Two-variable Problem Research Articles

Solvent Effects on ρ Values of the Hammett Equation

Abstract The effects of selvents on ρ values have been systematically treated as a two-variable problem within a linear free-energy relationship framework. It is assumed that the standard activation free energy is a continuous function of σ, a substituent variable, and ξ, a solvent variable. The applicability of the resultant equations is proved by a test of the dissociation constants (obtained from the literature) of benzoic acids in various solvents. As a result, it is proposed that these effects on ρ values can be explained in terms of the interaction or cross-term of σ and ξ. The application to the reaction rates of a typical Menschutkin reaction, obtained by experiments in ten nonaqueous solvents, suggests that the solvent dependence in these reactivities is much more complicated.

Quality Control Methods for Several Related Variables

The methods described in an earlier article devoted to control methods for two related variables are extended to the case of more than two related variables. The concept of matrix notation is introduced because of the resultant simplification in multivariate analysis and the original two-variable problem is restated in matrix form. The method of principal components is introduced both as a method of charscterizing a multivariate process and as a control tool associated with control procedures. These methods are illustrated with a numerical example from the field of ballistic missiles. Approximate multivariate techniques, designed to simplify the administration of these control programs, are also discussed.