630 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



u.v 



X + dx 



II. Theoretical Considerations. 



We can calculate the value of H in equation (2) provided we can 



calculate the values of the concentrations of the two ions at the point 



and at the time at which a ring begins to appear. We propose to attempt 



to obtain the values of these concentrations by the mathematical theory 



of diffusion. To avoid possible confusion from the use of general terms 



we shall take the specific case in which silver chromate is the precipitated 



substance ; the same analysis will, of course, apply to any other case. 



Suppose a capillary tube (Figure 3), containing in gelatine a dilute 



solution of potassium chromate, to be plunged at the time < == into a 



vessel containing a strong solution 



of silver nitrate; let the distance 



X be measured from the end of 



the tube, with its positive direction 



along the axis of the tube ; let u be 



+ 

 the concentration of the Ag ion at 



the point x at the time t, v the cor- 

 responding value for the Cr04 ion. 

 At the instant the tube is plunged 

 in, that is, at the time < = 0, m is 

 equal to Uq for all negative values 

 of a", and for all positive values 

 of X. Let and Vq be the corre- 

 sponding initial values of v. 



As a first approximation we may 

 assume that the dissociation of the 

 KoCr04 is complete, since this 

 substance is present in very dilute 

 solution for values of x and t such as we shall need to consider; the con- 

 centration of the silver salt in the tube will also be small, and we shall at 

 present consider it to be completely dissociated. Later we can see how to 



correct the error introduced by this assumption. Thus if we measure con- 



+ 

 centration in gram-molecules, the concentration of the Ag ion is the same 



as that of the AgNOs, **nd the concentration of the Cr04 ion is that of 

 the KoCrOi- We are to calculate the two concentrations from the diffu- 

 sion of AgNOa and KjCrOi. 



The diffusion constant is defined as the number of units of mass that 



x=o 



a 



Figure 3. 



