584 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



product of the densities of both. These densities must be equal because 

 when a positive ion is trapped the resulting ion pair is neutral so that a 

 trap is eliminated simultaneously. If these equal densities are designated 

 by n, we arrive at the second order rate law 



- f = '^^^ (^°-'> 



where /ca is a suitable constant, and t is time. 



This law would be perfectly valid if the mean free path of a mobile 

 positive ion were large compared to the distance between ions and the 

 probability of sticking on a first encounter were small. The trapping 

 cross-section rather than the movement prior to trapping would de- 

 termine the trapping rate. In this case the rate would certainly depend 

 on the concentrations of both the traps and the ions being trapped. 



On the other hand, in our case, not only is the mean free path of a 

 positive ion much smaller than the distance between ions, but the 

 sticking probability is high. A given ion must diffuse or make many ran- 

 dom jumps before encountering a trap and upon doing so is immediately 

 captured. Therefore, the rate of reaction is diffusion controlled. 



Because of the random jump process a given mobile ion is most likely 

 to be captured by its nearest neighbor during the first half of relaxation, 

 and relative to the degree of advancement of the trapping process, the 

 density of traps may be considered constant. This leads to first order 

 kinetics rather than second,* i.e., to 



- ^ = /cin (10.2) 



at 



where n is the density of untrapped ions. 



By definition ki is the fraction of ions captured in unit time, i.e., the 

 probability that one ion will be captured per unit time. Its reciprocal 

 must be the average lifetime of an ion. This lifetime 



r = I (10.3) 



ki 



shall be defined as the relaxation time for ion pairing. A rough calculation 

 of T can be made quickly. Thus, suppose that the initial concentrations 

 of donors and acceptors are equally A^. About each fixed acceptor can be 

 described a sphere of volume, 1/A^. On the average this sphere should be 

 occupied by one donor which according to what has been said above, will 

 eventually be captured by the acceptor at the center. In the mind, all 



* The phenomenon stems from the fact that first and second order processes are 

 almost indistinguishable during the first half of the reaction, but also from the 

 fact that the diffusion control prevents the process from being a ti-ue second or- 

 der one, although its departure from second order may be small. 



