The Apparent Ionic Charge

In Eqs. (4.172) and (4.186), two relations between the random movement of particles and directed drift (mobility under an electric field gradient) have been 

The relations derived were based upon very acceptable concepts, such as the random movement of particles in fluids, or their drift in the direction of an applied electric field. These ideas are so general that they cannot be doubted—they do not involve models later found to be wrong and to be displaced by other models, etc. For this reason, equations such as those named above are called phenomenological, meaning that they involve phenomena (drift and random movement) that cannot be doubted by the greatest skeptic. They happen. 

One gets a mild shock therefore when one looks into the experimental tests of these equations, for only under extremely simple conditions (in fact, very dilute solutions) do they work out to be correct. Although the results never hint that the equations are wrong, there is sufficient discrepancy (e.g., between tire diffusion coefficient calculated by using an experimental mobility substituted in the Einstein equation and that determined by direct experiment) for one to take notice and form some idea of a puzzle. How can simple mathematical reasoning based on the existence of movements undeniably present give rise to error  

This explanation is all very well for the liquid sodium chloride type of case, but deviations from the predictions of the Nernst-Einstein equation occur in dilute aqueous solutions also, and here the + and - ions are separated by stretches of water, and ion pairs do not form significantly until about 0.1 M. 

Because deviations from the Nernst-Einstein equation are so widespread, and because the reasoning that gives rise to the equation is phenomenological, it is better to work out a general kind of noncommittal response—one that is free of a specific model such as that suggested in the molten salt case (see Section 5.2). The response 

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