On the theory of diffusion of electrolytes

In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10−14 mole/cm3 of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.