Plurilinear improvement of the Hammett equation

Abstract

The original Hammett equation, Δ = ρσ, is transformed in a constrained tetralinear relationship where each straight line with variable intercept term correlates one of the following four groups or subsets of dipolar substituents: normal and special substituents (depending on the absence or the presence of a lone electron pair in their atom next to the aromatic ring) and, in each of these classes, separating meta and para derivatives. There are a total of four fitting parameters in the resulting plurilinear Hammettian transformation (PHT) from which the statistically corrected parameters λ and γ are derived; λ and γ are the asymptotic values in a hyperbolic model for the representation of Δ4 vs Δ4/Δ3. This meta–para interrelationship is assumed to hold in the absence of through‐resonance effects which, in turn, are allowed for by the use of alternative sigma scales of substituent constants. By applying the PHT to a large number of selected literature data, parameters λ and γ are determined for the ionization equilibria of 3‐ and 4‐monosubstituted benzoic acids, anilinium ions, phenols and pyridinium ions. In these reactions series, parameter λ, which measures the para/meta ratio of field/inductive effects, is lower than unity and shows a marked dependence on the basic molecular framework. It is best modelled in terms of a through‐space field effect approach. The ratio γ/γ0, where γ0 is referred to the unified sigma‐zero scale, is shown to correspond to the original Hammett's reaction constant ρ. It is concluded that the PHT constitutes an improved Hammett equation for the analysis of substituent effects in benzene derivatives taking into account statistical errors and making allowance for different transmission coefficients for the field/inductive effect from meta and para positions in different reaction series.

Range of validity of the Hammett equation: acidity of substituted ethynylbenzenes

“…7,9 Further improvement was searched in dividing the substituents into classes 10 or even modifying the linear form. 11 A complete failure of the Hammett equation was observed only in rare cases, mostly for reactions of uncharged molecules.…”

Improved Yukawa-Tsuno equation and the substituent effect on pyridine basicity

“…10 On the basis of this model, a constrained tetralinear version of the Hammett equation was developed 11 which yielded values for the para/meta ratio of the field or inductive effect in excellent agreement 11 with Dewar et al 's model. 12 Then we referred to difficulties encountered while fitting data for the pyridinium ion series with sigma-plus constants, and to Sawada et al 's contribution 13 to the controversy on the proper scale to describe the pyridine reactivity.…”

“…The existence of a general link between the effect of a substituent from the meta and para positions should permit us to introduce constraints relating the fitting parameters for these two series. We addressed this problem in terms of a model 9,11 which uses the hyperbolic relationship in the form it is encountered in chemistry. 20 The incorporation of the Yukawa-Tsuno analysis into this model is straightforward, leading to (5) and (7) retains its original meaning 1,2,17 as the through-conjugation or direct-resonance demand relative to a standard system.…”

“…The first is to obtain an improved Yukawa-Tsuno equation from our previous plurilinear extension 11 of the Hammett equation. This leads to a constrained tetralinear equation with five adjustable parameters that allows for the higher complexity of the para derivatives while treating meta and para derivatives interrelately.…”

“…After deducting direct resonance effects, meta and para substituent effects become related 11 by the following pair of conjugate rectangular hyperbolae with asymptotes defined by D 4 /D 3 = ! and D 4 = g:…”