362 Dr. H. J. S. Sand. The Role of Diffusion during [Nov. 22, 



The total amount of solute J? (X) dt flowing in the time dt towards the 

 centre of the sphere across a concentric spherical surface of radius x 

 is given by 



* dt ........................ (4), 



dx 



if K indicates the diffusion coefficient of the solute, and c its concentra- 

 tion at the points under consideration. From this equation we arrive 

 by considerations of a well-known kind* at the relation 



K 



For the permanent state we have 9c/3tf = 0, the limiting conditions 



being 



c = for x = r ................... ..... (6), 



c = D for x = R ........................ (7). 



The result of the integration under these conditions is 



n K x-r /ftx 



D R^^r ........................ 8)> 



and the amount removed at the surface of the small sphere in the time 

 dt follows from Equation 4 



Felt = bricryVdt .......................... (9), 



the quantity y being denned by 



and decreasing from infinity to 1 as R increases from r to infinity. 



We now have to show that in all the cases we are dealing with the 

 quantity D does not differ appreciably from the average concentration 

 C of the liquid. Indicating again by N the number of particles per 

 unit volume of solution, and by v their total volume, then 



&rrN = t; ........................... (11), 



and C, which is equal to the total amount of solute per unit volume 



of liquid, is given by 



R 



C = N fiirafafa + (1 - N|7rR3) D. 



Eliminating c and N from this expression by means of Equations 8 

 and 1 1 and simplifying, we obtain 



* See e.g., Fourier's " Analjtical Theory of Heat," 112 and 113. 



