The Rate-Process Approach to Ionic Migration

The fundamental equation for the current density (flux of charge) as a function of the drift velocity has been shown to be 

Hitherto, the drift velocity has been related to macroscopic forces (e.g., the Stokes viscous force F = 6nrrf or the electric force F = zc X) through the relation 

Another approach to the drift velocity is by molecular models. The drift velocity Vj is considered the net velocity, i.e., the difference of the velocity v of ions in the direction of the force field and the velocity v of ions in a direction opposite to the field (Fig. 4.67). In symbols, one writes 

Any velocity is given by the distance traveled divided by the time taken to travel that distance. In the present case, the distance is the jump distance I, i.e., the mean distance that an ion jumps in hopping from site to site in the course of its directed random walk, and the time is the mean time t between successive jumps. This mean time includes the time the ion may wait in a cell of surrounding particles as well as the actual time involved in jumping. Thus, 

The reciprocal of the mean time between jumps is the netjump frequency k, which is the number ofjumps per unit of time. Hence, 

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