Chapter XII). The constants may be expressed, as in the text,
in terms of (molar) ‹concentrations› of the ions, or in terms of the ‹osmotic pressures› of the ions, a molar solution at 0° producing an osmotic pressure of 22.4 atmospheres. Where osmotic pressure and concentration are not strictly proportional (‹e.g.› for concentrated solutions), the osmotic pressure, rather than the concentration, is the determining factor and, when known, is used in exact calculations. The plan, pursued in the text, is adopted in order to express these constants in the terms used for all the other equilibrium constants. It should be recalled (‹e.g.› p. 30) that in calculations, in general, where pressure and concentration are not strictly proportional, the pressure is the determining factor. A third method of expressing the solution-tension relations consists in giving the ‹potential differences›, which exist ‹between elements› and solutions of their ‹ions, in which the ions have unit (molar) concentration›. These potential differences are ‹functions› of the solution-tension constants, as will be discussed below, and the constants, in terms of concentrations or osmotic pressures, may be easily calculated, from the potential differences, with the aid of this function (see below, and see the table at the end of Chapter XV).
[526] According to Wilsmore's tabulation (‹Z. phys. Chem.›, «36», 92 (1901)), the potential difference ε_{Cu, Cu^{2+}} of copper against a 0.5 molar solution of cupric sulphate, in which [Cu^{2+}] = 0.11, is +0.584 volt. Inserting these values for [Cu^{2+}] and ε_{Cu, Cu^{2+}} in the equation ε_{Cu, Cu^{2+}} = (0.0575 / 2) log([Cu^{2+}] / K) (see below) and solving the equation for K, we find K = 8E−22. For [Cu^{2+}] = 0.24, ε_{Cu, Cu^{2+}} is +0.594 volt and K = 8E−22. In regard to the convention determining the signs used (in the present case ε_{Cu, Cu^{2+}} is ‹positive›), see the footnote below, p. 262, and in regard to the definition of zero potential, to which the potential differences used in this book refer, see the table and summary at the end of Chapter XV.
[527] Nernst, ‹loc. cit.›, p. 151.
[528] In other words, the greater the concentration of cupric-ion, the greater its osmotic pressure must be, and the repelling electric force, required to overcome the pressure of the cupric-ion, would be correspondingly greater.
[529] ‹Cf.› Nernst, ‹Theoretical Chemistry› (1904), pp. 720–723, in regard to the derivation and the general form of his formula.
[530] For elements that form ‹negative ions›, ‹e.g.› for chlorine, bromine, oxygen, etc., the ‹equation reads› (see pp. 273, 275 and the table at the end of Chapter XV):
ε_{Elem., Electrolyte} = −(0.0575 / ‹v›) log(C / K).
Note the ‹changed› sign of the expression on the right. The difference in sign expresses the fact that, when negative ions discharge on an electrode, they render it negative, and when they are formed by an electrode, they leave the latter positive; for positive ions, it will be recalled, the conditions are just the ‹reverse› (see above).
Where a ‹soluble› element (‹e.g.› chlorine) or a solution of a metal (‹e.g.› sodium amalgam) is used as an electrode, its concentration, in general, is not constant, as in the case of a pure, solid metal like copper (p. 258). In such cases, the quantity in the denominator of the ratio in the logarithm cannot be expressed by a constant K, but is expressed by K × C_{Element}, C_{Element} being used to indicate the concentration of the element in the experiment in question.
[531] The convention, adopted in the text, for the use of the positive and negative signs in expressing potentials, is that proposed by Luther (‹cf.› Le Blanc's ‹Lehrbuch der Elektrochemie› (third edition), p. 212). The ‹sign› always ‹denotes› the character of the ‹charge› on the ‹first component› written in the subscript to ε. Thus, for a copper plate in contact with a solution of cupric sulphate, when C > K, the logarithm, log(C / K), has a ‹positive› value and ε_{Cu, CuSO_{4}} is ‹positive›, which means that the ‹metal› will be ‹positive›, the electrolyte negative. For instance, for [Cu^{2+}] = 1, ε_{Cu, CuSO_{4}} is found to be +0.606 (see the table at the end of Chapter XV). ε_{Cu, CuSO_{4}} = −ε_{CuSO_{4}, Cu′}. By this use of the signs one is never in doubt as to their meaning. Unfortunately, widely different definitions of the signs have been used (‹cf.› Le Blanc, ‹Electrochemistry› (1896), pp. 209, 219, and Lehfeldt, ‹Electro-Chemistry› (1904), p. 159). Care must be taken, in using the data of original papers, to be informed as to the definition used.
In accordance with the convention as to signs, adopted in this book, the ratio of concentrations (C / K), used in the logarithm of Nernst's formula, is the ‹reciprocal› of the ratio usually given. The change has been made in order that the algebraic signs of the values obtained from the application of the formula should be the same as those observed in the experimental arrangements, as demanded by the convention.
[532] When two electrodes are combined to form an electric cell or couple, the potential difference of the couple is always the (algebraic) ‹difference› of the two individual electrode potentials, and hence these are ‹subtracted› from each other (algebraically). The electrode of the first term of the difference (the minuend) is named first in the subscript of the potential of the couple; then the sign of the difference represents the character of the charge on that electrode, in agreement with the convention (see footnote 2, p. 261). In illustration: two copper electrodes may be taken, each of which, considered by itself, carries a positive charge, because the concentrations of the cupric-ion in the solutions bathing them are both greater than K; when they are combined, each of the two electrodes will tend to send a positive current, in ‹opposite› directions, into the metal connecting them. But the potential of the electrode with the heavier charge (the one dipping into the solution containing the greater concentration of cupric-ion) will overcome the potential of the other electrode, and the current will flow, through the connecting metal, with a potential that represents the difference between the two values. If the electrode of the more concentrated solution is named first in the subscript of the potential of the couple, its individual electrode-potential appears as the first term of the difference (the minuend) and is reduced by the value of the electrode-potential of the second electrode; as this is numerically smaller than the value of the minuend, the difference will be positive, showing that the electrode in the stronger solution, named first in the subscript of the potential difference of the couple, carries a positive charge. Further, if the second electrode dips into a solution, in which the concentration of the cupric-ion is smaller than K, the logarithmic expression for its electrode-potential will be found to give a negative value; and the (algebraic) subtraction of this negative quantity from the electrode-potential of the first electrode will give a larger potential difference, for the couple, than that possessed by the first electrode alone—all of which agrees with the experimental results, when such combinations are made.
Where negative elements are concerned, the same convention holds, but the ‹logarithmic expression for the potential of such an electrode carries a negative sign› (see footnote 1, p. 261), which must be inserted, algebraically, when the expression is used as a term in the difference under discussion.
[533] If C′ > C″, the logarithm will be positive and ε_{Cu′, Cu″} will have a ‹positive› value, which means that the copper plate, Cu′, ‹which is named first in the subscript to ε›, will be charged positively, ‹when the system works›. If C′ < C″, the logarithm will be negative, which means that the first plate, Cu′, mentioned in the subscript, will receive a negative charge, ‹when the system works›. The sign is therefore intended, by the convention adopted (p. 261), to express any result for the ‹working system›, irrespective of the charge on the individual plates before they are combined. For instance, for C′ = 1 and C″ = 10^{−10}, both plates are positive, ‹before› they are connected with each other, since in each case C > K, and ε_{Cu, CuX} = (0.0575 / 2) log(C / K) = a positive value. When the plates are combined, we find from ε_{Cu′, Cu″} = (0.0575 / 2) log(C′ / C″) that the first plate, dipping in the more concentrated solution of cupric-ion, is ‹positive›, which is confirmed by experiment.
[534] (1 / 10)-molar cupric sulphate, 100 c.c., containing some sodium sulphate or nitrate, to reduce the resistance, is a convenient concentration.
[535] The copper plate is best freed from adhering sulphide by means of a strong cyanide solution, and re-introduced into the solution.
[536] Küster, ‹Z. Elecktrochem.›, «4», 110 and 503 (1897).
[537] In a solution of zinc sulphate in which [Zn^{2+}] = 0.114, the potential ε_{Zn, ZnSO_{4}}= −0.514 (the minus sign indicates that the metal named ‹first› in the subscript has a negative charge). Inserting the values for [Zn^{2+}] and ε_{Zn, ZnSO_{4}} in the general equation given on p. 261, and solving for K, we find K = 10^{17}. For [Zn^{2+}] = 0.022 and ε_{Zn, ZnSO_{4}} = −0.535, we find K = 10^{16.8}. (‹Cf.› Wilsmore's tables, ‹loc. cit.›)
[538] Equilibrium will be established whenever the potential of the system is equal to 0. The potential of the system may be calculated according to the equation (see footnote 1, p. 262)
ε_{Cu, Zn} = ε_{Cu, CuSO_{4}} − ε_{Zn, ZnSO_{4}} = (0.0575 / 2) (log(Cu^{2+} / K_{Cu}) − log(Zn^{2+} / K_{Zn})).
The potential ε_{Cu, Zn} is 0 whenever [Cu^{2+}] / K_{Cu} = [Zn^{2+}] / K_{Zn}, ‹i.e.› when [Zn^{2+}] / [Cu^{2+}] = K_{Zn} / K_{Cu}.
For ions of different ‹valence›, such as silver and cupric ions, the equilibrium equation assumes a somewhat less simple form. For Cu ↓ + 2 Ag^{+} ⇄ 2 Ag ↓ + Cu^{2+}, we have [Ag^{+}]^2 / [Cu^{2+}] = (K_{Ag})^2 / K_{Cu}.
[539] ‹Vide› Ostwald's ‹Lehrbuch der allgemeinen Chemie›, 2d Ed., Vol. II, p. 874, for the historical data on this action. ‹Vide› Küster's experiments, ‹Z. Elektrochem.›, «4», 503 (1897).
[540] See the footnote, p. 267, in regard to the form the equilibrium ratio assumes when metals producing ions of different ‹valence› are used.
[541] The value of the constant is calculated from the data given by Peters, ‹Z. phys. Chem.›, «26», 193 (1898).
[542] The fact that this equilibrium relation has been proved to hold for the action Fe^{2+} ⇄ Fe^{3+} and that it must be taken into account in all oxidation-reduction reactions involving these ‹ions›, in no wise excludes the possibility that other equilibrium relations can also exist between ferrous and ferric compounds. For instance, ferrous hydroxide Fe(OH)_{2} may well have a characteristic tendency of its own to assume a further positive charge (lose an electron) according to Fe(OH)_{2} ⇄ Fe(OH)_{2}^{+}, the potential of which action may, under given conditions, be a ‹main determining factor› in the course of an action, ‹e.g.› in alkaline mixtures. It is not impossible, even, that we also must consider negative ions FeO_{2}^{2−} and their tendency to be oxidized. Evidence would ‹suggest› that ferrous hydroxide, ‹or its negative ion› FeO_{2}^{2−}, may have, indeed, a very ‹great tendency› to be oxidized, possibly much greater than the tendency of Fe^{2+} to form Fe^{3+}. (‹Cf.› Manchot, ‹Z. anorg. Chem.›, «27», 419 (1901), and McCoy and Bunzel, ‹J. Am. Chem. Soc.›, «31», 370 (1909)). Closer investigations of these relations, from a quantitative viewpoint, would probably determine this question and bring exceedingly important relations to light.
[543] ‹E.g.› by the potential of the action Cl_{2} ⇄ 2 Cl^{−}.
[544] The potential of a solution of the iron salts is given by ε = 0.058 log(10^{17} × [Fe^{3+}] / [Fe^{2+}]). In a solution of a ferric salt, if [Fe^{2+}] = 0, the potential would obviously be ∞, which could not present a condition of equilibrium. Equilibrium is established in such a solution, as will be shown further on in the text, by the liberation of chlorine and the formation of ferro-salt, according to 2 Fe^{3+} + 2 Cl^{−} ⇄ 2 Fe^{2+} + Cl_{2}, until the potential, resulting from the tendency of chlorine to form chloride-ion, just balances the tendency of the ferric-ion to form ferro-ion. But when a ferric chloride solution is used as the source of supply of positive electricity, as in the experiment described in the text, ‹both› the ferric-ion and the chlorine tend to charge the platinum electrode with positive electricity and to revert to a condition of equilibrium in reference to their individual constants. The relations are much like those between a cupric salt solution and a copper plate: if [Cu^{2+}] > K_{Cu^{2+}}, equilibrium will be established, as we have seen, by the positive charging of the plate in sufficient degree to oppose the tendency of the cupric-ion to discharge (see p. 259). But when the solution and plate are used as the source of supply for an electric current (p. 264), both the positive charge on the plate, and the tendency of the cupric-ion to discharge and acquire the concentration [Cu^{2+}] = K_{Cu^{2+}}, will supply the positive current. In calculations we ignore the positive charge already deposited on the plate and deal only with the concentration of Cu^{2+}. The chlorine, liberated in a solution of ferric chloride, plays practically the same rôle as does the copper plate in a cupric salt solution, and it can be ignored in the discussion of the combination described in the text. In a ferrous salt solution, in a similar manner, some ferric-ion must always be formed by liberation of hydrogen (see p. 282), until equilibrium is reached according to 2 Fe^{2+} + 2 H^{+} ⇄ 2 Fe^{3+} + H_{2}. Hydrogen plays here the same rôle as chlorine does in the ferric chloride solution.
[545] The condition for equilibrium is [Fe^{2+}] : [Fe^{3+}] = 10^{17}, in a solution considered for itself.
[546] This ratio need not be 10^{17}, since we have two solutions combined with each other and the total potential will be expressed by:
ε = ε_{1} − ε_{2} = 0.058 (log(10^{17} × [Fe^{3+}]_{1} / [Fe^{2+}]_{1}) − log(10^{17} × [Fe^{3+}]_{2} / [Fe^{2+}]_{2}))
= 0.058 log([Fe^{3+}]_{1} × [Fe^{2+}]_{2} / ([Fe^{2+}]_{1} × [Fe^{3+}]_{2})).
Equilibrium is reached when the total potential is 0. Then
[Fe^{3+}]_{1} × [Fe^{2+}]_{2} / ([Fe^{2+}]_{1} × [Fe^{3+}]_{2}) = 1
and [Fe^{2+}]_{1} / [Fe^{3+}]_{1} = [Fe^{2+}]_{2} / [Fe^{3+}]_{2}.
[547] In order to have very decided differences in the speeds of the action in the absence and presence of fluoride, it is best to use an old ferrous sulphate, or ferrous ammonium sulphate, solution which contains considerable ferric salt.
[548] ‹Vide› Peters, ‹loc. cit.›, p. 236.
[549] Ostwald [‹Lehrbuch d. allgem. Chem.›, 2d Ed., II, 883 (1893)], first emphasized the fact that potential differences are a ‹measure› of oxidizing and reducing powers.
[550] The constant is calculated from the data of Küster and Crotogino on the potential of solutions of iodine in potassium iodide [‹Z. anorg. Chem.›, «23», 88 (1900)]. Owing to the formation of complex ions I_{3}^{−}, for which due allowance has not been made in the calculation, and owing to some uncertainty as to the vague definition of the concentration of iodine used, the estimation of the constant can only be considered a rough one. The value given expresses the order of the equilibrium ratio sufficiently well for our present purposes. In a recent paper, Bray and MacKay [‹J. Am. Chem. Soc.›, «32», 914 (1910)] have determined the constant for the formation of the complex ion according to I_{3}^{−} ⇄ I_{2} + I^{−}, which might be used to correct the data of Küster and Crotogino; but in view of other uncertainties and inaccuracies, the correction has not been considered advisable.
Several related methods may be used to calculate the equilibrium constant for [I^{−}]^2 : [I_{2}] = K from the data of Küster and Crotogino. Perhaps the simplest method is the following: A solution of iodine ([I] = 1 / 32 normal, and therefore [I_{2}] = 1 / 64 molar) in 1/8 molar potassium iodide, in which, the degree of ionization being taken into account, [I^{−}] = 0.109, was observed to show a potential ε_{I_{2}, I^{−}} = + 0.860 (the convention as to signs, discussed on p. 261, is used here and the potential, observed against a so-called "calomel electrode," is reduced to the so-called "absolute potential"; ‹cf.› Le Blanc, ‹Lehrbuch der Elektrochemie›, p. 214). Now, ‹there must be a certain concentration of iodide-ion›, which we will call [C], ‹with which iodine› of the above concentration ‹would be directly in equilibrium› and would give no potential at all (‹cf.› pp. 261 and 258 in regard to copper). With a change in the concentration of the iodide-ion, a potential would be produced according to ε_{I_{2}, I^{−}} = 0.0575 log([C] / [I^{−}]). This relation is of exactly the same nature as that developed for the potential of copper plates, immersed in solutions of cupric-ion of different concentrations (but see footnote 1, p. 261, concerning the ‹sign› of the new relation). In the present case, we are dealing with univalent ions, I^{−}, in place of bivalent ions Cu^{2+}, and the factor 0.0575 is used instead of 0.0575 / 2 (see p. 261). If we insert the observed values, [I^{−}] = 0.109 and ε = 0.860, of the experiment described above, into the equation ε_{I_{2}, I^{−}} = 0.0575 log([C] / [I^{−}]) and solve the equation for [C], we find [C] = 10^{14}. ‹That means›, 1 / 64 molar iodine ‹would be directly in equilibrium with a concentration of iodide-ion› = 10^{14} (if this value is inserted for [I^{−}] in the logarithmic equation, the potential is found to be 0). For the condition of equilibrium for I_{2} ⇄ 2 I^{−}, according to [I^{−}]^2 : [I_{2}] = K, we have then (10^{14})^2 : (1 / 64) = K = 6.4E29. Similarly, for [I^{−}] = 0.109 and [I_{2}] = 1 / 512 the potential ε = 0.831 is observed, and the equilibrium constant is found to be 5.1E29. When [I^{−}] = 0.109 and [I_{2}] = 1 / 128, the potential is 0.850 and the constant is calculated to be 5.3E29. The mean value for K is 5.6E29. In these calculations, the formation of ions I_{3}^{−}, affecting the values for [I^{−}] and [I_{2}], has not been considered, and there is some doubt whether the concentrations of iodine, given by Küster and Crotogino, do not represent [I_{2}] rather than [I], as assumed in the calculations. If the former be the case, the mean value of the above experiments would be 2.8E29. The value, used in the text, is considered sufficiently accurate for the purposes of this book.
[551] This relation of the equilibrium constant and the solution-tension constants may be deduced in a manner similar to that for the analogous equilibrium constant for the oxidation of zinc by the cupric-ion, as given in footnote 1, on page 267. The ‹exact› value of the equilibrium constant is uncertain, since K_{I^{−}, Iodine} has not yet been determined with a sufficient degree of accuracy; but the value, used, gives the order of the constant sufficiently well for our purposes, especially when it is considered in connection with the constant given below for the same relation, when the chloride-ion is substituted for the iodide-ion.
[552] This is the value of the constant as calculated from the data given by Wilsmore (‹Z. phys. Chem.›, «36», 91 (1900)) for the solution-tension of chlorine under atmospheric pressure at 18°. The calculation may be made exactly as in the case of the similar constant for iodine (p. 273). ‹There must be a concentration of chloride-ion›, which we will call [C], ‹with which chlorine, of one atmosphere pressure at 18°›, would be directly in equilibrium. The potential of chlorine, against any other concentration of chloride-ion, would be ε_{Cl_{2}, Cl^{−}} = 0.0575 log([C] / [Cl^{−}]). For [Cl^{−}] = 1, ε is +1.694 (see the table at the end of Chapter XV), and inserting these values in our equation and solving it for [C], we find [C] = 2.88E29. That means, that chlorine, at 18° and of atmospheric pressure, would be in equilibrium with chloride-ion of the concentration given. Since chlorine, at this temperature and pressure, has a concentration of 1 / 23.9 moles (at 18°, one mole is contained in 23.9 liters, instead of in 22.4 liters, at O°), we have for the condition of equilibrium: [Cl^{−}]^2 : [Cl_{2}] = (2.88E29)^2 : (1 / 23.9) = 2E60. [Cl_{2}] represents, thus, in the calculation of this constant, the concentration of chlorine gas (see Chapter XV concerning gas electrodes) and not the concentration of the dissolved chlorine; the latter, however, is proportional to the gas concentration (Chapter VII).
[p277]