CHAPTER VIII

«SIMULTANEOUS CHEMICAL AND PHYSICAL EQUILIBRIUM.—THE SOLUBILITY- OR ION-PRODUCT.»

It frequently happens that we have to deal, simultaneously, with conditions of chemical and of physical equilibrium, obtaining in the same system. For instance, a gas like carbon dioxide, in contact with its saturated solution in water, is in equilibrium with the dissolved carbon dioxide, and this, in turn, is in equilibrium with its hydrate, carbonic acid. A substance may be distributed between two solvents and show a different molecular weight in the two (see p. 18); it may exist, in the one, primarily in polymeric form, and only to a slight extent in the simple form, the two forms being in equilibrium (chemical equilibrium). In the other solvent, it may exist only in its simple molecular form, and this will be in equilibrium with the same simple molecular form in the first solvent (physical equilibrium). In matters dealing with the solubility of electrolytes in water, and, therefore, in questions of their ‹precipitation› or ‹solution›, such simultaneous conditions of chemical and physical equilibrium are constantly occurring. Since qualitative analysis deals to a very considerable extent with just such precipitates of salts, acids and bases, these cases are of particular importance to us.

«Earlier Derivation of the Solubility-Product Principle.»—A very simple relation was derived by Nernst[291] for the combined conditions of chemical and of physical equilibrium, where difficultly soluble electrolytes (precipitates) were concerned. We shall develop the relation first for a simple salt, such as silver acetate.

When water is added to solid silver acetate, the salt will dissolve. If an excess of the acetate is used, equilibrium will result between the solid salt and its solution, when the solution is saturated at the temperature used. As the salt dissolves, it is more or less ionized, and in the saturated solution we have a [p140] condition of chemical equilibrium between the salt and its ions:

CH_{3}COOAg ⇄ CH_{3}COO^{−} + Ag^{+}. (1)

If the law of chemical equilibrium is applied to this reversible action, we have (p. 98)

[CH_{3}COO^{−}] × [Ag^{+}] / [CH_{3}COOAg] = K_{Ionization}. I

The nonionized silver acetate is present in two phases, in the solid phase and also in solution:

CH_{3}COOAg ↓ ⇄ CH_{3}COOAg. (2)

Applying the law of physical equilibrium to this system, we have further (p. 121)

[CH_{3}COOAg] / [CH_{3}COOAg]_{solid} = K. II

The concentration of a pure solid, as we have seen, may be considered a constant at a given temperature. Consequently, if we consider the question of the size of the solid particles as a minor factor and negligible, we shall conclude, that, for saturated solutions of silver acetate, the concentration of the solid silver acetate being a constant, the concentration of the nonionized or molecular silver acetate, the first term of our constant ratio II, must also have some definite, constant value at a given temperature. We may call this concentration the "molecular solubility" of silver acetate and may put

[CH_{3}COOAg] = K_{mol. sol.} III

for a ‹saturated› solution of silver acetate in water at the given temperature. Now, since the concentration of the nonionized silver acetate, [CH_{3}COOAg], in the saturated solution also forms the second term of equation I, representing the condition of chemical equilibrium between the acetate and its ions, we obtain, by combining I and III,

[CH_{3}COO^{−}] × [Ag^{+}] = K_{Ionization} × K_{mol. sol.} = K_{Solubility-Product}. IV

Further, if the assumption is made that the presence of foreign electrolytes, in not too concentrated solutions, does not affect either the molecular solubility, K_{mol. sol.}, of silver acetate or its tendency to ionize—as expressed in K_{Ionization}—then, the [p141] relation, which has been developed, would hold for saturated aqueous solutions of silver acetate in the presence of foreign electrolytes, as well as for a saturated, pure, aqueous solution. A single, simple equation would thus express the conditions for simultaneous chemical and physical equilibrium between a difficultly soluble ionogen, of the type of silver acetate, and its saturated solutions, at a given temperature, in the presence or the absence of foreign electrolytes.

«The Solubility- or Ion-Product Principle.»—We may formulate this important conclusion by stating, that, ‹in saturated solutions of silver acetate, the product of the concentrations of its ions has a constant value at a given temperature›. Analogous relations may be developed for the saturated solutions of other difficultly soluble ionogens. The constant has been called the ‹solubility-product constant› or the ‹ion-product constant› of the ‹ionogen›. For salts like lead iodide PbI_{2}, silver chromate Ag_{2}CrO_{4}, etc., each molecule of which forms more than one of a given ion, the concentration of such an ion is raised, in the solubility-product, to the power corresponding to the number of ions of this kind formed from a single molecule of the electrolyte.[292] Thus, lead iodide ionizes according to the equation PbI_{2} ⇄ Pb^{2+} + 2 I^{−} and, for a saturated solution of lead iodide[293] at a given temperature, [Pb^{2+}] × [I^{−}]^2 = K. For silver chromate, ionizing according to the equation Ag_{2}CrO_{4} ⇄ 2 Ag^{+} + CrO_{4}^{2−}, the form of the solubility-product equation is [Ag^{+}]^2 × [CrO_{4}^{2−}] = K. In general, for a saturated solution of a difficultly soluble salt at a given temperature ‹the product of the ion concentrations, each raised to the power corresponding to the number of that kind of ion formed from the ionization of one molecule of the salt, is a constant›.

«Criticism of the Derivation of the Principle.»—Nernst developed this important relation in 1889, shortly after the theory of ionization was formulated. Since then, however, the soundness of the theoretical development, on which it was based, has been rendered open to question in a way that could hardly have been foreseen at that early stage in the development of the theory of ionization. In the first place, it is known now that the ionization of easily ionizing substances (strong electrolytes) does [p142] not conform to the law of chemical equilibrium (‹vide› p. 108); as far as our present knowledge goes, the ratio in equation I is not a constant, but grows ‹larger› with an increasing total concentration of good electrolytes. In the present case, this total concentration may be increased by the introduction of ‹foreign salts›.[294] In the second place, the second fundamental principle used, the principle of the constant solubility of the dissolved molecular or non-ionized salt, as expressed in equation III, was questioned and disproved by Arrhenius in 1899. The molecular solubility depends on the total concentration of salts in the solution and, in general, decreases with increasing concentration of the total dissolved salts. This result does not invalidate the law of physical equilibrium; it merely means that the presence of salts, especially in appreciable quantities, modifies the nature of the solvent and changes its dissolving power, much as we have different dissolving power shown by different pure solvents, such as water and alcohol.

It appears, however, that while the soundness of this theoretical development of the relations expressed by the solubility-product must be questioned, nevertheless as a ‹matter of experiment›, the ‹product of the ion concentrations› of a difficultly soluble salt is found, in dilute solutions, to be a constant, or sufficiently close to a constant to satisfy all but the most rigorous requirements.[295]

It is, in fact, quite evident, that a ‹decreasing value for the second term of the ratio I›—namely, for [CH_{3}COOAg], the molecular solubility of the salt—as the total concentration of the electrolytes present increases, together with an ‹increasing value for the whole ratio I› under the same conditions, are ‹not incompatible with a constant value› of the first term of the ratio. That is, ‹the product of the ion concentrations›, [CH_{3}COO^{−}] × [Ag^{+}], ‹may well remain constant› (equation IV), or approximately constant, in dilute salt solutions, even if equations I and III do not hold for salt solutions. [p143]

Whether in the case of all difficultly soluble salts, as the total salt concentration increases, the increasing values of the chemical equilibrium ratio (equation I) will be so nicely balanced by the decreasing values of the molecular solubility, that the first term of the first ratio (the solubility-product) will always be constant, is a question demanding further extended investigation.[296] The range of the investigation must be extensive, because it must include several other classes[297] of salts (‹e.g.› Me_{2}X, MeY_{2}, etc.), for which the first equation has a different form; for instance, for Me_{2}X,

[Me^{+}]^2 × [X^{2−}] / [Me_{2}X] = K.

For the present we must remain content with the result of the past investigations and consider the principle of the constant solubility-product to be essentially an empirical one. It is an extremely convenient condensation, into a very simple mathematical form, of the main factors involved in the precipitation and solution of difficultly soluble salts, acids, and bases. It should be used with due knowledge of its character and limitations.

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.»—For a saturated aqueous solution of silver acetate at a given temperature, the product of [p145] the ion concentrations may be considered a constant, [CH_{3}COO^{−}] × [Ag^{+}] = K_{S.P.}.

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 K_{S.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› [CH_{3}COO^{−}] × [Ag^{+}] > K_{S.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′› [CH_{3}COO^{−}] × [Ag^{+}] / ‹x′› = K_{S.P.}.

«Precipitation.»—We see thus that ‹precipitation› of a difficultly soluble ionogen ‹will result when the product of the ion concentrations› [p146] ‹is made greater than the value of the solubility-product constant for that substance›. In the second place, it is seen that the concentration of an ion of an insoluble salt, which can be present in the saturated solution of the salt, ‹is dependent on the concentration of the other ion (or ions) of the salt›. This fact is a very important one in analytical chemistry and it is taken advantage of in many ways, as we shall presently see.

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.› [CH_{3}COO^{−}] × ‹y› [Ag^{+}] > K_{S.P.}, and precipitation results. Silver acetate therefore crystallizes out, until

([CH_{3}COO^{−}] / ‹y′›) × ‹y′› [Ag^{+}] = K_{S.P.}.

The following table shows the relations when sodium acetate is added to the saturated solution of silver acetate. Column 1 gives the concentration of the sodium acetate in the solution saturated with silver acetate, column 2 the percentage of the sodium acetate that is ionized, column 3 the total concentration of silver acetate in the saturated solution, column 4 the percentage of it which is ionized, column 5 the concentration of the acetate-ion, column 6 the concentration of the silver-ion and column 7 the value of the solubility-product.

1 2 3 4 5 6 7 Na-Acet. 100 p. Ag-Acet. 100 p′. acetate. [Ag^{+}]. K_{S.P.}. 0 .... 0.0603 70.8 0.0427 0.0427 0.00182 0.061 78.6 0.0392 64.5 0.0735 0.0258 0.00185 0.119 75.8 0.028 59.7 0.1065 0.0167 0.00179 0.239 70.8 0.0208 52.3 0.1727 0.0109 0.00188

The second table shows the relations when an excess of the silver-ion is present, silver nitrate having been added to the saturated silver acetate solution. The columns have the same significance as in the first table, excepting that the first column gives the concentration of silver nitrate present and the second column its degree of ionization.[306]

It is clear from these results that a difficultly soluble salt is rendered less soluble (see column 3 of the tables) by the presence of another salt, when the [p147] latter has an ‹ion in common with the former›. This conclusion has been well established[307] for a considerable number of salts.[308]

1 2 3 4 5 6 7 AgNO_{2}. 100 p. Ag-Acet. 100 p′. acetate. [Ag^{+}]. K_{S.P.}. 0. .... 0.0603 70.8 0.0427 0.0427 0.00182 0.061 82.0 0.0417 64.0 0.0267 0.0767 0.00204 0.119 78.4 0.0341 58.6 0.0200 0.1142 0.00227 0.239 74.0 0.0195 51.7 0.0100 0.1809 0.00182

«Applications in Analysis.»—A few instances of the application of this relation in analysis follow. The determination of the sulphate-ion is based on the precipitation of barium sulphate from solutions of sulphates. The solubility of barium sulphate in water at 18° is 0.0023 gram, or 0.0023 / 233 = 1E−5 mole per liter. In such an extremely dilute solution, the salt may be considered to be completely ionized, and the value of the solubility-product constant is found from K_{S.P} = [Ba^{2+}] × [SO_{4}^{2−}] = (1E−5)^2, or 1E−10. Now, the amount of sulphate-ion left in solution, which would be about one milligram per liter of the aqueous solution, may be reduced by the use of a small excess of the precipitant, barium chloride. An excess of as little as 0.2 gram or 0.001 mole of BaCl_{2} per liter would increase the concentration of the barium-ion about one hundredfold, and barium sulphate would be precipitated, until the concentration of the sulphate-ion had been reduced about one hundredfold. The loss of the sulphate-ion is thus reduced to approximately 0.01 milligram per liter.

In passing, we may ask what the approximate loss of dissolved nonionized barium sulphate would amount to. The value of the ratio ([Ba^{2+}] × [SO_{4}^{2−}]) : [BaSO_{4}], representing the ionization of barium sulphate, is unknown for the extreme dilution represented [p148] by the saturated solution. If we assume it to be roughly of the order 2000 : 1,[309] the solubility of nonionized barium sulphate at 18° would be roughly 0.05 milligram per liter.

As a rule, then, in the absence of complicating conditions,[310] an excess of the precipitant promotes the complete precipitation of an ionogen.

«Washing of Precipitates.»[311]—When barium sulphate has been brought on the filter and the excess of precipitant is to be washed out, then, as the excess of barium chloride is removed, the sulphate becomes more soluble again. It is advisable, therefore, to wash the precipitate as effectively as possible with a very small volume of water—as a rule, the water is used in a very fine stream or is applied drop by drop.

Such precautions are still more important in the case of precipitates which are somewhat more soluble than is barium sulphate, and in such cases the question must be considered, whether as a washing fluid, some solution may not be used, which contains an ion in common with the precipitate, and which has, therefore, according to the principle of the solubility-product, a very much smaller dissolving power for the precipitate in question than pure water. That is a resource of the analyst to which recourse is occasionally taken. Lead sulphate, for instance, is washed with a very dilute solution of sulphuric acid, rather than with pure water;[312] potassium cobaltinitrite, K_{3}Co(NO_{2})_{6}, which is used in the separation of cobalt from nickel, is washed[313] with a ten per cent solution of potassium acetate, containing a little potassium nitrite. Ammonium phosphomolybdate, used in determinations of phosphates, is washed with a solution of ammonium nitrate. [p149]

The use of such solutions for washing precipitates is limited by the necessity, first, of avoiding salts which interfere with subsequent operations (‹e.g.› which would leave nonvolatile residues in the subsequent ignition of a precipitate, that is to be weighed after ignition) and, second, of avoiding the loss of precipitate by the formation of complex ions between an ion of the precipitate and a component of the washing mixture (see p. 148). But wherever such complications can be excluded, the method is a desirable one.

It has sometimes been recommended to wash a precipitate with a ‹saturated› aqueous solution of the precipitate itself, in place of with pure water. It was reasoned that the solution, being already saturated with the salt, would not be able to dissolve any of the precipitate obtained. That is true; but if a saturated solution of a salt, MeX, is placed on a filter still holding an excess of the precipitant, ‹i.e.› one of the ions, say X, of the precipitate, then this excess may cause supersaturation of the saturated washing fluid and some of the salt may be precipitated out of the washing fluid. The method, as commonly employed, has therefore the inherent fault, theoretically at least, of being liable to give too high results. If it is to be employed without error, precautions must be taken first to remove, from the precipitate and filter, the mother liquor (containing the excess of precipitant) as completely as possible. If in a given case this can be accomplished, then the danger of precipitating any of the salt (MeX) from the saturated solution is avoided, and the precipitate (MeX ↓) may then be further washed with a saturated solution of the same salt (MeX), with advantage, in certain cases. Thus, in the Lindo-Gladding method[314] of determining potassium in the form of potassium chloroplatinate, the source of error, just discussed, has been avoided in the following way: the excess of precipitant, chloroplatinic acid H_{2}PtCl_{6}, which has the ion PtCl_{6}^{2−} in common with the precipitate, is ‹first removed› from the precipitate by thorough washing of the precipitate with alcohol; subsequently, other impurities, ‹e.g.› sulphates, soluble in water but not in alcohol, are washed out with an aqueous solution of ammonium chloride[315] that has been saturated with potassium chloroplatinate. The method gives good results.

«The Solubility-Product in Volumetric Analysis.»—A final instance of the application of the solubility-product principle to the ordinary methods of analytical chemistry, may be taken from the field of quantitative, volumetric analysis. A particularly accurate method of determining silver consists in precipitating it as silver chloride by means of a standardized solution of sodium chloride. The aim of the method is to recognize, as exactly as [p150] possible, the point where the action AgNO_{3} + NaCl → AgCl ↓ + NaNO_{3} has just completed itself, ‹i.e.› where one equivalent of sodium chloride has been added for just one equivalent of silver nitrate present. A very sensitive method[316] depends on the fact that silver chloride can be made to coagulate by the vigorous shaking of its suspensions, and that the coagulated chloride settles rapidly, leaving a clear supernatant liquid, in which the appearance of the faintest turbidity may be recognized, when the sodium chloride solution is carefully added to the silver nitrate solution under investigation. Now, when sodium chloride solution is added in this way to silver nitrate, a point is reached (called the "neutral point"),[317] where the addition of a further drop or two of sodium chloride solution will still produce a precipitate, and where one would be inclined to decide that too little of the chloride had been used to complete the action. But, at the same time, the addition of a few drops of silver nitrate solution to the solution at the "neutral point" also produces a precipitate of silver chloride, seemingly indicating that an excess of sodium chloride had been used, and apparently contradicting the previous result. As a matter of fact, such a behavior is exactly what is to be expected from a solution, when exactly equivalent quantities of silver nitrate and sodium chloride have been brought together in solution. The solution is then saturated with silver chloride, K_{S.P.} = [Ag^{+}] × [Cl^{−}] and [Ag^{+}] = [Cl^{−}], and the further addition ‹either› of the silver-ion (silver nitrate) ‹or› of the chloride-ion (sodium chloride) should produce a precipitate, according to the principle of the solubility-product. The correct end-point in the determination is, thus, the "neutral point," for at that point the quantity of silver present is equivalent to the quantity of sodium chloride added.

«Effect of Electrolytes with No Ion in Common with the Precipitate.»—In the precipitation of silver acetate from its saturated solution by the addition of sodium acetate (p. 145), it is the acetate-ion, according to the principle of the solubility-product, which is effective—the sodium-ion has no part in the action. Also, in a similar way, the precipitation produced by the addition of silver nitrate is ascribed to the increased concentration of the silver-ion—the nitrate-ion has no share in the action. We may ask, what the effect on the solubility of silver acetate will be, if a salt, sodium nitrate, yielding ‹only foreign ions›, and no common one, is added to its saturated solution. The presence of sodium and of nitrate ions will not increase the concentration of either of the ions of silver acetate, will not increase the value of either factor in the product of ion concentrations; the addition of sodium nitrate, therefore, should not lead to the precipitation of any silver acetate, and as a matter of experiment it does not (‹exp.›). A closer study [p151] of the conditions will show, in fact, that it renders silver acetate somewhat ‹more soluble›. The addition of the sodium-ion will lead to the suppression of some of the acetate-ion, nonionized sodium acetate being formed to a certain extent; the nitrate-ion will combine with some of the silver-ion to form nonionized silver nitrate: thus the concentrations of both of the ions of silver acetate ‹are reduced›, and the product of ion concentrations is rendered smaller than the solubility-product constant, ([CH_{3}COO^{−}] − ‹x›) × ([Ag^{+}] − ‹y›) < K_{S.P.}.

Since the concentrations of both the silver and the acetate ions are reduced, they will not combine as rapidly as before to form nonionized silver acetate, and the conditions of equilibrium between the latter and its ions must be disturbed. The nonionized salt ionizes, for a moment, more rapidly than it is formed and its concentration will thus be reduced. We, therefore, might expect the solution to become ‹unsaturated› in respect to the nonionized form, and the solid salt, if present, should go into solution. In other words, the addition of a salt with two foreign ions should ‹increase the solubility of a difficultly soluble salt› (it is understood that no salt is used which would precipitate a new, less soluble salt). This expectation has also been fully confirmed by careful quantitative determinations, especially by A. A. Noyes and his collaborators. The effect may be demonstrated more easily by the addition to silver acetate of an electrolyte which will very thoroughly suppress one of its ions. Nitric acid is such an agent. The hydrogen-ion will very decidedly reduce the concentration of the acetate-ion, acetic acid being a weak acid (table, p. 104). There is no difficulty in recognizing the anticipated effect (‹exp.›).

«Solution of Precipitates.»—We find, then, that ‹when the product of the ion concentrations of a difficultly soluble salt becomes smaller than the solubility-product constant, the solution is unsaturated› in regard to the salt, and ‹the solid salt›, if present, ‹will go into solution›.

«Summary.»—The conclusions concerning the application of the solubility-product constant may be summarized as follows: A solution is saturated with a difficultly soluble ionogen when the product of the concentrations of the ions of the ionogen is equal to the characteristic solubility-product constant (at the given temperature). A solution is supersaturated and precipitation will follow, if the product of the ion concentrations is greater than the constant. [p152] (See p. 122 in regard to precautions against prolonged supersaturation.) A solution is unsaturated, and the ionogen, if present, will dissolve when the product of the ion concentrations is smaller than the constant.

«Further Considerations Concerning Precipitation and Solution.»—It is further evident that precipitation is favored, if the precipitating agent contains an electrolyte which produces the precipitating ion readily; for instance, carbonic acid does not precipitate any barium carbonate from a solution of barium chloride (‹exp.›)—carbonic acid being an exceedingly weak acid and producing only a minute concentration of the carbonate-ion CO_{3}^{2−}, necessary for the precipitation; but sodium carbonate, a readily ionized salt, will precipitate the barium carbonate quantitatively (p. 90). If some alkali is added to the mixture of barium chloride and carbonic acid (‹exp.›), the latter is converted into a readily ionized salt, the concentration of the carbonate-ion is thus decidedly increased, and the precipitate forms instantly. In the same way, if hydrogen sulphide—a still weaker acid (table, p. 104), which forms only minute quantities of sulphide and hydrosulphide ions (S^{2−} and HS^{−})—is passed through a solution of ferrous sulphate, it fails to precipitate ferrous sulphide; but a salt of hydrogen sulphide, ammonium sulphide, for instance, precipitates ferrous sulphide quantitatively (‹exp.›).

On the other hand, solution of an ionogen is evidently favored and its precipitation rendered more difficult, if we suppress one (or both) of its ions. Thus barium phosphate, calcium carbonate, silver borate, and many salts of weak acids, that are very difficultly soluble in water, are quite easily soluble in strong acids, which suppress the anions by converting them into little ionized acids. When a precipitate is dissolved by the addition of a reagent, such as an acid, an alkali, ammonia, ammonium sulphide—chemical solvents most frequently used in analytical work—we may, as a general principle, consider that the reagent must affect one or both of the ions of the precipitate in question, suppressing it (or them) and thereby making solution possible. The problem of determining in what way the suppression of the ion is effected, must then be faced. Many occasions to determine such questions[318] will arise. [p153]

We have, therefore, a certain degree of control over the precipitation and solution of electrolytes, the control depending upon, and being limited by, the fact that the ‹factors of the product of ion concentrations are variables›.

On the other hand, we have little control, in a given solvent, over the question of solution or precipitation as affected by the value of the ion product ‹constant›, the remaining term in the equation of the solubility-product for saturated solutions. These constants cover a very wide range of values for the various salts, which are most frequently used in analytical work for the precipitation of the common ions.[319] They are subject to variation with the temperature, and, as a rule, as most salts are more soluble at higher than at lower temperatures, the values of the constants increase with the temperature. For exceedingly difficultly soluble salts, the increase is commonly of no practical moment in analytical work, when, by an excess of the precipitant, the ion, which is to be precipitated, can be precipitated quantitatively; the solubility of the nonionized salt, that is precipitated, is so minute (see p. 148) in this case, even at high temperatures, that it is altogether negligible for ordinary purposes.[320] On the other hand, precipitates are often used which are not at all extremely insoluble but merely rather difficultly soluble; they are used in spite of their relatively slight insolubility because they are the best available forms for our purposes. Such salts are, for instance, lead chloride, magnesium-ammonium phosphate, potassium chloroplatinate. When these are precipitated, not only must the fact that they are appreciably soluble at ordinary temperature be taken into account, but also the fact that they are very much more soluble at higher temperatures. Lead chloride and potassium chloroplatinate are, for instance, quite soluble in hot water.

As a rule, we select for the form in which a given ion is to be precipitated, a form which, in a saturated aqueous solution, shows the ‹smallest concentration of the ion in question›. But if no form is [p154] known which is sufficiently insoluble to give satisfactory quantitative results, then we have recourse to a ‹change› in the ‹solvent›.

«Solubility and Solvent.»—For instance, a mixture of alcohol and water may be used, or water be excluded altogether; and either absolute (water-free) alcohol or a mixture of alcohol and ether may be employed. In the quantitative treatment of potassium chloroplatinate, the last-named mixture is used in place of water. The change of solvent affects the solubility by a change both in the solubility of the ionized portion of a salt and in that of the nonionized salt. An important quantitative relation between the solubility of a given ionogen in different solvents and the ionizing powers of the solvents, as determined by their dielectric constants (p. 63), was predicted, on the basis of theoretical considerations, by Malström[321] and by Baur.[322] Walden[323] has furnished experimental confirmation of the relation: ‹The degree of ionization of a salt is found to be the same in its saturated solutions in different solvents›, when the solutions are saturated at the same temperature.

If this relation is combined with that discussed on page 63, according to which the degree of ionization of a given salt, in different solvents, ‹is the same›, when the cube roots of its concentrations are directly proportional to the dielectric constants of the solvents (‹e›_{1} : ∛‹c›_{1} = ‹e›_{2} : ∛‹c›_{2} = a constant), then we find, that in ‹saturated solutions of a given salt, in different solvents, the cube roots of the concentrations, or solubilities, are directly proportional to the dielectric constants of the solvents›, or, ‹the solubilities are proportional to the third powers of the dielectric constants›.

‹e›_{1} : ∛‹c›_{1} = ‹e›_{2} :∛‹c›_{2} = a constant, or ‹e›_{1}^3 : ‹e›_{2}^3 = ‹c›_{1} : ‹c›_{2},

‹c›_{1} and ‹c›_{2} representing the solubilities, in molar concentrations, in two solvents of dielectric constants ‹e›_{1} and ‹e›_{2}.

The following table illustrates the relations for a salt examined by Walden, a derivative of ammonium iodide, namely tetraethyl ammonium iodide (C_{2}H_{5})_{4}NI. The first column gives the name of the solvent, the second the solubility or concentration in the saturated solution, in terms of the proportion of moles of the solute to the total number of moles present[324] [p155] (solute + solvent); the third column gives the dielectric constant, under comparable conditions, and the last column gives the relation ‹e› : ∛‹c›.

Solvent. Solubility. ‹e›_{5} ‹e› : ∛‹c› Water 0.0332 75.0 50.5 Nitrobenzene 0.0020 32.2 54.8 Ethyl alcohol 0.00201 26.6 45.5 Acetone 0.00072 21.8 52.8 Amyl alcohol 0.00031 15.0 48.

In view of the difficulties in determining the values for the dielectric constant, the agreement in the values of the last column must be considered satisfactory.[325]

This important principle forms another striking instance of the ‹supreme influence of electrical relations in determining the behavior of ionogens in solution› (see p. 111).

Since, in solutions saturated at the same temperature with a given ionogen, the degree of ionization of the ionogen is the same in both solvents, the proportion of nonionized to ionized salt is also the same. If a salt, ‹e.g.› calcium sulphate, is less soluble in alcohol than in water, the alcohol must hold less of the nonionized form, as well as less of the ionized salt, than does an equal volume of water at the same temperature.

The development of further relations, of fundamental importance to analytical chemistry, with the aid of the laws of chemical and physical equilibrium and of the principle of the solubility-product, will be taken up in the study of the reactions of the various analytical groups of ions.

FOOTNOTES:

[291] ‹Z. phys. Chem.›, «4», 372 (1889). See also van 't Hoff, ‹ibid.›, «3», 484 (1889).

[292] ‹Cf.› page 94.

[293] A. A. Noyes, ‹Z. phys. Chem.›, «9», 618 (1892); Findlay, ‹ibid.›, «34», 409 (1900).

[294] As foreign salts affect the ionization of poor «electrolytes» (p. 109), the ratio of equation I would hold as little for poor electrolytes, and would grow larger with an increased concentration of the foreign salts.

[295] ‹Cf.› A. A. Noyes, ‹Congress of Arts and Sciences› (St. Louis), «4», 321 (1904) and Stieglitz, ‹J. Am. Chem. Soc.›, «30», 946 (1908) («Stud.»), and the references to literature given there. The empirical relation seems to hold for dilute solutions, the total electrolyte concentration of which is not greater than 0.2 to 0.3 gram-equivalent per liter, and, roughly, for concentrations not greater than 0.5 gram-equivalent per liter.

[296] See Stieglitz, ‹loc. cit.›

[297] Since the writing of this it has been learned that such investigations have been carried out by Harkins. ‹Cf.› ‹J. Am. Chem. Soc.›, 1911.

[298] ‹J. Am. Chem. Soc.›, «32», 488 (1910).

[299] Otherwise a ‹perpetuum mobile› of the ‹second class› (footnote 3, p. 12) could be constructed, which is at variance with experience.

[300] This sentence is quoted from a letter from Dr. Washburn, who is at present investigating moderately concentrated solutions of electrolytes, to determine the range of concentrations in which it is possible to apply the laws of ideal solution.

[301] ‹Cf.› Geffcken, ‹Z. Phys. Chem.›, «49», 257 (1907), and the references given there.

[302] ‹Cf.› Arrhenius, ‹loc. cit.›, and similar investigations on the "salt effect" (p. 109).

[303] ‹Vide› Geffcken, ‹loc. cit.›, 295, and Stieglitz, ‹loc. cit.›, and p. 142.

[304] See also Hill, ‹J. Am. Chem. Soc.›, «32», 1186 (1910). Hill attacks the principle as a whole, but brings no evidence against its validity for solutions of concentrations up to 0.3.

[305] The limit of concentration depends, for constancy, upon the nature of the salts. The calculations, on which the data in the tables on pp. 146–7 are based, involve extrapolations which prevent the results, especially for the more concentrated solutions, being considered as final.

[306] For further illustrations, ‹vide› Stieglitz, ‹J. Am. Chem. Soc.›, «30», p. 947 (1908), and the references given there to the work of Noyes, Findlay, etc.

[307] Some instances are known where the solubility of a salt is ‹increased› by the addition of a salt with a common ion. In such cases it is extremely likely that an ion of the salt in question forms a ‹complex ion› with a component of the solution. ‹Vide› A. A. Noyes, ‹Z. phys. Chem.›, «6», 241 (1890), and «9», 603 (1892). In Chapter XII we shall discuss, in detail, instances of this nature where the formation of complex ions is particularly susceptible of ‹exact experimental verification›.

[308] Especially by Noyes, ‹loc. cit.›, and later papers; Findlay, ‹loc. cit.›

[309] This is the value for a similar ratio for KCl of the same concentration as found, by extrapolation, from the data in the table on p. 108.

[310] Owing to the possibility of the formation of complex ions (Chapter XII), each individual case must be considered by itself and the most favorable conditions for the complete precipitation determined experimentally. The rule mentioned is to be used as a guide, and the reference to the possibility of the formation of complex ions considered as a warning, in the planning of such investigations.

[311] ‹Cf.› p. 136, concerning precautions used to prevent precipitates from assuming the ‹colloidal› state.

[312] Fresenius, ‹Quantitative Analysis›, I, 355 (1904).

[313] ‹Ibid.›, I, 307.

[314] Official Methods of Analysis, Bulletin 107, p. 11, U. S. Dept. of Agriculture.

[315] The excess of chloroplatinic acid is first washed out of the precipitate primarily to avoid subsequent precipitation of ammonium chloroplatinate, but its removal also avoids the error discussed in the text.

[316] Gay-Lussac's method.

[317] Mulder. See Sutton's ‹Volumetric Analysis›, p. 304 (1904).

[318] ‹Vide› Chapters XII and XIII.

[319] A table of exact solubilities is given at the end of the Lab. Manual, ‹q.v.›

[320] In the most exact quantitative work, as demanded in the determinations of atomic weights, every known loss must, as far as possible, be measured and taken into account. Beautiful instances of such work are found in T. W. Richards' classic determinations of atomic weights. See, for instance, Richards, ‹Carnegie Institution Publications›, No. 125 (1910), Determinations of the Atomic Weights of Silver, Lithium and Chlorine («Stud.»).

[321] ‹Z. Elektrochem.›, «11», 797 (1905).

[322] ‹Ibid.›, «11», 936 (1905), and «12», 725 (1906).

[323] ‹Z. phys. Chem.›, «55», 707 (1906), and «61», 638 (1907).

[324] If ‹n› is the number of moles of solute dissolved in ‹N› moles of solute, the concentration of the solute may be expressed as ‹n› / (‹n› + ‹N›), which is called its "mole fraction." This form of expressing concentrations is in many particulars preferable to the mole / liter form. For very dilute solutions (‹n› is very small compared with ‹N›) the two forms become practically identical, but they are not so for more concentrated solutions, and the ‹mole-fraction› expression is then easier to treat rigorously.

[325] See Walden, ‹loc cit.›, for more extended data.

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