Monte Carlo analysis of error propagation in the potentiometric determination of equilibrium constants

Abstract

The calculation of formation constants of metal complexes is an important and active area of modern solution chemistry. The precision with which these constants are calculated can be quantified by a Monte Carlo technique from realistic estimates of the experimental errors in (potentiometric) titration parameters. Moreover, having better estimates of the standard deviations, this capability of Monte Carlo analysis has potential implications for model selection. A Matlab® program suite has been developed in order to quantify the statistical uncertainty on the optimized stability constants in complex models. The suite consists of a data generation program and a refinement program. The free concentrations of the individual components in the model are obtained by iteratively solving the set of non-linear mass balance equations using Newton’s method. The core of the refinement algorithm is a Gauss-Newton minimization of a sum of weighted squared residuals in electrode potential based on implicit differentiation (see also the standard program SuperQuad in Fortran 77). The present work shows a detailed error analysis in the determination of stability constants that is rarely done. Analysis of simulated data demonstrates that in certain cases the risk of accepting a false model is real! Verification of residual plots shows that the 2 test is the least convincing and most controversial criterion for model selection. In addition, we have formulated some new criteria to increase the reliability of potentiometric data.