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, [CH3COO−] × [Ag+] = KS.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 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.

—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. [CH3COO−] × y [Ag+] > KS.P., and precipitation results. Silver acetate therefore crystallizes out, until

([CH3COO−] / y′) × y′ [Ag+] = KS.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
Na-Acet.		 2
100 p.		 3
Ag-Acet.		 4
100 p′.		 5
[CH3COO−].		 6
[Ag+].		 7
KS.P..
... 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 established307 for a considerable number of salts.308

1
AgNO2.		 2
100 p.		 3
Ag-Acet.		 4
100 p′.		 5
[CH3COO−].		 6
[Ag+].		 7
KS.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 KS.P = [Ba2+] × [SO42−] = (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 BaCl2 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 ([Ba2+] × [SO42−]) : [BaSO4], 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, K3Co(NO2)6, which is used in the separation of cobalt from nickel, is washed313 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 method314 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 H2PtCl6, which has the ion PtCl62− 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 chloride315 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 AgNO3 + NaCl → AgCl ↓ + NaNO3 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 method316 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, KS.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, ([CH3COO−] − x) × ([Ag+] − y) < KS.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 CO32−, 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 (S2− 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 questions318 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 Malström321 and by Baur.322 Walden323 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 (e1 : ∛c1 = e2 : ∛c2 = 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.

e1 : ∛c1 = e2 :∛c2 = a constant, or e13 : e23 = c1 : c2,

c1 and c2 representing the solubilities, in molar concentrations, in two solvents of dielectric constants e1 and e2.

The following table illustrates the relations for a salt examined by Walden, a derivative of ammonium iodide, namely tetraethyl ammonium iodide (C2H5)4NI. 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 present324 [p155] (solute + solvent); the third column gives the dielectric constant, under comparable conditions, and the last column gives the relation e : ∛c.

Solvent. Solubility. e5 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.

Chapter VIII 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.

PART II SYSTEMATIC ANALYSIS AND THE APPLICATION OF FUNDAMENTAL PRINCIPLES

CHAPTER IX SYSTEMATIC ANALYSIS FOR THE COMMON METAL IONS. THE IONS OF THE ALKALIES AND OF THE ALKALINE EARTHS. ORDER OF PRECIPITATION OF DIFFICULTLY SOLUBLE SALTS WITH A COMMON ION

[p157] TOC

In systematic analysis it is most convenient to make separate examinations for the metal and for the acid ions. The examination for metal ions usually precedes that for the acid ions, and the scheme of analysis for the former will be considered first.

The analytical grouping of the metallic elements is not a natural one, as far as their chemical behavior is concerned. Such a grouping is found in the Periodic System of Mendeléeff and is used in systematic inorganic chemistry.326 The groups in analysis are based chiefly, but not exclusively, on the physical property of greater or smaller solubility of certain salts of the metals. According to the salts chosen, different systems vary somewhat in detail. Frequently elements of the same natural family are also found in the same analytical group, relationship in chemical properties being often coincident with relationship in the physical behavior of the salts of the metals.

In the following list, the common metal ions are arranged in groups, which are given in the order in which they are precipitated in the method of systematic analysis adopted. In each case, a group name and the characteristic reagents used in separating a group from those following it, are given.