Linear correlations

The presented algorithm was applied to 4 proteins (lysozyme, ribonuclease A, ovomucid and bovine pancreatic trypsin inhibitor) containing 51 titratable residues with experimentally known pKaS [32, 33]. Fig. 2 shows the correlation between the experimental and calculated pKaS. The linear correlation coefficient is r = 0.952 the slope of the line is A = 1.028 and the intercept is B = -0.104. This shows that the overall agreement between the experimental and predicted pKaS is good. 

Solvents exert their influence on organic reactions through a complicated mixture of all possible types of noncovalent interactions. Chemists have tried to unravel this entanglement and, ideally, want to assess the relative importance of all interactions separately. In a typical approach, a property of a reaction (e.g. its rate or selectivity) is measured in a laige number of different solvents. All these solvents have unique characteristics, quantified by their physical properties (i.e. refractive index, dielectric constant) or empirical parameters (e.g. ET(30)-value, AN). Linear correlations between a reaction property and one or more of these solvent properties (Linear Free Energy Relationships - LFER) reveal which noncovalent interactions are of major importance. The major drawback of this approach lies in the fact that the solvent parameters are often not independent. Alternatively, theoretical models and computer simulations can provide valuable information. Both methods have been applied successfully in studies of the solvent effects on Diels-Alder reactions. 

Grieco , Lubineau and Gajewski . Desimoni has observed linear correlations of the rate of the... 

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. ... 

The results of kinetic studies of the quatemization of 4-alkyl-2,5-dimethylthiazoles do not give a linear correlation between log(fc/ko) and Tafts Es parameter, such as is found for 2-alkyl-, 4-alkyl-, and 2,4-dialkylthiazoles (258). 

H-nmr chemical shifts of N-1—H and N-3—H signals have been used as a criterion for distinguishing between N-l-substituted and N-3-substituted hydantoin derivatives (22). They can often be related to electronic properties, and thus good linear correlations have been found between the shifts of N—H and Hammett parameters of the substituents attached to the aryl group of 5-arylmethylenehydantoins (23). 

C-nmr data have been recorded and assigned for a great number of hydantoin derivatives (24). As in the case of H-nmr, useful correlations between chemical shifts and electronic parameters have been found. For example, Hammett constants of substituents in the aromatic portion of the molecule correlate weU to chemical shifts of C-5 and C-a in 5-arylmethylenehydantoins (23). Comparison between C-nmr spectra of hydantoins and those of their conjugate bases has been used for the calculation of their piC values (12,25). N-nmr spectra of hydantoins and their thio analogues have been studied (26). The N -nmr chemical shifts show a linear correlation with the frequencies of the N—H stretching vibrations in the infrared spectra. 

Subsequent studies (63,64) suggested that the nature of the chemical activation process was a one-electron oxidation of the fluorescer by (27) followed by decomposition of the dioxetanedione radical anion to a carbon dioxide radical anion. Back electron transfer to the radical cation of the fluorescer produced the excited state which emitted the luminescence characteristic of the fluorescent state of the emitter. The chemical activation mechanism was patterned after the CIEEL mechanism proposed for dioxetanones and dioxetanes discussed earher (65). Additional support for the CIEEL mechanism, was furnished by demonstration (66) that a linear correlation existed between the singlet excitation energy of the fluorescer and the chemiluminescence intensity which had been shown earher with dimethyl dioxetanone (67). 

Surface Area. This property is of paramount importance to catalyst performance because in general catalyst activity increases as the surface area of the catalyst increases. However because some reaction rates are strongly dependent on the nature of the stmcture of the catalytic surface, a linear correlation of catalyst activity with surface area should not be expected. As the catalyst surface area increases, for many reactions the selectivity of the catalyst is found to decrease. If the support material is completely inert to the reactants and products, this effect may be diminished somewhat. 

Much can be predicted from these parameters. Linear correlations have been estabhshed between the R value and the drawabhity parameter, LDR. Thus higher R values are associated with better deep drawabhity. AR, on the other hand, predicts the height and location of ears. Eor example, larger absolute values of AR predict higher ears during deep drawing cup shapes. 

For a linear correlation in Y with one independent variable, x, and constants a and b ... 

The least-squares technique can be extended to any number of variables as long as the equation is linear in its coefficients. The linear correlation ofj vs X can be extended to the correlation ofj vs multiple independent variables generating an equation of the form ... 

The indazole molecular structure (Figure 11) shows the tautomeric proton bonded to N-1 (1//-indazole. Section 4.04.1.5.1). A linear correlation between the bond lengths and the bond orders calculated by the CNDO/2 method was observed (74T2903). 

According to a kinetic study which included (56), (56a) and some oxaziridines derived from aliphatic aldehydes, hydrolysis follows exactly first order kinetics in 4M HCIO4. Proton catalysis was observed, and there is a linear correlation with Hammett s Ho function. Since only protonated molecules are hydrolyzed, basicities of oxaziridines ranging from pii A = +0.13 to -1.81 were found from the acidity rate profile. Hydrolysis rates were 1.49X 10 min for (56) and 43.4x 10 min for (56a) (7UCS(B)778). O-Protonation is assumed to occur, followed by polar C—O bond cleavage. The question of the place of protonation is independent of the predominant IV-protonation observed spectroscopically under equilibrium conditions all protonated species are thermodynamically equivalent. 

We thus get the values of a and h with maximum likelihood as well as the variances of a and h Using the value of yj for this a and h, we can also calculate the goodness of fit, P In addition, the linear correlation coefficient / is related by... 

A differential variant of the tangent method was used for the processing of the kinetic data, because a linear correlation exists between the absorbance at 700 nm and time during the first 6 min after mixing. 

In addition to analyzing the residuals, it may be desirable to determine the degree of agreement between sets of paired measurements and estimates. The linear correlation coefficient is... 

Comparisons (49) of measured concentrations of SFg tracer released from a 36-m stack, and those estimated by the PTMPT model for 133 data pairs over PasquiU stabilities varying from B through F, had a linear correlation coefficient of 0.81. Here 89% of the estimated values were within a factor of 3 of the measured concentrations. The calculations were most sensitive to the selection of stability class. Changing the stability classification by one varies the concentration by a factor of 2 to 4. 

Gasoline working capacity (GWC) also shows a strong relationship with the pore volume in the mesopores. Similar to BWC, GWC is a measure of adsorption capacity in which actual gasoline vapors are used as the adsorbate. The relationship between the BWC and GWC is shown in Fig. 12. The data shows a strong relationship between the BWC and GWC. The relationship would be expected since both the BWC and GWC have excellent linear correlations with the pore volume in the small mesopores. 

Analogous plots for many other reactions of aromatic compounds show a similar linear correlation with the acid dissociation constants of the corresponding benzoic acids. 

Comparison of the data for methoxide with those for t-butoxide in Table 6.4 illustrates a second general trend Stronger bases favor formation of the less substituted alkene. " A stronger base leads to an increase in the carbanion character at the transition state and thus shifts the transition state in the Elcb direction. A linear correlation between the strength of the base and the difference in AG for the formation of 1-butene versus 2-butene has been established. Some of the data are given in Table 6.5. 

The kinetics of the hydrolysis of some imines derived from benzophenone anc primary amines revealed the normal dependence of mechanism on pH with ratedetermining nucleophilic attack at high pH and rate-determining decomposition of the tetrahedral intermediate at low pH. The simple primary amines show a linear correlation between the rate of nucleophilic addition and the basicity of the amine Several diamines which were included in the study, in particular A, B, and C, al showed a positive (more reactive) deviation from the correlation line for the simple amines. Why might these amines be more reactive than predicted on the basis of thei ... 

This simply relates to how linear the relationship between the peak molecular weight of narrow polystyrene standards versus elution volume fits a straight line. This is typically measured with the linear correlation coefficient, r. ... 

The second use of activation parameters is as criteria for mechanistic interpretation. In this application the activation parameters of a single reaction are, by themselves, of little use such quantities acquire meaning primarily by comparison with other values. Thus, the trend of activation parameters in a reaction series may be suggestive. For example, many linear correlations have been reported between AT/ and A5 within a reaction series such behavior is called an isokinetic relationship, and its significance is discussed in Chapter 7. In Section 5.3 we commented on the use of AS to determine the molecularity of a reaction. Carpenter has described examples of mechanistic deductions from activation parameters of organic reactions. 

The most common manifestation of extrathermodynamic relationships is a linear correlation between the logarithms of rate or equilibrium constants for one reaction series and the logarithms of rate or equilibrium constants of a second reaction series, both sets being subjected to the same variation, usually of structure. For illustration, suppose the logarithm of the rate constants for a reaction series B is linearly correlated with the logarithm of the equilibrium constants for a reaction series A, with substituent changes being made in both series. The empirical correlation is... 

Correlations with o in carboxylic acid derivative reactions have been most successful for variations in the acyl portion, R in RCOX. Variation in the alkyl portion of esters, R in RCOOR, has not led to many good correlations, although use of relative rates of alkaline and acidic reactions, as in the defining relation, can generate linear correlations. The failure to achieve satisfactory correlations with cr for such substrates may be a consequence of the different steric effects of substituents in the acyl and alkyl locations. It has been shown that solvolysis rates of some acetates are related to the pA", of the leaving group, that is, of the parent alcohol. The pK of alcohols has been correlated with but this relationship... 

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