Ion Speeds and Conductivity:
The Einstein and Stokes-Einstein Equations
Movement of an ion through a solution under the influence of a potential gradient (electric field) is the result of the acceleration of the charged particle in the field and the opposing forces of assymetry and the electrophoretic effect. Thus the ion moves at a constant rate determined by a balance of these forces. This leads to the definition of mobility ui of the ion i.
Ion mobilities at 298 K in aqueous solution
| Ion | u0(m2s-1V-1) |
| H3O+ | 36.3 x 10-8 |
| OH- | 20.5 x 10-8 |
| Li+ | 4.0 x 10-8 |
| Na+ | 5.2 x 10-8 |
| K+ | 7.6 x 10-8 |
| Ag+ | 6.4 x 10-8 |
| Mg2+ | 5.5 x 10-8 |
| Zn2+ | 5.5 x 10-8 |
| Cl- | 7.9 x 10-8 |
| Br- | 8.1 x 10-8 |
| NO3- | 7.4 x 10-8 |
| SO42- | 8.3 x 10-8 |
The units are m2 s-1 V-1. The relationship between the diffusion constant and mobility is expressed in the Einstein equation
The diffusion constant may be related to the viscosity by the Stokes-Einstein equation
where ri is the radius of the solvated ion and N is Avogadro’s number.
Sample calculations
Given the mobility of the silver ion in aqueous solution at 298K is 6.40 X 10-8 m2 s-1 V-1, and the viscosity of water at 298K of 8.94 x 10-4 kg m-1 s-1:
Calculate the diffusion coefficient of the silver ion.
Using the Einstein equation, D = (uRT)/zF = 6.4 x 10-8/38.95 = 1.64 x 10-9 m2 s-1.
Calculate its molar ionic conductivity.
Using the Nernst-Einstein equation, l = (z2F2D)/RT = 38.95 x 96500 x 1.64 x 10-9 = 0.616 x 10-2 W-1 m2 mol-1.
Calculate its effective radius.
Using the Stokes-Einstein equation, D = (RT)/(6p rhN) and solve for r. Yields r = 0.149 nm.