Washburn[298] derives the principle of the constancy of the solubility-product, without involving in his derivation the relation between ions and nonionized molecules—a relation which, as was stated above, deviates from the law of chemical equilibrium. The deviation, it will be recalled (p. [109]), is generally supposed to be due to the fact that the fundamental kinetic assumption which must be made to derive the law of chemical equilibrium from the kinetic theory, the assumption that there should be none but negligible forces of attraction and repulsion between the molecules (of a gas or solute) which are in equilibrium, is not fulfilled in the case of solutions of strong electrolytes (p. [109]). According to Washburn, if it is assumed that the ions of an electrolyte fulfill this fundamental condition and that only the nonionized molecules do not—the latter causing the deviation from the law of chemical equilibrium—then the principle of the solubility-product constant follows.[299] He sees an approximate confirmation of the assumption made, in the fact that the principle is found, empirically, to be true, and that other relations, developed on the basis of the same assumption, agree with the observations made. [p144]

This theoretical derivation of the principle, like the derivations of the law of chemical equilibrium and of all our laws of dilute solutions, assumes[300] that the nature of the solvent, and consequently of the solution-process, is not changed by added substances, for instance by an excess of the precipitating ionogen. There can be no question, however, that the nature of the solvent must change, as a continuous function, by the addition of electrolytes to solutions. The changed solubilities of inert gases in salt solutions,[301] and a mass of other evidence,[302] lead to this conclusion. The addition of a half mole of sodium chloride to a liter of water reduces the dissolving power of the liquid towards oxygen at 25° by 15%, i.e. by 30% per mole of salt. A weak electrolyte, such as acetic acid, has practically no effect at this concentration, and so the effect must be chiefly due to the ions of sodium chloride; since the salt, in half-molar solution, is ionized 73%, the reduction in the dissolving power would be 30 / 0.73 = 41% per mole of fully ionized salt. The principle of the constant solubility-product cannot be considered as established for solutions more concentrated than 0.2 to 0.3 molar; but it is evident that, in any comprehensive theoretical formulation of the principle for the range in which it is found empirically to hold, the change in the nature of the solvent, which in some cases is conspicuous in 0.5 molar solution, must be taken into consideration as a factor even in more dilute solutions (say 0.05 to 0.3 molar). It seems at present, quite possible, perhaps even probable, that the constancy, in all but the most dilute solutions, is the result of the approximate balancing of two (or more) opposing factors.[303] When we leave the range of concentrations mentioned, and go to more concentrated solutions, these factors seem to be less well balanced and the validity of the principle ceases.[304] For the present it will be safe to consider the principle as an empirical one, holding for solutions of total salt content up to 0.25 or 0.3 molar.[305] For quite dilute solutions the effect of the electrolyte on the solvent would be negligible, and only to such solutions would the theoretical derivation brought forward by Washburn be applicable.

Influence of a Common Ion.

In such an aqueous solution, containing no foreign salts, the concentration of the silver-ion is equal to the concentration of the acetate-ion, since a molecule of silver acetate, when it ionizes, gives one silver ion for every acetate ion formed. The numerical value of the solubility-product may then be calculated, if the solubility of the salt and its degree of ionization are known. For instance, at 16° one liter of water dissolves 10.07 grams of silver acetate, that is, 10.07 / 167, or 0.0603 gram-molecule (mole). Conductivity measurements show that 70.8% of the salt is ionized in such a solution, and consequently the concentration of the silver-ion is 0.708 × 0.0603, or 0.0427. The concentration of the acetate-ion is the same, and the value of the solubility-product constant, obtained by inserting these quantities in the above equation, is KS.P. = 0.0427 × 0.0427 = 0.00182.

Now, if, to the saturated solution of the silver acetate, there are added a few drops of a concentrated solution of sodium acetate or some crystals of solid sodium acetate, the concentration of the acetate-ion is thereby increased and the condition of equilibrium in the solution is disturbed:

x [CH3COO−] × [Ag+] > KS.P..

The concentration of the acetate-ion having been increased, the ion will combine more rapidly than before with the silver-ion, and the concentration of the nonionized salt will be increased. The solution being already saturated with nonionized silver acetate, the excess formed must be precipitated. As a matter of fact, a precipitate of silver acetate is readily obtained in this way (exp.). Precipitation will cease when sufficient silver acetate has crystallized out to make the product of the concentrations of the ions again equal to the solubility-product constant. If, after the crystallization is complete and equilibrium has been reëstablished, the acetate-ion is x′ times as concentrated as it was in the pure aqueous solution, the concentration of the silver-ion must be reduced to 1 / x′ its original value:

x′ [CH3COO−] × [Ag+] / x′ = KS.P..

Precipitation.

It is clear that a corresponding result should be obtained when, to the saturated aqueous solution of silver acetate, an excess of the silver-ion is added—for instance by the addition of solid silver nitrate or of a little of a concentrated solution of this salt (exp.). Here again, the product of the ion concentrations is greater than the constant, i.e. [CH3COO−] × y [Ag+] > KS.P., and precipitation results. Silver acetate therefore crystallizes out, until