other substances that its undissociated amount can be quite accurately estimated by a short approximation process. Since now at least one of the three bodies CD, AD, and CB must be a salt, it is always possible to simplify in this manner. The simplest cases are those in which all three of the substances CD, AD, and CB, or the last two of these, are strongly dissociated substances, so that the quantities b, c, and d, or the last two of them, can be determined from the conductivities of the salts in pure water. The solubility of thallium chloride in potassium nitrate, referred to at the beginning of this article, is one example of this kind. That of silver benzoate in sodium nitrate would be another. This case, is not, however, of present interest, as we have no new experiments of the kind to communicate.' The next simplest case is that where CD and AD (or CB) are strongly dissociating substances, while CB (or AD) is only slightly or moderately dissociated. In this case the quantities b and c (or d) can be determined from conductivity measurements, and can therefore be regarded as known quantities in the derivation of the final equation, so that equations (7) and (8) (or (9)), which contain them, can be dispensed with. The solubility of the silver benzoate in nitric acid is a case of this kind; for nitric acid and silver nitrate are strongly dissociated, and benzoic acid is weakly dissociated. In order, now, to derive this equation, we assume b and c to be known, so as to have a definite case under consideration, and proceed as follows: For brevity we put I being a known quantity, and x the quantity to be determined. By elimination between equations (1), (2), and (6), we obtain : Furthermore we obtain by combining equations (2), (4) and (9): d=4(x+ 1 + k¿ ± √ x2 + 12 + kå2+ 2xk¿ + 2lką — 2lx). 1 For a full discussion of the theory in this case see Noyes: Ztschr. phys. Chem., 27. If now we place these two values of d equal to each other and simplify, we obtain the following expression : x2 + (r — 2c) x2 + (c2 — rc — rn + rb − k1) x + (ren — rcb+ in which r= ka By means of this equation we can calculate the value of x, since all the other values are known, and from this we obtain the solubility m by equation (10). . In the case where the substance AB is difficultly soluble and the value n is not very great, the quantities b and c are negligible. Then the following equation applies : x2 + rx2 - (rn + k.)x− k ̧ro. (13) If further x is very small relatively to r, the cubic term may be neglected. In this case: x2 — (n+ka)x-ko. (14) The remaining case to be considered is that where only the substance AD (or CB) is strongly dissociated, both CD and CB (or AD) being only slightly or moderately dissociated, so that they follow the theoretical dissociation-law. In this case only the quantity (or d) can be regarded as known, and only equation (8) (or (9)) can be dispensed with in the derivation of the solubility equation. For the sake of definiteness we assume c to be known. The solubility of silver benzoate in chloracetic acid is an example of this case, for chloracetic acid and the benzoic acid. produced by the metathesis are weakly dissociated, and the silver chloracetate is much dissociated. To derive the solubility equation, equations (3), (4), and (7) are solved for b with elimination of C and D, whereby we obtain : 2b= (c+d― 2n― k1) ± √ (c + d − 2n − ks)2 + 4 (cn + dn — n2 — cd) . Eliminating B and C from equations (2), (4), and (9), we obtain : If now we place these values of 26 equal to each other, simplify, and substitute in the resulting equation the value of d derived x2 — cx — k. from equations (1), (2), and (6), viz., d= simplification we obtain : x-c after rx3 + (1 − 3cr) x2 + (3c2r — 3c + ko−2k¿) x3 + (4ck¿—¿r+ 3c2 — 2k ̧ — 2ck,+s) x2 + (kaka—2c2ka+3cka—¿a3 — kaks +čk ̧— ns— cs) x + (ka — ckaka — c3ka + ck ̧k+cns — k ̧s) = 0, (15) lutions, or in general as an approximation, c may be placed equal to o, when follows: rx3+ x*+ (k ̧— 2k¿) x3 + (s — 2k ̧)x2 + (k„ką — k ̧k ̧ — ns)x+ (16) (ka-kas) = o. In certain cases, for example, when k, is negligible in comparison with k, or the reverse, a further simplification may easily be made. COMPARISON OF THE EXPERIMENTAL AND THEORETICAL VALUES. 0 We have, now, by means of equation (12), calculated the theoretical solubility of silver benzoate in the various solutions of nitric acid, making use of the following data: k„=m'a' = 0.0000932 (since the solubility in pure water, m = 0.01144, and the corresponding dissociation-value, a, = 0.844, according to conductivity measurements of our own); the dissociation constant of benzoic acid, k = 0.000060, according to Ostwald's measurements, whence the ratio = 1.553. The values of c in the different cases were deduced from the conductivity of silver nitrate as determined by Kohlrausch. The equations were solved in this case and also in that of the chloracetic acid, by repeated substitution of estimated values of x until the exact root was found.' The solubility of silver benzoate in chloracetic acid has been calculated by equation (15), using the same values of k, and k¿, as before, putting the dissociation constant of chloracetic acid, 1 For further details of the method of calculation, reference is made to a more extended article on the theory in the current volume of the Zeitschrift für physikalische Chemie. k = 0.00155, in accordance with the measurements of Ostwald, and determining c from conductivity measurements. In the following tables are given the experimental and the theoretical values, together with the percentage differences between them. The agreement between the found and the calculated values is very satisfactory, especially considering the large number of dissociation values used in the calculation; consequently the solubility principles and the validity of the theoretical laws of dissociation in the case of weakly dissociated acids are again. confirmed. It is specifically shown by this investigation that the solubility of silver benzoate is increased both by nitric acid and chloracetic acid in such a way that the product of the concentrations of the Ag and the C,H,CO, ions remains constant. [CONTRIBUTIONS FROM THE CHEMICAL LABORATORIES OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY.] THE SOLUBILITY OF ACIDS IN SOLUTIONS OF THE BY ARTHUR A. NOYES AND EDWARD S. CHAPIN. I. PURPOSE OF THE INVESTIGATION. 2 N an article published elsewhere by one of us, the theory of the influence of one di-ionic electrolyte on the solubility of 1 Read at the Boston meeting of the American Chemical Society, August, 1898. 2 Ztschr. phys. Chem., 27. another with different ions, was developed for the different cases actually occurring. Certain experiments made with the object of testing the theoretical equations so derived have been presented in the preceding paper; but there remains an especially important case to which the application of the theory has not yet been experimentally tested,—that involving the effect of a neutral salt of a partially dissociated acid on the solubility of another acid, likewise only partially dissociated. An example of this kind would be the effect of sodium acetate or of sodium formate on the solubility of benzoic acid, and we have in fact investigated this case, using each of the salts at different concentrations. 2. DESCRIPTION OF THE EXPERIMENTS. In the first place the preparation of the substances and of the solutions employed will be described. The benzoic acid was prepared from a commercial sample by dissolving in the sodium carbonate solution, partially precipitating with dilute hydrochloric acid, and twice recrystallizing from boiling-water. As powdered benzoic acid probably saturates a solution much more quickly than does the crystallized form, the acid was then melted and finely pulverized. It was shown by qualitative test to be free from chlorine. The sodium acetate and sodium formate used were prepared by twice recrystallizing the commercial salts. Solutions of these were made up by weighing out roughly the air-dried salts; and their concentrations were then accurately determined by evaporating measured portions with hydrochloric acid in platinum dishes, moistening the residue with the acid, and igniting gently until a constant weight was obtained. The three check determinations made in each case agreed within three-tenths per cent. The solubility determinations were made by rotating for about sixteen hours at 25° in the previously described apparatus' bottles containing pure water or the solutions of sodium acetate or sodium formate and an excess of the solid benzoic acid. In one-half of the experiments the state of saturation was approached by cooling the solutions from a higher temperature, 1 Ztschr. phys. Chem., 9, 606. |