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on them do not depend on the application of a potential gradient, it follows that the only way in which the current density can be affected by varying gradients is in the velocity acquired by the ions. It is concluded, therefore, that the ionic velocities are proportional to the potential gradients producing them, and this may be taken as the true physical meaning of Ohm's law.
This conclusion has been verified by direct measurement of ionic velocities by the methods described below.
In accordance with Ohm's law we may express the actual velocity u of any ion as the product of two factors, the actual potential gradient, and the velocity under a gradient of one volt per centimetre; for the latter we shall use the symbol U, and refer to UA and Ug as the mobility of the anion or cation.
The mobilities of various ions are different, according to their specific character, and methods have been worked out for measuring them. We shall begin with a method, invented by Hittorf, for comparing the mobility of an anion and a cation. The method may be explained by the aid of an experiment, as follows :
Take a rectangular glass jar (Fig. 9): fix in it (by means of paraffin wax) two porous plates, in such a way as to divide it into
three chambers. Put copper plates, for electrodes, into the two end compartments, fill up the whole with copper sulphate solution, and pass a current (of several amperes) through it for a few minutes. If specimens of the liquid be taken out and analysed it will be found
that the solution in the middle chamber 1 )
is unaltered in strength, but that the
cathode liquid is weaker, the anode Fig. 9.
stronger than before. Qualitatively the change in colour of the electrolyte will be sufficient to show this.
In order to account for these changes of concentration, let us make a list of the actions occurring in the three chambers. We will assume that x is the fraction of the whole current conveyed by the anion, so that 1- x is conveyed by
the cation. Hence, when one faraday of electricity is passed through the liquid, requiring on the whole one gram-equivalent of ions to convey it, there migrate across any section in the interior of the liquid, 1 – x equivalents of cations in the direction of the current, and x equivalents of anions in the opposite direction. x is called the migration ratio, or Hittorf's number (for the anion). Its relation to the mobilities of the ions is easily found, for we have seen that the currents produced by movement of the cation and anion are proportional to their mobilities. Hence
Cationic current Uc
Anionic current UA The actions in the three chambers during the passage of one faraday are thenAnode Chamber.
Middle Chamber. Cathode Chamber. Formation of i equiv. | Import of 1 – x equiv. Discharge of i equiv. Cu" from anode.
Cu“ from anode Cu" on cathode. Export of 1 – x equiv. chamber.
Import of 1 -- x equiv. Cu" to middle cham- Export of I x equiv. Cu" from middle ber.
Cu" to cathode chamber. Import of x equiv. chamber.
Export of x equiv. SO," from middle Export of x equiv. SO," to middle chamchamber.
So," to anode ber.
chamber. Net result:
Import of x equiv. Net result :Gain of x equiv. Cu" SO," from cathode Loss of x equiv. Cu“ and So", i.e. x equiv. chamber.
and SO.", i.e. x equiv. CuSO4
CuSO, No change. In the experiment, if correctly carried out, there ought to be no change in the middle chamber, while of the others, the anode chamber should gain just as much salt as the cathode loses. If, e.g., o 2 faraday be used (conveniently measured by a copper voltameter put in series with the migration cell) and it be found that the change in each chamber is o‘124 equivalents, it follows that
Again, suppose in the tripartite cell, the electrodes of platinum, the electrolyte dilute sulphuric acid ; the action in anode
and cathode chambers is as follows (the middle chamber is always unchanged) : Anode Chamber.
Cathode Chamber. Discharge of i equiv. SO,", which Discharge of i equiv. H:.
reacts with water, forming i equiv. Import of 1 - x equiv. H: from gaseous oxygen + i equiv. H', middle chamber. and reforming the SO,"
Export of x equiv. SO," to middle [SO, + H,0 = 2H: + So," + 0] chamber. Export of I - x equiv. H. to middle
chamber. Import of x equiv. SO," from middle
chamber. Net result :
Net result:Gain of x equiv. H. and the same of Loss of x equiv. H' and the same SO,", i... of x equiv. H SO, of SO,", i.e. of x equiv. H2SO,.
In this case, too, the anode solution increases in strength at the expense of the cathode ; the change is easily followed by titration. It is found that about 0'19 equivalents of acid migrate for each faraday, so that x = 0'19.
On comparing the results of such experiments with the changes occurring at the electrodes, we find that quite different conditions regulate the flow of current in the two cases, although the total current must be the same at each part of the circuit. At the electrodes when there are several available ions, the current will be conveyed by that whose formation or discharge is the least difficult, i.e. involves the least expenditure of energy. The whole current may be, and often is, conveyed by a single kind of ion, so that it may be exclusively anionic or exclusively cationic. In the interior of the solution, on the other hand, all the ions present necessarily share in the current, and the share that each takes is determined merely by the amount of it present and by its specific mobility. This is true, not merely in considering the relation of anion to cation, but also with regard to the shares taken by the various ions of the same sign, if more than one be present; for the expressions of p. 21 still hold, however many ions may exist in the solution. Thus, in the case of copper sulphate, besides Cu" there is H: in the liquid, although in very small proportion; hence a small part of the cationic current is carried by the hydrogen ions, and similarly a small part of the anionic by hydroxyl ions.
In passing from an electrode to the interior, then, each
partial current changes, while the total remains the same. Let i be the total current; then
į = ictia where ic and in are the cationic and anionic parts respectively.
E.3. in CuSO4 between copper electrodes the current at the anode is entirely cationic, in the solution only partially so; ic therefore decreases, while ia increases by the same amount. In electrolysing H,SO, between platinum plates the current at the anode is exclusively anionic, so that ic increases and ia decreases in passing towards the interior.
The chief results of measurements on migration ratios are given on the table, p. 256. The numbers are interpolated by Kohlrausch and Holborn (Leitvermögen der Elektrolyte) from experiments by Hittorf, Kuschel, Loeb and Nernst, Kirmis, Bein, Hopfgartner, and Kümmell.
The apparatus required varies somewhat according to the case studied-concentration of the liquid, presence or absence of precipitates or gaseous products at the electrodes, etc. Many forms have been adopted by different experimenters : that of Nernst and
Fig. 10. Loeb 1 in their experiments on silver salts may be described as typical (Fig. 10).
Zeitschr. phys. Chem., 2. 948 (1888).
The anode A is a silver wire, insulated for the greater part of its length by a glass tube, and opened out into a flat spiral below; the upper end of the wire is tightly enclosed by rubber tubing so as to be gas-tight. The cathode B is hung from a hook of platinum, and enclosed in a small side bulb, so that fragments of silver that fail to adhere to it may not fall on to the anode and so disturb the results. The whole is filled with the silver solution to be investigated, and as the action of the current is to make the solution denser near the anode and lighter near the cathode, it is necessary to avoid convection currents by placing the latter at a higher level than the former. In making an experiment, the apparatus is weighed; the liquid introduced by sucking at the small tube inserted by the side of the cathode—40 to 60 с.c. being required. The apparatus is then placed in a thermostat, and the electrolysis begun. On account of the long column of electrolyte needed in migration experiments the resistance is high, and a large voltage is required; nowadays the electric lighting supply is commonly used for the purpose. When the electrolysis is finished, portions are removed for analysis by blowing through the tube at B. The weight of each portion must be determined in order to find its total content of salt. The first portion should consist of so much of the liquid near the anode as to contain all the increase in salt due to the current. The next portion will then represent the middle of the electrolyte, and should be unchanged in concentration. The cathode portion will remain in the electrolyser, and may be weighed in that vessel, and then washed out and analysed.
In the above instance there are no secondary reactions at the electrodes to cause complications. It is only necessary to avoid convection currents, inequalities of temperature, and not to prolong the experiment so that diffusion sensibly interferes with it. In most cases formation of new ions at the electrodes and their migration introduces new difficulties. To see how these are met, we may take the case of sodium chloride, studied by Bein. In electrolysing sodium chloride between platinum plates, the discharge at the anode would be of chlorine
Bein, Wied., 46. 29 (1892); Zeitschr. phys. Chem., 27. 1-54 (1898).