[521] Rigorous quantitative examination of the relations shows (p. [275]) that these reductions and oxidations do take place, but equilibrium is reached when they have proceeded to so slight an extent, that, qualitatively, they are not always obvious or discernible.
[522] A change in the nature of the solvent changes the value of the equilibrium constant, just as it changes the ionization constant of electrolytes. See p. [61] and see remarks by Sackur, Z. Elektrochem., 11, 387 (1905).
[523] For exceedingly thin films of copper we cannot make this assumption, and for such films the conclusions, that follow, are, in fact, found not to hold. (Overbeck. Vide Le Blanc, Electrochemistry, p. 252 (1896)).
[524] Z. phys. Chem., 4, 129 (1889).
[525] The values of this and similar equilibrium constants are derived by means of Nernst's formula (see below) for the potential difference between an element and solutions of its ions. The derivation involves the assumption that this formula expresses correctly the relation between the potential change and the concentration change at all concentrations. This assumption appears to be justified by all experimental indications thus far observed. The constants are of importance, primarily, for the calculations which can be made with their aid (see below), and may, conservatively, be considered to be essentially "calculation factors" ("Rechengrössen," according to Haber. See pp. [232]–[7], 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 εCu, Cu2+ of copper against a 0.5 molar solution of cupric sulphate, in which [Cu2+] = 0.11, is +0.584 volt. Inserting these values for [Cu2+] and εCu, Cu2+ in the equation εCu, Cu2+ = (0.0575 / 2) log([Cu2+] / K) (see below) and solving the equation for K, we find K = 8E−22. For [Cu2+] = 0.24, εCu, Cu2+ is +0.594 volt and K = 8E−22. In regard to the convention determining the signs used (in the present case εCu, Cu2+ 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):