M 



and anion in the solulion, tlieji n = is the quantity which 



HiTTORF has called ^'Ueberführungszahl" of the cation. 



Of a cm-rent i the part « . i is carried by the cation, the part 

 (1 — u) . i by the anion. So the number of gram equivalents of the 

 cation in the unity of volume will increase per unity of time by : 



1 1 



— div (n . i) — — (1 , yn), 

 e 8 



as div.i = is; s represents the charge of a univalent gram ion. 

 In the same way the number of gram equivalents of the anion will 

 increase by the same amount per unity of time, so that the solution 

 will remain neidral. If k is the valency of the molecule, and ni the 

 molecular weight, the mass of the salt will increase per unity of 

 time by an amount : 



TO 



If the quantity of electricity e^ passes through the unity of surface, 

 and if ij represents the unity vector in the direction i, the increment is: 



dv=^--^{\,,Tjn) (11) 



k . e 



In the volume elements which lie on the surface, the increment 

 of the mass of salt will be per unity of surface : 



dv =: . LVi . w (12) 



k . s 



when xV is the direction of the normal directed inward. The total 

 quantity of salt inside the solution will now increase by an amount : 



I dv + I c/j' = - — I «1 . ('i, xyn) dx -f I ^i . iiVi • *t • da 

 V s 'v. . s 



when we apply Gauss's theorem and make use of the equation 

 div[ = 0. The quantity of salt, therefore, does not change. 



The only change consists in this that tlie concentration in the 

 different volume elements is modified, and that a quantity of elec- 

 tricity e dissolved at the anode, has deposited at the cathode. 



We shall examine what change the total free energy of the 

 system has undergone in consequence of what has taken place in 

 the electrolyte. Above we found the expression (8) for the free 

 energy, (9) holding for the ''magnetic" part of it. We further chose 

 the variation so that the magnetic induction did not undergo any 

 change. In the first place we must now take into account that at 



50 



Proceedings Royal Acad Amsterdam. Vol. XVII. 



= 0, 



