﻿the Kinetics of the Process of Coagulation. 307 



%ti denotes the total number of particles in all stages 

 of coalescence in unit volume when the time is " t " ; 

 n l denotes the number of the primary particles whose 

 original number was n Q at the time " t " ; n k denotes the 

 number of particles of the ki\\ stage of coalescence — that is, 

 the number of aggregates each of which consists of u k " 

 of the primary particles. " k" is evidently an integer. 

 In 1918 Zsig.nondy published the results of an investi- 

 gation to test this theory. He restricted his investigation 

 to the rate of decrease in the primary particles (green in 

 the ultramicroscope) in a colloidal gold sol when the 

 minimum time of coagulation has been reached. He found 

 that Ra = 2*2 times r, the radius of the particles. Similar 

 values were obtained by Westgren and Reitstotter (Zeitschr. 

 Phijs. Chem. vol. xcii. p. 600, 1918) with more coarsely 

 dispersed gold sols. The value of Ra/r, however, varied 

 in one experiment from 1'4 to 3*8. The recent experiments 

 of Kruvt and Van Arkel (Rec. Trav. Chim. Pays-Bas, 

 vol. xxsisc. [4] p. 056, vol. xl. p. 169, 1920) show greater 

 variations. They are of opinion that there is some regularity 

 in these variations. They could not observe a maximum 

 value of Ra/r equal to 2. They found a maximum value 

 equal to 0'73. 



iSmoluchowski, assuming from the data of Zsigmondy 

 available at that time that Ra/r = 2, points out that the 

 maximum rate of coagulation is reached when each collision 

 between two par tides is successful in bringing about a 

 coalescence. When the rate of coagulation is slower, 

 all the collisions are not successful in bringing about a 

 coalescence of the particles. If " e " is the fraction of the 

 collisions that are successful in bringing about coalescence, 

 then " T ,J in equations (1) and (2) takes the form 



T _3 Np.iy r , 



4Ra,rco.0.e' ' [ ° J 



where !N" , Ra, n , 6, and rj have the same meaning as in 

 equations (1) and (2). 



Putting 



we nave 



^^v__l (6) 



n n 



t l + /3.e.t 



(?) 



X 2 



