Modelling complex solution equilibria. I. Fast, worry-free least-squares refinement of equilibrium constants

The methods of Steepest Descent, Gauss–Newton (first-order Taylor series) and Newton–Raphson (second-order Taylor series) least-squares iteration were examined in the context of the refinement of estimated equilibrium constants β in solution. Under certain conditions, all three can produce corrections to the parameters that overshoot the global minimum and diverge therefrom, owing to the shape of the parameter surface. The latter two methods are problematic when the β parameters are overestimated, or when their logarithms are underestimated, whence a useful approximation to the analytical second and higher derivatives was found for any data type. This reduces an exact, infinite-order Taylor series expression of any observable to a simple first-order expression. As illustrated with experimental pH data, faster, more reliable refinement results without overshoot or divergence problems, and without resort to computationally onerous algorithms, such as the Marquardt–Levenberg, Fletcher–Powell, or Hartley–Wentworth methods. Keywords: equilibrium constants, least-squares, Gauss–Newton, complex solution equilibria.