The conductivity of a solution must be made up, therefore, of the sum of the shares which the positive ions and the negative ions, respectively, take in carrying the current. This principle was first advanced by Kohlrausch. The share of each kind of ion in conducting a current may be determined, for hydrochloric acid for instance, in the following way: A porous diaphragm may be used to divide the solution in an electrolytic cell into two halves, the concentration of the acid being the same in both halves (represented, as indicated in Fig. 11, by 15 molecules[89] of ionized acid in each half). A measured current is passed through the solution, say, sufficient to liberate 3 molecules of hydrogen H2, and 3 of chlorine Cl2, corresponding to 6 ions of each, and the concentration of the acid in each half is then again determined by analysis. Say it is found to correspond to 14 molecules of hydrochloric acid in the half of the solution on the side of the cathode and 10 molecules in the half on the side of the anode (see Fig. 12). Then the anode half has lost 5 ions of hydrogen, which must have passed through the diaphragm toward the cathode and taken the place of five of the six hydrogen ions discharged at the cathode. Similarly, the solution around the cathode has lost one chloride ion, which must have passed through the diaphragm toward the anode, and the hydrogen-ion corresponding to it, remaining on the right side without a compensating negative ion, must be the sixth hydrogen-ion discharged at the cathode. In other words, five hydrogen ions passed to the right, while one chloride ion passed to the left. The hydrogen ions then carried five-sixths of the current through the diaphragm, and consequently through the solution, and the chloride ions only one-sixth of the current. Since the solutions were of equal concentration to start with, the hydrogen ions have moved five times as fast toward the cathode as the chloride ions have moved toward the anode.
Fig. 11.
Fig. 12.
The equivalent conductivity of 0.1-molar hydrochloric acid is 351 at 18°, and experiment shows that the hydrogen-ion carries 84% of the current, the chloride-ion only 16%. The conductivity may then be considered to be the [p056] sum of the share the hydrogen-ion has in carrying the current, i.e. 0.84 × 351, or 295, and of the share of the chloride-ion, 0.16 × 352.5, or 56. These values may be called the equivalent partial conductivities or mobilities of the ions in this solution.
In a similar way, the conductivity of every solution of an electrolyte may be shown to represent the sum of the mobilities of the ions carrying the current (principle of Kohlrausch). The limit of the conductivity of one equivalent of an electrolyte is the sum of the mobilities of the ions composing the electrolyte. The frictional forces being constant for infinitely dilute solutions, at a given temperature, an ion will always show the same mobility, irrespective of the nature of the ion of opposite charge, with which it forms the electrolyte. We may then put Λ∞ = (l+∞ + l−∞), if l+∞ and l−∞ are used to designate the limits of the mobilities of gram-equivalents of the positive and negative ions forming the electrolyte. The following table[90] gives the limits of the mobilities for gram equivalents of some of the most important ions at 18°.
| Limits of Mobilities of Common Ions at 18°. | ||
|---|---|---|
| K: 65.3 | ½ Ca: 53.0 | I: 66.7 |
| Na: 44.4 | H: 318.0 | NO3: 60.8 |
| (NH4): 64.2 | OH: 174.0 | C2H3O2: 33.7 |
| Ag: 55.7 | Cl: 65.9 | ½ SO4: 69.7 |
For quite dilute solutions, in which the friction may be assumed to be approximately constant, the conductivity will depend, not only on the mobilities of the ions, which may be taken to be the same as for solutions of extreme dilution, but also on the proportion of electrolyte that is ionized, i.e. on the degree of ionization, α. Then Λv = α (l+∞ + l−∞), which is an elaboration of the original equation given on page [50].
Now, Kohlrausch discovered the principle of the summation of the mobilities of ions a number of years before the theory of Arrhenius was advanced, and the proportion in which the ion is present in a given solution being unknown, the effect of what is here known as the degree of ionization was included empirically in the value of the mobility. It is not surprising, then, that an ion was found to have approximately the same mobility only in solutions of the same concentration of strictly analogous and closely related salts, which, according to present methods of investigation, are now found to have approximately the same degree of ionization. For instance, the mobility of the gram-equivalent of the chloride-ion was found to be approximately the same, 47.3 and 50.5 respectively, in molar solutions of sodium and potassium chloride at 18°, no account being taken of the degrees of ionization. However, the degrees of ionization of the two salts are approximately the same, 66.9% and 74.9% respectively, and might be ignored in a comparison of the conductivities, without affecting the result of the comparison in any marked way. [p057]