Free-energy relationships

Linear Free Energy Relationships. - Kibby and Hall studied the dehydration of fifteen acyclic alcohols on a stoicheiometric (HA) hydroxyapatite [Caio(P04)6(OH)2] and a non-stoicheiometric (NHA) hydroxyapatatite for which Ca/P= 1.58. The former gave both dehydrogenation and dehydration but the latter gave only dehydration. In the case of the NHA catalyst the dehydration rate constants correlated with the Taft constants for a-carbon substitution giving p = — 5 at 230 °C, a-propanol being the reference alcohol so that the Taft equation was of the form (equation 2). 

Kochloefl and Knozinger observed, for deuteriopropan-2-ol dehydration over a number of oxides (AI2O3, Zi02, Ti02, and Si02) that the kinetic isotope values correlated with p values. The rate constants for dehydration ki (propan-2-ol), kji [ Hi]propan-2-ol and [l,3- H6]propan-2-ol were measured and the kinetic isotope values ai = an = i/ m 

It appears, therefore, that meaningful correlations can be obtained, but should be examined with some care. 

Substituting for k and K with the appropriate terms for free energy of activation (from transition-state theory) and free energy of reaction gives  

From this analysis, it is apparent that a relationship between k and K (at constant temperature) is essentially a relationship between free energies. The LFER therefore indicates that the change in free energy of activation (AG ) exerted by a series of substituents is directly proportional to the change in free energy of reaction (AG). 

The use of LFERs constitutes one of the most powerful means for the elucidation of reaction mechanisms. LFERs also provide us a means to predict reaction rates or bioactivity from more easily measured equilibrium constants such as octanol-water partition coefficients (Ko, ), ionization constants (KJ, or acidity constants (Khb)-Brezonik (1990) has summarized the major classes of LFERs applicable to reactions in aquatic ecosystems (Table 1.2). These empirical correlations pertain to a variety 

Relationship Types of Reaction or Reactants Basis of LFER 

Br nsted Acid- or base-catalyzed reactions hydrolysis, dissociation, association Rate related to or of product or catalyst 

There are many techniques of varying degrees of generality for the study of mechanisms and that of free energy relationships is the most readily applicable and general. Free energy relationships comprise the simplest and easiest of techniques to use but the results are probably the trickiest to interpret of all the mechanistic tools. 

We shall be discussing the following main attributes of the technique of free energy relationships throughout the text  

Linear free energy relationships are empirical observations which can be derived when the shapes of the potential energy surfaces of a reaction are not substantially altered by varying the substituent. 

The slopes of linear free energy relationships for rate constants are related to transition structure. 

Transition structure is obtained from knowledge of the bond orders of the major bonding changes as well as of solvation. 

5-thiazolyl (174) 4-thiazoIyl (25) 2-thiazolyl (1), reflects well the tt net charges calculated by the PPP method (133) for the three positions of the thiazole molecule [5(-0.086)/4(+0.029)/2(+0.100)]. Tlie authors noted that the same calculated charges did not agree with the relative reactivity induced by a phenyl group such a discrepancy between two different types of substrate is not unexpected because of the crude approximation of the method, but the interesting point is that the right experimental order is reproduced within the same thiazole molecule. 

The activation of halothiazoles toward nucleophilic displacement is discussed in Chapter V no unique conclusion can be drawn because of the various possible interactions between the halothiazole base and the electrophilic counterpart of the nucleophile. 

Only few attempts have been made to apply free-energy relations to thiazole reactivity. 

The first was proposed by Iraoto and Otsuji (511) and Otsuji et al (512) and concerned the pK of substituted 2-, 4-, and 5-carboxylic acids and the alkaline hydrolysis rate k of their respective ethyl esters (259, 260, and 261, where Y = Et). When Hammett cr , values were used for 

4- and Tp values for 2,5-disubstituted thiazoles, a linear relation existed between log k and cr (except for the tautomerisable amino compounds) with p values of 1.856 for 2-substituted-5-carboxylic acid ethyl esters (1, Y = Et), 1.618 for 5-substituted-2-carboxylic acid ethyl esters (3, Y = Et), and 1.551 for 2-substituted-4-carboxylic acid ethyl esters (2, Y = Et). A linear relationship between pK and a was found, with p values of 0.83, 2.35, and 1.34, respectively, for the corresponding acids (259, 260, and 261, where Y = H). 

One of the first reaction series studied involved triethylamine reacting with a series of methyl esters of substituted benzoic acids. A plot of the logarithm of the rate constant (Infe) versus the logarithm of the acid equilibrium constant (InRa) was linear. In mathematical form, this is Eq. [2] where m is the slope and b the intercept. 

From a heuristic (suggestive, not rigorous) viewpoint, this linear relationship between In and In k is not surprising given the relation between the equilibrium constant K for an elementary reaction and its forward and backward rate constants, k and k, respectively. 

The free energy terminology can be interpreted as coming from the familiar relations 

In a broader context, LFER and similar approaches are subsets of correlation analyses. Exner defines correlation analysis as a mathematical treatment starting from experimental data and seeking empirical relationships which can subsequently be interpreted theoretically. Although certainly not restricted to chemistry, correlation analysis has been developed extensively in physical organic chemistry. In addition to LFER, LSER, QSAR, and QSPR involve empirical models and, hence, fall in the category of correlation analysis. 

Erom an operational standpoint, the LFER, LSER, QSAR, and QSPR approaches can be quite similar, with distinctions based on their applications. QSAR is usually applied to biological properties, especially those important to pharmacology and toxicology. QSPR usually dwells on physicochemical properties in general. LSER focuses on solute-solvent systems. For organizational purposes, we like to view LSER and some applications of QSAR and QSPR (along with related methods) as subsets of LFER. Each approach typically uses some form of regression analysis (statistics) to help find a mathematical relationship between a property and a set of descriptors. 

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Such linear free energy relationships are available for alkyl sulphates and for tire C4 to C9 homologues of tire dialkanoyl lecitliins (see table C2.3.3 for stmcture). Most of tire naturally occurring phospholipids are too insoluble to fonn micelles, but tire lower alkanoyl lecitliins, also known as phosphotidylcholines, do fonn micelles. The ernes for tliese homologues are listed in table C2.3.6. The approximately linear free energy relationship between tire alkyl chain iengtli and log cmc is given by ... 

The ernes of ionic surfactants are usually depressed by tire addition of inert salts. Electrostatic repulsion between headgroups is screened by tire added electrolyte. This screening effectively makes tire surfactants more hydrophobic and tliis increased hydrophobicity induces micellization at lower concentrations. A linear free energy relationship expressing such a salt effect is given by ... 

A quantitative treatment of surfactant solubility has been successfully made empirically using linear free energy relationships. An important relation is that for the linear free energy of transfer of alkanes to water [23] ... 

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. 

Hammett [7] was the first to develop an approach that was later subsumed under Linear Free Energy Relationships (LFER). He showed that the addity constants of a... 

This shows that Eqs. (1) and 2) are basically relationships between the Gibbs free energies of the reactions under consideration, and explains why such relationships have been termed linear free energy relationships (LEER). 

N.B. Chapman, J. Shorter (Eds.), Advances in Linear Free Energy Relationships, Plenum Press, London, 1972. po] N.B. Chapman, J. Shorter (Eds.), Correlation Analysis in Chemistry, Plenum Press, London, 1978. pi] J. Shorter, Linear Free Energy Relationships (LEER), in Encyclopedia of Computational Chemistry, Vol. 2, P.v.R. Schleyer, N.L. Ailinger, T. Clark,... 

Two approaches to quantify/fQ, i.e., to establish a quantitative relationship between the structural features of a compoimd and its properties, are described in this section quantitative structure-property relationships (QSPR) and linear free energy relationships (LFER) cf. Section 3.4.2.2). The LFER approach is important for historical reasons because it contributed the first attempt to predict the property of a compound from an analysis of its structure. LFERs can be established only for congeneric series of compounds, i.e., sets of compounds that share the same skeleton and only have variations in the substituents attached to this skeleton. As examples of a QSPR approach, currently available methods for the prediction of the octanol/water partition coefficient, log P, and of aqueous solubility, log S, of organic compoimds are described in Section 10.1.4 and Section 10.15, respectively. 

N. B. Chapman. J. Shorter, Advances in Linear Free Energy Relationships, Plenum Press, London, 1972. 

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. 

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. ... 

Brown developed the selectivity relationship before the introduction of aromatic reactivities following the Hammett model. The former, less direct approach to linear free-energy relationships was necessary because of lack of data at the time. 

The Van t Hoff isotherm identifies the free energy relationship for bulk chemical reactions. 

J-K Elwang, A Warshel. Microscopic examination of free-energy relationships for electron transfer m polar solvents. J Am Chem Soc 109 715-720, 1987. 

Table 3.2. Composition-Equilibriuin-Free-Energy Relationships"...

Substituent Effects and Linear Free-Energy Relationships... 

SECTION 4.3. SUBSTITUENT EFFECTS AND LINEAR FREE-ENERGY RELATIONSHIPS... 

Since AG and AG are combinations of enthalpy and entropy terms, a linear free-energy relationship between two reaction series can result from one of three circumstances (1) AH is constant and the AS terms are proportional for the series, (2) AS is constant and the AH terms are proportional, or (3) AH and AS are linearly related. Dissection of the free-energy changes into enthalpy and entropy components has often shown the third case to be true. °... 

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