Page:Elementary Text-book of Physics (Anthony, 1897).djvu/347

§ 285] Now in order to determine the values of $$u$$ and $$v,$$ we need them combined in another relation; this relation may be obtained from a study of the migration of the ions. For, consider a row of molecules in the electrolyte stretching between the electrodes, of which the ions are moving independently, the positive ions to the right with the velocity $$u,$$ and the negative ions to the left with the velocity $$v.$$ Let $$n$$ represent the number of ions of either sort in unit length of this line. At the end of the short time $$t$$ the relative displacement of the rows of ions will be $$(u + v)t,$$ and the number of ions freed at either end will be the same and equal to $$n (u + v)t.$$ Though the number of ions which are freed at either end is the same, the loss of molecules or of pairs of associated ions is different at the two ends. If a line be drawn perpendicularly across the line of molecules, the number of ions which pass to the right, and therefore the number of molecules lost on the left of this line is $$nut,$$ while the number of molecules lost on its right is $$nvt.$$ If, therefore, we measure the diminution of the substance decomposed at each electrode, the ratio of the values found will be the ratio of the velocities $$u$$ and $$v$$ of the constituent ions. The ratio of one of these losses or diminutions to the sum of them both, or the ratio of the velocity of the corresponding ion to the sum of the velocities of the two ions, is called the migration constant of the ion. The migration constants have been determined for many ions by Hittorf, Nernst, and others. By combining the ratios of the velocities thus found with the sums of the velocities found by Kohlrausch, the velocities may be separately determined. It is found that the velocity of any one ion is the same, whatever be the electrolyte of which it forms a part, provided the solution be sufficiently dilute. This result is a strong confirmation of the theory of the independent motion of the ions upon which the calculations are based.

In many cases, especially when the solution is not very dilute, the molecular conductivity is found to be less than that assigned by theory on the assumption that all the ions of the electrolyte are dissociated. This discrepancy is explained by Arrhenius by the