Calculation of equilibrium distributions of chemical species in aqueous solutions by means of monotone sequences

Abstract

A numerical method for the calculation of equilibrium distributions of chemical species in aqueous solutions of electrolytes is presented. This method is constructed by transforming the problem of determining the set of unknown concentrations satisfying the mass balance and mass action equations into the equivalent problem of finding the limits of certain well-behaved mathematical sequences in a multivariable direct iteration scheme. The total (analytical) concentrations are taken as the starting estimates, and the recursive equations are constructed from the starting equations. It is shown that the sequences so constructed are both monotonic and bounded, hence convergent in an orderly fashion by a mathematical axiom. Each unknown free ion concentration (that is, each limit) is approached simultaneously from above and from below, being effectively “sandwiched” in an ever tightening manner. Strict error bounds therefore are easily constructed. The method has been found to be exceedingly efficient in practice, with the error bound (maximum fractional error in taking the upper estimate as the final answer) decreasing in an approximately exponential fashion with respect to iteration number, commonly 0.5–1.5 orders of magnitude per iteration. Correction for nonideality is presented, and the possibility of this giving rise to more than one solution is discussed in connection with three examples of natural waters of low-, medium-, and high-ionic strengths.