Average Value of the Drift Velocity

Any particular ion starts off after a collision with a velocity that may be in any direction this is the randomness in its walk. The initial velocity can be ignored precisely because it can take place in any direction and therefore does not contribute to the drift (preferred motion) of the ion. But the ion is all the time under the influence of the applied-force field. This force imparts a component to the velocity of the ion, an extra velocity component in the same dkection as the force vector F. It is this additional velocity component due to the force F that is called the drift velocity Vj. What is its average value  

From Newton s second law, it is known that the force divided by the mass of the particle is equal to the acceleration. Thus, 

Now the time between collisions is a random quantity. Sometimes the collisions may occur in rapid succession at others, there may be fairly long intervals. It is possible, however, to talk of a mean time between collisions, t. In Section 4.2.5, it was shown that the number of collisions (steps) is proportional to the time. If Acollisions occur in a time t, then the average time between collisions is t/N. Hence, 

The average value of that component of the velocity of an ion picked up from the externally applied force is the product of the acceleration due to this force and the average time between collisions. Hence, the drift velocity Vj is given by 

This is an important relation. It opens up many vistas. For example, through the mean time t, one can relate the drift velocity to the details of ionic jumps between sites, as was done in the case of diffusion (Section 4.2.15). 

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