Linear free energy relationship equations

A useful technique for analyzing structure-activity relationships which has come into use within the past five or so years is that pioneered by Corwin Hansch (19). Its underlying concept is that the relative biological activity of a derivative depends on the difference in hydrophobic, electronic, or steric factors between the derivative and the parent compound. The contribution of each effect to activity is assumed to be independently additive so that one has a linear free energy relationship. Equations of the form below are expected ... 

Data from the first nine entries in Table 5 may be expressed in a linear free energy relationship, equation (21), where k2 is the second order rate constant for oxygenation of the metal complex and a(X) is the Hammett constant for the para substituent in the triarylphosphine ligand [7]. 

Let us illustrate this with the example of the bromination of monosubstituted benzene derivatives. Observations on the product distributions and relative reaction rates compared with unsubstituted benzene led chemists to conceive the notion of inductive and resonance effects that made it possible to explain" the experimental observations. On an even more quantitative basis, linear free energy relationships of the form of the Hammett equation allowed the estimation of relative rates. It has to be emphasized that inductive and resonance effects were conceived, not from theoretical calculations, but as constructs to order observations. The explanation" is built on analogy, not on any theoretical method. 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

C. D. Johnson, The Hammett Equation, Camhndge University Press, Cambridge, 1973. P. R. Wells, Linear Free Energy Relationships, Academic Press, New bik, 1968. 

The second aspect is more fundamental. It is related to the very nature of chemistry (quantum chemistry is physics). Chemistry deals with fuzzy objects, like solvent or substituent effects, that are of paramount importance in tautomerism. These effects can be modeled using LFER (Linear Free Energy Relationships), like the famous Hammett and Taft equations, with considerable success. Quantum calculations apply to individual molecules and perturbations remain relatively difficult to consider (an exception is general solvation using an Onsager-type approach). However, preliminary attempts have been made to treat families of compounds in a variational way [81AQ(C)105]. 

Linear free energy relationships, see Bronsted equation, Dual substituent parameter (equations), Hammett equation(s), Quantitative structure-activity relationships, Ritchie nucleophilicity equation... 

The Hammett equation is the best-known example of a linear free-energy relationship (LFER), that is, an equation which implies a linear relationship between free energies of reaction or activation for two related processes48. It describes the influence of polar meta-or para-substituents on reactivity for side-chain reactions of benzene derivatives. 

In contrast to the steric effoits, the purely electronic influences of substituents are less clear. They are test documented by linear free-energy relationships, which, for the cases in question, are for the most part only plots of voltammetrically obtained peak oxidation potentials of corresponding monomers against their respective Hammett substituent constant As a rule, the linear correlations are very good for all systems, and prove, in aax>rdance with the Hammett-Taft equation, the dominance of electronic effects in the primary oxidation step. But the effects of identical substituents on the respective system s tendency to polymerize differ from parent monomer to parent monomer. Whereas thiophenes which receive electron-withdrawing substituents in the, as such, favourable P-position do not polymerize at all indoles with the same substituents polymerize particularly well... 

The Hammett equation is a linear free energy relationship (LFER). This can be demonstrated as follows for the case of equilibrium constants (for rate constants a similar demonstration can be made with AG instead of AG). For each reaction, where X is any group,... 

As proximity electrical effects are a function of the a and Or constants, and secondary bonding interactions when present may be a function of the Oi constants the effect of an ortho- or a ds-vinylene substituent may be represented by an equation including electrical and steric terms. The presence or absence of a steric effect may be ascertained in the following manner (72). There are four major cases of interest, (a) The steric effect obeys a linear free energy relationship. Then we may write the equation... 

Ledaal and co-workers (101-105) have proposed a linear free energy relationship for predicting the percent zwitterion formed at the rth carbon atom in substituted quinones, substituted dibenzoylethylenes, and substituted acetylenes. Fliszar and Granger (106) have proposed the equation... 

Therefore, reaction series with constant entropy have been accorded great significance and have been investigated thoroughly. The condition in eq. (8) was even considered necessary for any linear free energy relationship to hold (16). However, as experimental data accumulated and precision improved, it was clear that for many theoretically important reaction series, this condition is not fulfilled (1, 17). It was also proved that a LFER can hold if entropy is not constant, but linearly related to enthalpy (18, 19). The linear equation... 

If equation 5 is valid, if a linear relationship exists between AH and the calculated AE(t) parameters, and if a linear free energy relationship exists between AH and EA , we might expect that the following linear relationship might hold for the decomposition of reactant Y to produce free radicals R(Y) ... 

It should be emphasized that the above equations, which relate reaction temperatures to calculated reactant or product energies, are equivalent to the more conventional linear free energy relationships, which relate logarithms of rate constants to calculated energies. It was felt that reactant temperatures would be more convenient to potential users of the present approach -those seeking possible new free radical initiators for polymerizations. 

Nucleophilic reactivity toward Pt(II) complexes may be conveniently systematized via linear free energy relationships established between reactions of trans Ptpy2Cl2 (py = pyridine) with various nucleophiles and reactions of other Pt(II) complexes with the same nucleophiles. First, each nucleophile is characterized by a nucleophilicity parameter, derived from its reactivity toward the common substrate, trans Ptpy2Cl2. Reactivity toward other Pt(II) substrates is then quite satisfactorily represented by an equation of the form (21), wherein ky is the value of in the reaction with nucleophile Y... 

The above Hansch equations are also generally referred to as linear free energy relationships (LFER) as they are derived from the free energy concept of the drug-receptor complex. They also assume that biological activity is linearly related to the electronic and lipophilic contributions of the various substituents on the parent molecule. 

In a broad sense, one may include the Free-Wilson equation within the class of linear free energy relationships (LFER). It is also subjected to the assumption of additivity of the contributions to the biological activity by substituent groups at different substitution sites. The assumption requires, for example, that there is no hydrogen bonding interaction between the various substitution groups. 

Before terminating our discussion of the Hammett equation, we should note that the existence of linear correlations of the type indicated by equation 7.4.20 implies a linear free energy relationship. The rate or equilibrium constants can be eliminated from this equation using equation 7.4.1 that is,... 

Equation (12) is a linear free-energy relationship, since activity coefficients/can be represented as AG° values. The reason for defining the slope parameter as in equation (12) (subscript e for equilibrium) is that a little rearranging of equations (11) and (12) leads to the easy-to-use Bunnett-Olsen equation for equilibria, equation (13) 30... 

Equation (9.72) is known as a linear free energy relationship, and it shows that there should be a linear relationship between the logarithm of the rate constant for a reaction and the free energy for the dissociation of the acid. 

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