50 Dr. H, J. S. Sand on the Concentration 



copper from which would require the same voltage as to 

 liberate hydrogen from a normal solution of sulphuric acid at 

 15°, we find a value of about 2 X 10 -18 normal. This means, 

 according to the theoretical considerations advanced, that 

 hydrogen will not be given oft' from an acid solution of copper 

 sulphate till the concentration of the copper has practically 

 gone down to zero at the electrode. 



In apparent opposition to this, the experimental liberation 

 of hydrogen from a not too concentrated acid solution of 

 copper sulphate is a matter which can be accomplished by 

 comparatively low current-densities, and doubt might well 

 arise, whether the suppositions on which our deductions are 

 based were not at fault. It might be supposed thot diffusion 

 of the copper sulphate in the liquid would effectually hinder 

 its concentration from going down to zero at the electrode 

 with the rapidity which the experiments require in order 

 to justify the theory. Indeed, as far as I can see, the idea 

 that the concentration of the copper is zero at parts of an 

 electrode from which hydrogen is being given off has not 

 been entertained by any of the experimenters who have 

 published work on a mixture of copper sulphate and sulphuric 

 acid. 



It must be remarked here that changes of concentration in 

 a liquid are in most cases no doubt effaced to a much greater 

 extent by convection-currents than by diffusion. But in 

 order to test the theory, it is possible experimentally to do 

 away with convection-currents almost entirely by the methods 

 described in the experimental part ; besides from its very 

 nature convection must be an erratic phenomenon, not 

 amenable to calculation and which could not be depended on, 

 continuously and uniformly, to neutralize changes of con- 

 centration arising on the total surface of an electrode. In 

 order to test the theory experimentally, it would be necessary 

 to calculate the concentration of the copper ions at the 

 electrode of an acid solution of copper sulphate from which 

 copper alone was being deposited, under the supposition that 

 only diffusion, and no convection, neutralized differences of 

 concentration brought about by the current. Unfortunately 

 neither the laws of conduction of the current nor those of 

 diffusion in a mixture are completely known, and if they were, 

 their application would probably involve extremely great 

 mathematical difficulties. It is therefore not possible accu- 

 rately to solve the problem just stated. As will be seen later 

 on, we can, however, solve it within certain limits by first 

 turning our attention to the following simpler case. 



Problem : Let us suppose electrolysis to take place in 



