u + v = 1.036×107×1119×10-13 = 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 experiments have been made on the subject (see Das Leitvermögen der Elektrolyte).
By combining the results thus obtained with the sum of the velocities, as determined from the conductivities, Kohlrausch calculated the absolute velocities of different ions under stated conditions. Thus, in the case of the solution of potassium chloride considered above, Hittorf’s experiments show us that the ratio of the velocity of the anion to that of the cation in this solution is .51 : .49. The absolute velocity of the potassium ion under unit potential gradient is therefore 0.000567 cm. per sec., and that of the chlorine ion 0.000592 cm. per sec. Similar calculations can be made for solutions of other concentrations, and of different substances.
Table IX. shows Kohlrausch’s values for the ionic velocities of three chlorides of alkali metals at 18° C, calculated for a potential gradient of 1 volt per cm.; the numbers are in terms of a unit equal to 10-6 cm. per sec.:—
Table IX.
| KCl | NaCl | LiCl | |||||||
| m | u + v | u | v | u + v | u | v | u + v | u | v |
| 0 | 1350 | 660 | 690 | 1140 | 450 | 690 | 1050 | 360 | 690 |
| 0.0001 | 1335 | 654 | 681 | 1129 | 448 | 681 | 1037 | 356 | 681 |
| .001 | 1313 | 643 | 670 | 1110 | 440 | 670 | 1013 | 343 | 670 |
| .01 | 1263 | 619 | 644 | 1059 | 415 | 644 | 962 | 318 | 644 |
| .03 | 1218 | 597 | 621 | 1013 | 390 | 623 | 917 | 298 | 619 |
| .1 | 1153 | 564 | 589 | 952 | 360 | 592 | 853 | 259 | 594 |
| .3 | 1088 | 531 | 557 | 876 | 324 | 552 | 774 | 217 | 557 |
| 1.0 | 1011 | 491 | 520 | 765 | 278 | 487 | 651 | 169 | 482 |
| 3.0 | 911 | 442 | 469 | 582 | 206 | 376 | 463 | 115 | 348 |
| 5.0 | 438 | 153 | 285 | 334 | 80 | 254 | |||
| 10.0 | 117 | 25 | 92 | ||||||
These numbers show clearly that there is an increase in ionic velocity as the dilution proceeds. Moreover, if we compare the values for the chlorine ion obtained from observations on these three different salts, we see that as the concentrations diminish the velocity of the chlorine ion becomes the same in all of them. A similar relation appears in other cases, and, in general, we may say that at great dilution the velocity of an ion is independent of the nature of the other ion present. This introduces the conception of specific ionic velocities, for which some values at 18° C. are given by Kohlrausch in Table X.:—
Table X.
| K | 66 | × 10-5 cms. | per sec. | Cl | 69 | × 10-5 cms. | per sec. |
| Na | 45 | ” | ” | I | 69 | ” | ” |
| Li | 36 | ” | ” | NO3 | 64 | ” | ” |
| NH4 | 66 | ” | ” | OH | 162 | ” | ” |
| H | 320 | ” | ” | C2H3O2 | 36 | ” | ” |
| Ag | 57 | ” | ” | C3H5O2 | 33 | ” | ” |
Having obtained these numbers we can deduce the conductivity of the dilute solution of any salt, and the comparison of the calculated with the observed values furnished the first confirmation of Kohlrausch’s theory. Some exceptions, however, are known. Thus acetic acid and ammonia give solutions of much lower conductivity than is indicated by the sum of the specific ionic velocities of their ions as determined from other compounds. An attempt to find in Kohlrausch’s theory some explanation of this discrepancy shows that it could be due to one of two causes. Either the velocities of the ions must be much less in these solutions than in others, or else only a fractional part of the number of molecules present can be actively concerned in conveying the current. We shall return to this point later.