when this ceases to hold, the concentration of the solution has in general become so great that the conductivity of the solvent may be neglected. The general result of these experiments can be represented graphically by plotting k/m as ordinates and 3√m as abscissae, 3√m being a number proportional to the reciprocal of the average distance between the molecules, to which it seems likely that the molecular conductivity may be related. The general types of curve for a simple neutral salt like potassium or sodium chloride and for a caustic alkali or acid are shown in fig. 4. The curve for the neutral salt comes to a limiting value; that for the acid attains a maximum at a certain very small concentration, and falls again when the dilution is carried farther. It has usually been considered that this destruction of conductivity is due to chemical action between the acid and the residual impurities in the water. At such great dilution these impurities are present in quantities comparable with the amount of acid which they convert into a less highly conducting neutral salt. In the case of acids, then, the maximum must be taken as the limiting value. The decrease in equivalent conductivity at great dilution is, however, so constant that this explanation seems insufficient. The true cause of the phenomenon may perhaps be connected with the fact that the bodies in which it occurs, acids and alkalis, contain the ions, hydrogen in the one case, hydroxyl in the other, which are present in the solvent, water, and have, perhaps because of this relation, velocities higher than those of any other ions. The values of the molecular conductivities of all neutral salts are, at great dilution, of the same order of magnitude, while those of acids at their maxima are about three times as large. The influence of increasing concentration is greater in the case of salts containing divalent ions, and greatest of all in such cases as solutions of ammonia and acetic acid, which are substances of very low conductivity.
Theory of Moving Ions.—Kohlrausch found that, when the polarization at the electrodes was eliminated, the resistance of a solution was constant however determined, and thus established Ohm’s Law for electrolytes. The law was confirmed in the case of strong currents by G. F. Fitzgerald and F. T. Trouton (B.A. Report, 1886, p. 312). Now, Ohm’s Law implies that no work is done by the current in overcoming reversible electromotive forces such as those of polarization. Thus the molecular interchange of ions, which must occur in order that the products may be able to work their way through the liquid and appear at the electrodes, continues throughout the solution whether a current is flowing or not. The influence of the current on the ions is merely directive, and, when it flows, streams of electrified ions travel in opposite directions, and, if the applied electromotive force is enough to overcome the local polarization, give up their charges to the electrodes. We may therefore represent the facts by considering the process of electrolysis to be a kind of convection. Faraday’s classical experiments proved that when a current flows through an electrolyte the quantity of substance liberated at each electrode is proportional to its chemical equivalent weight, and to the total amount of electricity passed. Accurate determinations have since shown that the mass of an ion deposited by one electromagnetic unit of electricity, i.e. its electro-chemical equivalent, is 1.036×10−4×its chemical equivalent weight. Thus the amount of electricity associated with one gram-equivalent of any ion is 104/1.036 = 9653 units. Each monovalent ion must therefore be associated with a certain definite charge, which we may take to be a natural unit of electricity; a divalent ion carries two such units, and so on. A cation, i.e. an ion giving up its charge at the cathode, as the electrode at which the current leaves the solution is called, carries a positive charge of electricity; an anion, travelling in the opposite direction, carries a negative charge. It will now be seen that the quantity of electricity flowing per second, i.e. the current through the solution, depends on (1) the number of the ions concerned, (2) the charge on each ion, and (3) the velocity with which the ions travel past each other. Now, the number of ions is given by the concentration of the solution, for even if all the ions are not actively engaged in carrying the current at the same instant, they must, on any dynamical idea of chemical equilibrium, be all active in turn. The charge on each, as we have seen, can be expressed in absolute units, and therefore the velocity with which they move past each other can be calculated. This was first done by Kohlrausch (Göttingen Nachrichten, 1876, p. 213, and Das Leitvermögen der Elektrolyte, Leipzig, 1898) about 1879.
In order to develop Kohlrausch’s theory, let us take, as an example, the case of an aqueous solution of potassium chloride, of concentration n gram-equivalents per cubic centimetre. There will then be n gram-equivalents of potassium ions and the same number of chlorine ions in this volume. Let us suppose that on each gram-equivalent of potassium there reside +e units of electricity, and on each gram-equivalent of chlorine ions −e units. If u denotes the average velocity of the potassium ion, the positive charge carried per second across unit area normal to the flow is n e u. Similarly, if v be the average velocity of the chlorine ions, the negative charge carried in the opposite direction is n e v. But positive electricity moving in one direction is equivalent to negative electricity moving in the other, so that, before changes in concentration sensibly supervene, the total current, C, is ne(u + v). Now let us consider the amounts of potassium and chlorine liberated at the electrodes by this current. At the cathode, if the chlorine ions were at rest, the excess of potassium ions would be simply those arriving in one second, namely, nu. But since the chlorine ions move also, a further separation occurs, and nv potassium ions are left without partners. The total number of gram-equivalents liberated is therefore n(u + v). By Faradays law, the number of grams liberated is equal to the product of the current and the electro-chemical equivalent of the ion; the number of gram-equivalents therefore must be equal to ηC, where η denotes the electro-chemical equivalent of hydrogen in C.G.S. units. Thus we get
and it follows that the charge, e, on 1 gram-equivalent of each kind of ion is equal to 1/η. We know that Ohm’s Law holds good for electrolytes, so that the current C is also given by k·dP/dx, where k denotes the conductivity of the solution, and dP/dx the potential gradient, i.e. the change in potential per unit length along the lines of current flow. Thus
|u + v = η||k||dP||.|
Now η is 1.036×10−4, and the concentration of a solution is usually expressed in terms of the number, m, of gram-equivalents per litre instead of per cubic centimetre. Therefore
|u + v = 1.036×10−1||k||dP||.|
When the potential gradient is one volt (108 C.G.S. units) per centimetre this becomes
Thus by measuring the value of k/m, which is known as the equivalent conductivity of the solution, we can find u + v, the velocity of the ions relative to each other. For instance, the equivalent conductivity of a solution of potassium chloride containing one-tenth of a gram-equivalent per litre is 1119×10−13 C.G.S. units at 18° C. Therefore
= 1.159×10−3 = 0.001159 cm. per sec.
In order to obtain the absolute velocities u and v, we must find some other relation between them. Let us resolve u into ½(u + v) in one direction, say to the right, and ½(u − v) to the left. Similarly v can be resolved into ½(v + u) to the left and ½(v − u) to the right. On pairing these velocities we have a combined movement of the ions to the right, with a speed of ½(u − v) and a drift right and left, past each other, each ion travelling with a speed of ½(u + v), constituting the electrolytic separation. If u is greater than v, the combined movement involves a concentration of salt at the cathode, and a corresponding dilution at the anode, and vice versa. The rate at which salt is electrolysed, and thus removed from the solution at each electrode, is ½(u + v). Thus the total loss of salt at the cathode is ½(u + v) − ½(u − v) or v, and at the anode, ½(v + u) − ½(v − u), or u. Therefore, as is explained in the article Electrolysis, by measuring the dilution of the liquid round the electrodes when a current passed, W. Hittorf (Pogg. Ann., 1853–1859, 89, p. 177; 98, p. 1; 103, p. 1; 106, pp. 337 and 513) was able to deduce the ratio of the two velocities,for simple salts when no complex ions are present, and many further