Free energy correlation plots

Figure 7.11 Linear free energy correlation plots for inhibition of subtilisin BPN mutants by wild type (open circles) and mutant (close circles) chymotrypsin inhibitor 2. Left panel Correlation between AGbinding for the inhibitor and AGm. Right panel Correlation between AGbinding for the inhibitor and AGES.

Hansch [66] and his co-workers soon found that curved free-energy correlation plots were frequent and equations of the type shown in Eqn. 81 were found. 

The relative shift of the resonances of the dihydride nuclei listed in Table 12.2 follow a free energy correlation, as is outlined in the Hammett plots shown in Figure 12.14. 

Fig. 12.14 Free energy correlation and Hammett plot of chemical shift data of the intermediate dihydrides.

The existence of linear free energy correlations is well known for numerous series of organic reactions. Such correlations are particularly useful for systematic studies of substituent effects. A necessary consequence for the linearity of such plots for two reaction series is that for any two members (i, j) in either reaction series... 

Free energy correlations often exhibit scatter plots where the deviations from a linear regression fall outside experimental error. These deviations may be attributable to differences in microscopic environment between the standard equilibrium and the reaction being studied and are called microscopic medium effects. 

Alternative definitions parallel mechanisms can be diagnosed when the observed rate on one side of the break-point in a free energy plot is greater than that calculated from the correlation on the other side. Parallel mechanisms are diagnosed if the non-linear free energy correlation exhibits a concave upwards curvature. 

Fig. 29. Decomposition of acetophenone-bisulphite complexes [105c]. Inequality of inductive and resonance interactions causes a U-shaped free energy correlation in the simple Hammett approach. Yukawa-Tsuno type correction of the mesomerically donating groups is illustrated by the horizontal lines leading to a linear plot.

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. ... 

Another method for studying solvent effects is the extrathermodynamic approach that we described in Chapter 7 for the study of structure-reactivity relationships. For example, we might seek a correlation between og(,kA/l ) for a reaction A carried out in a series of solvents and log(/ R/A R) for a reference or model reaction carried out in the same series of solvents. A linear plot of og(k/iJk ) against log(/ R/ linear free energy relationship (LFER). Such plots have in fact been made. As with structure-reactivity relationships, these solvent-reactivity relationships can be useful to us, but they have limitations. 

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... 

When put into an appropriate model [N0rskov et al., 2004], the binding energy correlations directly define a limit to t/o on the metals obeying the linear relations shown in Fig. 3.7. Since all intermediates are dependent on Eq, it is possible to plot the heights of all the steps AGi 4 as functions of Eq at zero potential. The step with the smallest free energy change wUl define I/ork (Fig. 3.8) ... 

Having said all that, it is equally important to remember that the number and variety of useful correlations to which Hammett plots have given rise is quite astonishing, particularly when we consider the simplicity and convenience of the approach. Indeed, linear free energy relationships in general constitute a testament to the theoretical utility of concepts that are purely empirical in their genesis ... 

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... 

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... 

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