Page:Scientific Papers of Josiah Willard Gibbs.djvu/466

430 the cation (1) has the charge $$c_{1}$$, the force necessary to prevent its migration would be For an anion (2) the force would be  Now we may suppose that the same ion in different parts of a dilute solution will have velocities proportional to the forces which would be required to prevent its motion. We may therefore write for the velocity of the cation (1), and for the flux of the cation (1),  for the flux of the anion (2),  where $$k_{1}, k_{2}$$ are constants ('migration velocities') depending on the solvent, the temperature, and the ion. Now whatever the number of ions the flux of electricity is given by the equation where the upper sign is for cations and the lower for anions, and the summation for all ions. This gives That is,  The form of this equation shows that since $$\phi$$ is the current, $$\frac{dx}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma_{1}}$$ is the "resistance" of an elementary slice of the cell, and the next term the (internal) electromotive force of that slice.