252 HYDROGEN ION CONCENTRATION 



vicinity of the upper or lower chamber wall, then the velocity of the 

 interior layers must rapidly decrease, become zero, and then reverse 

 itself. The velocity v is hence a function of the distance x of the 

 water lamella or layer from the chamber wall, or simpler, it is a func- 

 tion of the "depth," If the total depth of the chamber is d, than this 

 function must become sjTimietrically arranged at one half of the 

 depth of the chamber, where x = d/2. 



Therefore, the observed motion of a particle depends upon the 

 value of X, and it is the algebraic sum of its own cataphoretic motion 

 and of the streaming of the water occurring at the distance x. Hence 

 the true velocity of cataphoresis is observed only at that depth x 

 at which the water streaming = 0, and this, according to Smoluchow- 

 ski, is the case for both layers at 



"(}^7t^ 



or 



X = approximately 0.2 d and 0,8 d 



It is to be noted that the depth d of the chamber may have any 

 value, provided it remains quite small, so that only lamellar currents 

 without whirls are possible.'-^ In macroscopic transference apparatus 

 this condition does not prevail, and in these the streaming of the 

 water plays no important role. 



If the true velocity of electrophoresis is ascertained in this way, 

 then, on the basis of Helmholtz's assmnptions and Smoluchowski's 

 development of Helmholtz's theory, we can calculate the wall poten- 

 tial f , and hence also the potential difference between the solution and 

 the surface of the moving particle, from the following equation: 



4 X • V • 77 f • K • H 



f = or V = 



K • H 4 X • 77 



where v is the electrophoresis velocity in cm. per second, and 77, K and 

 H have the designations given above. The most important aspect 

 of this equation is that (with constant viscosity) the potential is 

 proportional to the electrophoresis velocity. (If f is expressed in 

 volts, V in cm./sec, H in volts/cm., then in order to reduce the ab- 



2* The conditions under which this occurs were worked out by O. Reynolds. 

 Phil. Transact. 174, 935 (1883). 



