FREE METAL AND FREE LIGAND CONCENTRATIONS DETERMINED FROM TITRATIONS USING ONLY A PH ELECTRODE - PARTIAL DERIVATIVES IN EQUILIBRIUM STUDIES

In the studies of metal complex equilibria in aqueous solutions, the variation techniques presented by Osterberg, Sarkar, Kruck, and McBryde allow one to calculate free metal and free (unassociated) ligand concentrations from titration experiments using only a pH electrode. The method makes no assumptions about the associated species present in solution and is thus model independent. We present an alternate general derivation of the expressions and attempt to clarify the technique. Although the approach described below is applicable to any number of components (metals, ligands), in a manner analogous to Hedstrom's treatment of two-component systems we treat the three-component system by introducing six Jacobian unit determinants for the six variables (M, L, H, m, I, h), that is, the three total and three free concentrations of the metal, ligand, and hydrogen respectively:[Formulla Omitted] These Jacobians are very useful in deriving numerous variational expressions. Using computer-generated data, we test the techniques under a variety of conditions. A useful Fortran computer program is discussed. A simple technique for analytically evaluating implicit functions such as (ӘpH/ӘM)/, or (ӘpH/ӘL)M is presented. This enables one to avoid the use of numerical methods in the least-squares refinement of formation constants, thus leading to substantial reduction in the computational effort. © 1979, American Chemical Society. All rights reserved.