The Stokes Einstein Relation

During the course of diffusion, the individual particles are executing the complicated starts, accelerations, collisions, stops, and zigzags that have come to be known as random walk. When a particle is engaged in its random walk, it is of course subject to the viscous drag force exerted by its environment. The application of Stokes' law to these detailed random motions is no easy matter because of the haphazard variation in the speed and direction of the particles. Instead, one can apply Stokes' law to the diffusional movements of ions by adopting the following artifice suggested by Einstein.

When diffusion is occurring, it can be considered that there is a driving force -dfi/dx operating on the particles. This driving force produces a steady-state diffusion flux J, corresponding to which [cf. Eq. (4.14)] one can imagine a drift velocity vd for the diffusing particles.27 Since this velocity vd is a steady-state velocity, the diffusional driving force -dfildx must be opposed by an equal resistive force, which will be taken to be the Stokes viscous force 6nnjvd. Hence, dfi

~dx= rtlVd

The existence of a charge on a moving body has the following effect on a polar solvent: It tends to produce an orientation of solvent dipoles in the vicinity of the ion. Since, however, the charge is moving, once oriented, the dipoles take some finite relaxation time r to disorient. During this relaxation time, a relaxation force operates on the ion; this relaxation force is equivalent to an additional frictional force on the ion and results in an expression for the drag force of the form where s is (4/9)(x/6nt])el// and e is the dielectric constant of the medium. The correction may be as much as 25% but will be neglected here in the interest of deriving the classical Stokes-Einstein relation.

One can therefore define the absolute mobility uabs for the diffusing particles by dividing the drift velocity by either the diffusional driving force or the equal and opposite Stokes viscous force

27The hypothetical nature of the argument lies in the fact that in diffusion, there is no actual force exerted on the particles. Consequently, there is not the actual force-derived component of the velocity; i.e., there is no actual drift velocity (see Section 4.2.1). Thus, the drift velocity enters the argument only as a device.

The fundamental expression (4.172) relating the diffusion coefficient and the absolute mobility can be written thus: 