εElem., 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 × CElement, CElement 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 εCu, CuSO4 is positive, which means that the metal will be positive, the electrolyte negative. For instance, for [Cu2+] = 1, εCu, CuSO4 is found to be +0.606 (see the table at the end of Chapter XV). εCu, CuSO4 = −εCuSO4, 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 εCu′, 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 εCu, CuX = (0.0575 / 2) log(C / K) = a positive value. When the plates are combined, we find from εCu′, 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.