Linear Gibbs energy relationships

It is assumed that all the explanatory variables are independent of each other and truly additive as well as relevant to the problem under study [144], MRA has been widely used to establish linear Gibbs energy (LGE) relationships [144, 149, 150], The Hammett equation is an example of the simplest form of MRA, namely bivariate statistical analysis. For applications of MRA to solvent effects on chemical reactions, see Chapter 7.7. 

The Hammett equation is the best-known example of a linear free-energy relationship (LEER), that is an equation which implies a linear relationship between free energies (Gibbs energies) of reaction or activation for two related processes71. It describes the influence of polar meta- or para-substituents on reactivity for side-chain reactions of benzene derivatives. 

Some reactions are characterized by straight-line plots of TAS versus having a slope of approximately one, where this linearity results from compensatory, or off-setting, changes of AH and TAS. For this reason, the change in the Gibbs energy of activation, AG = AH - TAS, is a better description of the variation in the reaction than either AH or alone . See also Isokinetic Relationship... 

Organic functional groups exert characteristic electronic effects upon other groups to which they are attached. The quantitative expression of such effects can sometimes be correlated by linear Gibbs energy relationships. The best known of these is the Hammett equation, which deals with the transmission of electronic effects across a benzene or other aromatic ring. Consider the acid dissociation constants of three classes of compounds ... 

Many other linear Gibbs energy relationships have been proposed for example, the acid strengths of aliphatic compounds can be correlated using the "Taft polar substituent constants" o. ... 

An example of a linear Gibbs energy relationship that is widely used in discussing mechanisms of enzymatic reactions is the Bronsted plot (Eqs. 9-90 and 9-91). 

The standard molar Gibbs energy of solvation can also be derived from pure component data using spectroscopic information for determining solvatochromic parameters in respect of activity, basicity, polarity, etc. There exists a number of linear solvatochromic scales, the most widely used of which is the linear solvation energy relationship (LSER) devised by Kamlet and Taft [37, 38]. The Nernst distribution of solute i according to Kamlet is ... 

Linear free energy relationship (LFER) — For various series of similar chemical reactions it has been empirically found that linear relationships hold between the series of free energies (-> Gibbs energy) of activation AG and the series of the standard free energies of reactions AGf, i.e., between the series of log fc (k -rate constants) and log K (Kt - equilibrium constants) (z labels the compounds of a series). Such relations correlate the - kinetics and -> thermodynamics of these reactions, and thus they are of fundamental importance. The LFER s can be formulated with the so-called Leffler-Grunwald operator dR ... 

J. A. Jeneson, H. V. Westerhoff, T. R. Brown, C. J. Van Echteld, and R. Berger. Quasi-linear relationship between Gibbs free energy of ATP hydrolysis and power output in human forearm muscle. Am. J. Physiol., 268 C1474-1484, 1995. 

The quasi-linear variation of power with ATP hydrolysis is observed experimentally, as the contraction is being activated at the level of actinomyocin activity. The kinetic approach suggests that the muscle power output varies hyperbolically with the ADP concentration. Both the ADP control and the Gibbs energy of ATP hydrolysis control are similar, and when muscle power is varied voluntarily, muscle energetics may be represented by the linear flow-force relationships. 

Medium effects on acid/base equihbria in aqueous solutions of strong acids have been analyzed not only in terms of Hammett acidity functions, but also in terms of linear Gibbs energy relationships, first developed by Bunnett et al. [225] see [226] for a... 

It would appear from these observations that the solvation capability might be better characterized using a linear Gibbs energy relationship approach than functions of relative permittivity. There are now numerous examples known, for which the correlation between the rates of different reactions and the solvation capability of the solvent can be satisfactorily described with the help of semiempirical parameters of solvent polarity [cf. Chapter 7). 

According to Lutskii, even for quite simple molecules, acceptably precise ealeula-tions of Av/v° still present insuperable difficulties. This explains the growing praetiee of eorrelating Av/v° with empirieal parameters of solvent polarity within the framework of linear Gibbs energy relationships. Some of these empirical parameters are even derived from solvent-dependent IR absorptions as reference processes as, for example, the G-values of Sehleyer et al. [154] cf. Section 7.4. 

This kind of procedure, i. e. empirical estimation of solvent polarity with the aid of actual chemical or physical reference processes, is very common in chemistry. The well-known Hammett equation for the calculation of substituent effects on reaction rates and chemical equilibria, was introduced in 1937 by Hammett using the ionization of meta-ox /iflra-substituted benzoic acids in water at 25 °C as a reference process in much the same way [10]. Usually, the functional relationships between substituent or solvent parameters and various substituent- or solvent-dependent processes take the form of a linear Gibbs energy relationship, frequently still referred to as a linear free-energy (LFE) relationship [11-15, 125-127]. 

Eq. (7-1) essentially describes a relationship between standard molar Gibbs energies . It is often convenient to express linear Gibbs energy relationships in terms of ratios of constants by referring all members of a reaction series to a reference member of the series thus, the correlation in Eq. (7-1) can also be expressed by Eq. (7-4). 

Such relationships are useful in two ways. The first application is in the study of reaction mechanisms. The correlation of data for a new reaction series by means of a linear Gibbs energy relationship establishes a similarity between the new series and the reference series. The second use of linear Gibbs energy equations is in the prediction of reaction rates or equilibrium constants dependent on substituent or solvent changes. Let us consider a reaction between a substrate and a reagent in a medium M, which leads, via an activated complex, to the products . ... 

In principle, the same considerations as in Eq. (7-5) can be made for the spectral excitation of a substrate, dissolved in a medium M, with photons h v. Although linear Gibbs energy relationships usually deal only with relative reactivities, in the form of reaction-rate and equilibrium data, this approach can be extended to various physical... 

In the early use of linear Gibbs energy relationships, simple single-term equations sueh as the Hammett equation were eonsidered suffieient to fit given sets of experimental data from reaetion series. Later on, more eomplieated multi-term equations with more than one produet term were formulated in order to model the simultaneous influence of several effects on chemical reactions or optical excitations one product term per effect [15], The connection between such multiparameter relationships and solvent effects will be described in Section 7.7. 

Simple and multiple linear Gibbs energy relationships can be generally interpreted in two distinct ways [126, 127] ... 

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