A rigorous statistical analysis of errors incurred in the study of intermolecular fast-exchange equilibria by NMR spectroscopy

A detailed analysis of the algorithms used in the analysis of NMR fast-exchange equilibria is made. It is shown that, in a system of step-wise equilibria of the type A + B ⇌ AB, AB + B ⇌ AB 2, …, AB n−1 + B ⇌ AB n , the most efficient way to solve the resulting system of equations is to solve for [B], the amount of uncomplexed B. It is then shown that in analyzing the equilibria, the equilibrium constants must be obtained via nonlinear regression for the greatest reliability. The problem is novel in that the other important parameters, the bound shifts, are obtained by simultaneous linear regression. A detailed analysis of the standard deviations of the various parameters is then made with particular emphasis being placed upon the type of data obtained with lanthanide shift reagents.