Hammett free-energy equation

Other authors found that the enthalpies for the hydrogen bonding also give a good correlation against Hammett values. The linear free energy equations between Avq , Av q and AH obtained in their investigations were as follows ... 

The simplest calculations utilise a free energy equation and the appropriate substituent constant both obtained from tables. Thus the Hammett, Taft and Charton relationships (see Appendix 4) may be combined with a, o and cr, values (see Appendix 1) to arrive at a calculated pA for a substrate. 

The most popular of these approaches is multiple regression analysis, introduced by Hansch in 1968. In this method, the most favoured variables are (i) partition coefficients (P), from the system octanol/water (ii) the sigma (cr) and rho (p) values derived from Hammett s Linear Free Energy Equation (Section 17.2) and (iii) Taft s steric factors E ) which are used to determine what space for substituents exists between a nominated small region of the lead molecule and its receptor (Section 16.2). These variables are correlated in the following equation ... 

The Hammett free-energy relationship is expressed as in the following equations for equilibria and for rate data, respectively ... 

A knowledge of the effects of various substituents on any ionizable group allows each substituent to be classed as electron-attracting or electron-releasinga and placed in ranking order. This operation was quantified about 194O3 by the introduction of Hammett s Linear Free Energy Equation ... 

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. 

Linear Free Energy—Linear Solvation Energy Relationships. Linear free energy (LFER) and linear solvation energy (LSER) relationships are used to develop correlations between selected properties of similar compounds. These are fundamentally a collection of techniques whereby properties can be predicted from other properties for which linear dependency has been observed. Linear relationships include not only simple y = rax + b relationships, but also more compHcated expressions such as the Hammett equation (254) which correlates equiUbrium constants for ben2enes,... 

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

Finally, two sets of physical properties have been correlated by the Hammett equation. Sharpe and Walker have shown that changes in dipole moment are approximately linearly correlated with ct-values, and Snyder has recently correlated the free energies of adsorption of a series of substituted pyridines with u-values. All the reaction constants for the series discussed are summarized in Table V. 

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,... 

The Hammett equation is the best-known and most widely studied of the various linear free energy relations for correlating reaction rate and equilibrium constant data. It was first proposed to correlate the rate constants and equilibrium constants for the side chain reactions of para and meta substituted benzene derivatives. Hammett (37-39) noted that for a large number of reactions of these compounds plots of log k (or log K) for one reaction versus log k (or log K) for a second reaction of the corresponding member of a series of such derivatives was reasonably linear. Figure 7.5 is a plot of this type involving the ionization constants for phenylacetic acid derivatives and for benzoic acid derivatives. The point labeled p-Cl has for its ordinate log Ka for p-chlorophenylacetic acid and for its abscissa log Ka for p-chloroben-zoic acid. The points approximate a straight line, which can be expressed as... 

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,... 

The sulfur compounds RSH and disulfides are highly reactive, but RS and RSR are not. Nitro compounds usually react at diffusion-controlled rates. Aromatic compounds also fit into the Hammett (1940) equation when log k is plotted against free-energy change due to polar effects log(k/kg) = ap. Anbar... 

It is of interest to note that almost fifty years ago physical organic chemists began to derive the corresponding relationships between substituents and chemical reactivity. These are the well known free energy relationships such as the Hammett Equation (6) in which a is a substituent constant, p a reaction constant, kR the rate of the reaction with substituent R present and k0 the rate of reaction with a standard substituent (usually hydrogen) (33). 

Relative activation enthalpies (Aif) in Table 2 were converted to o% kx k ) at 298 K, and were plotted against Hammett a constants. Here, we used enthalpies, because the size of the entropy and hence the free energy depend much on low frequencies, which are less reliable than higher frequencies, especially for compounds with weak interactions such as TS (8). The use of free energy (AG ) gave similar correlations with more scattered points. As for the Hammett o constant, we used dual-parameter o constants in the form of the Yukawa-Tsuno equation (LArSR equation) (9) as defined in eq 3. Here, the apparent a constant (aapp) has a variable resonance contribution parameter (r), which varies depending on the nature of the reaction examined for t-cumyl... 

The structure-reactivity relationship is a concept familiar to every organic chemist. As commonly used it refers to a linear free energy relationship, such as the Bronsted or Hammett equations, or some more general measure of the effect of changing substituent on the rate or equilibrium of a reaction. A substituent constant is conveniently defined as the effect of the substituent on the free-energy change for a control reaction. So the so-called structure-reactivity relationship is in fact usually a reactivity-reactivity relationship. 

Extensive collections of pK values are available in the literature, e.g., [98-101]. It is also possible to predict pK values for a broad range of organic acids and bases using linear free energy relationships based on a systematic treatment of electronic (inductive, electrostatic, etc.) effects of substituents which modify the charge on the acidic and basic center. Quantitative treatment of these effects involves the use of the Hammett Equation which has been a real landmark in mechanistic organic chemistry. A Hammett parameter (a), defined as follows ... 

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