Necessary and sufficient conditions for the appearance of extrema on concentration distribution curves in complex equilibrium systems

It is shown that species distribution curves with at least one minimum may be found in most equilibrium systems with three or more components. Whether such concentration minima are actually observed then depends on the values of the equilibrium constants and on the total (analytical) concentrations of the components. A general algorithm is given for the necessary and sufficient conditions for the appearance of extrema in multicomponent systems. Three-component systems are studied in more detail and special attention is given to the limiting case of a horizontal inflection, i.e., the point where the concentration minimum just disappears. Two well-studied chemical examples, the Cu(2+)-diethylenetriamine-OH(-) and Hg(2+)-Cl(-)-OH(-) systems are discussed, along with a simple model system showing as many as five extrema on a single distribution curve.