Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/340

carefully to consider the data on which it is based. The only objection which can be raised is to the assumption that all the molecules are active, and on this the accuracy of the results deduced apparently depends, for if we suppose only a part of the salt to be active, it seems necessary to allow a greater ionic velocity in order that the same current may be carried. Now the work of Arrhenius, Van’t Hoff, Ostwald, and others, gives very strong evidence that only a certain fraction of the number of dissolved molecules are active (whatever we may assign as the cause of this activity), and it seems to have been generally supposed that this was quite inconsistent with Kohlrausch’s hypothesis (see Lodge, ‘B. A. Report,’ 1886, p. 391, et seq.). I have shown, however, that by examining the matter a little more closely the two ideas can be reconciled. The proof has already been published (‘Phil. Mag.,’ July, 1891), but it will be convenient to reproduce it here.

Suppose that the ratio of the number of the active to the whole number of molecules, which measures the “ionization” (to use Professor Fitzgerald’s convenient name), represents in reality the fraction of each second during which each molecule is on the average active. Each molecule is in turn active, but at any instant only a certain fraction of the whole number of molecules are active. (In terms of the dissociation hypothesis this ratio measures the “mean free time” of each ion.) As far as statical effects, such as osmotic pressure are concerned, this is, of course, equivalent to supposing a certain fixed fraction of the whole number of molecules to be active, but when we consider the velocities of the ions the case is different.

Kohlrausch gets the relative veloctity of the ions from the relation

$\mathrm {U} _{1}=u+v=k/m$;

but if we suppose that only $1/n^{\mathrm {th} }$ of the molecules are active, we should apparently have to put $\mathrm {U} _{2}=kn/m$ to get the same current through the solution, which would give $\mathrm {U} _{2}=n\mathrm {U} _{1}$.

But this $\mathrm {U} _{2}$ represents the actual velocity of the ions while they are active (or “free”), and if we take a dynamical view of the ionization equilibrium, they are active only for $1/n^{\mathrm {th} }$ part of their time. While inactive (or “combined”) they have no relative velocity, and so their average velocity for any long time is $1/n^{\mathrm {th} }\,\mathrm {U} _{2}=\mathrm {U} _{1}$—the same as in Kohlrausch’s hypothesis. The view that all the salt is active supposes a uniform ionic velocity—the view above detailed supposes a series of rests interposed with a series of intervals during which the ion is moving forward with a velocity which is, while it lasts, on the average $=n\mathrm {U} _{1}$; but the final result is the same, the effective velocity is $\mathrm {U} _{1}$.

We can, therefore, combine Kohlrausch’s theory with the supposition that some of the molecules present in the solution are inactive, the result being that the presence of the inactive or non-electrolytic molecules, which decreases the molecular conductivity of the solution, shows itself by diminishing the effective velocities of the ions.