Relationship between the osmotic coefficient and the solute mean activity coefficient for strong electrolyte solutions

1. Introduction
The need for formally expressing any colligative property of solutions in terms of either solute or solvent properties faces both practical and theoretical problems. Here, we show that a simple extension of the Debye-Hückel model provides an useful and simple solution which, at least in the case of 1:1 strong electrolytes, accounts for the relationship between concentration, mean ionic activity and osmotic coefficient. A two-parameter model provides a fairly accurate numerical relationship within a concentration range which extends much beyond the concentration range where the Debye-Hückel model has a sound basis.
2. The Gibbs-Duhem equation
The general Gibbs-Duhem relationship1 relating the chemical potentials of components in a mixture
(1)
can be written in the form proposed by Niels Bjerrum for electrolyte solutions, with the mean ionic activity for the solute, and the osmotic coefficient  for the solvent; in the case of one solute2,
(2...