Distribution of ions in electrolyte solutions and plasmas. Part 1

Abstract

In a series of 3 papers the distribution of ions in dilute electrolyte solutions and plasmas is investigated. Representing the electrolyte with a Chemical Model, in which allowance is made for the existence of bound states between ions of various species, mutually consistent expressions are obtained for the binary distribution function (F_ji), the potential of mean force between any two ions (W_ji) and the electrostatic potential about an ion (ψ_j) in a solution containing any number of ionic species of any valency type. Expanding all functions as power series in the plasma parameter (ε=e²κ/DkT) and using the relationship F_jik=(F_jiF_jkF_ik)G_jik for the ternary distribution function, the second equation in the BBGY hierarchy is solved to obtain F_ji to the order ε³ for symmetrical and ε² for unsymmetrical electrolytes. From F_ji, W_ji and ψ_j are obtained independently. The Poisson–Boltzmann (PB) equation is examined in both its linearized and non-linearized forms and the order of error in each is found. It is shown that W_ji=e_iψ_j+(εⁿ) and thus that the “central approximation” upon which the PB equation is based has an error of the order εⁿ, where n= 3 for symmetrical and 2 for unsymmetrical salts. It is found that both linearized and non-linearized PB equations have errors of this order, the terms in εⁿ are found and it is shown that the error is more severe in the linearized equation. The Kirkwood superposition principle, which is equivalent to the assumption that G_jk= 1, is investigated and it is shown that G_{j k}= 1 +(εⁿ), where again n= 3 for symmetrical and 2 for unsymmetrical electrolytes. By solving the third equation in the BBGY hierarchy, differential equations are obtained for the higher terms in G_jik. The ground is prepared for the later investigation of more concentrated solutions using the Chemical Model.

In Part 1 the model used is discussed and the statistical mechanical methods to be used are developed.