Relationships linear free energy

If the free-energy change on dissociation of unsubstituted benzoic acid (X = H) is designated as AG%, the free energy on dissociation of a substituted benzoic acid (A G°) can be considered to be AG plus an increment, A A G°, con- 

In order to bring the relationship (2.4) into more convenient form, a parameter a is defined for each substituent according to Equation 2.5, so that equation 2.4 becomes Equation 2.6. 

By using the relationship 2.7 between free energy and equilibrium, Equation 2.6 can be rewritten as Equation 2.8, which in turn simplifies to Equation 2.9. 

If we now examine the effect of substituents on another reaction, for example acid dissociation of phenylacetic acids (Equation 2.10), we can anticipate that the 

AG° is the free-energy change for the new reaction with substituent X and 

Linear free energy relationships describe the proportionality between rates and equilibria observed in many reactions of structurally related compounds. They have the general form of Equation 9.18 where kQ and k are rate constants of a parent compound and a second compound in the reaction under test, and K0 and K are the equilibrium constants for a (different) reversible reference reaction of compounds having the same structural relationship as the first pair  

In the Bronsted relationship, the reference equilibria are dissociations of protic acids, used directly as ApK i, and the constant of proportionality r = fi (or a, see Chapter 11). In the Hammett relationship, acid dissociation constants are also used for the reference equilibria, but indirectly. The dissociation constants of substituted benzoic acids in water at 25°C are used to define a set of substituent parameters, cr, which are then used in the equation of correlation, and r = p in Equation 9.18. 

Although introduced empirically, a detailed theoretical analysis of the shapes of computed PE surfaces suggests that such relationships are expected to describe the response of reactivity 

The principle here is general, but the absence of a break in a linear correlation does not exclude an intermediate. Values of A/9 or Ap maybe so small, or errors in determining the slopes of the components so large, that the break point is undetectable. Equally, prediction of the values of the parameters at which the break point occurs (i = k2) is not straightforward and it may lie outside the range of available reactants. The technique has been applied most often in the demonstration of non-accumulating intermediates in substitution processes, both at carbon and at other elements, in quasi-symmetrical reactions [50]. 

(a) Special edition of Chemical Reviews, prefaced by Dantus, M. andZewail, A.H. (2004) Chemical Reviews, 104, 1717 (b) Polany, J.C. and Zewail, A.H. (1995) Accounts of Chemical Research, 28, 119. 

The term linear free energy relationship (LFER) appHes to a variety of relationships between kinetic and thermodynamic quantities that are important in both organic and inorganic reactions. About 80 years ago, J. N. Bronsted found a relationship between the dissociation constant of an acid, Ka, and its abihty to function as a catalyst in reactions that have rates that are accelerated by an acid. The Bronsted relationship can be written in the form 

From this equation, we can see that a plot of log k versus pKa should be hnear. However, the anion, A , of the acid HA is capable of functioning as a base that reacts with water. 

Therefore, we can write the equilibrium constant for this reaction, Kt, as 

Some reactions are catalyzed by bases so we can obtain relationships that are analogous to Eqs. (5.95) and (5.96), and the first is written as 

In aqueous solutions, Kb can be written as K r/Ka where is the ion product constant for water. Therefore, Eq. (5.101) can also be written as 

The rate of reaction correlates with the metal-ligand bond strength of the leaving group, in a linear free-energy relationship (LEER, Section 12.4.2). 

Kinetic effects are related to thermodynamic effects by linear free-energy relationships (LFER). A LEER can be observed when the bond strength of a metal-ligand bond (correlated to a thermodynamic parameter) plays a major role in determining the dissociation rate of a ligand (correlated to a kinetic parameter). When this is true, a plot of the logarithm of the rate constants for [MLsX] -I- Y substitution reactions, where X is varied but Y is not, ver- 

FIGURE 12.5 Linear Free Energy Relationship for [CofNHjjjX] Hydrolysis at 25.0 °C (X indicated at each point). The log of the rate constant is plotted against the log of the equilibrium constant for the acid hydrolysis reaction of [Co(NH3)5X] ions. Data for F from S. C. Chan, J. Chem. Soc., 1964,2375, and for I from R. G. Yalman, Inorg. Chem., 1962,1,16. All other data from A. Haim, H. Taube, Inorg. Chem., 1964,2,1199. 

Langford argued that X is dissociated and acts as a solvated anion in the transition state for [Co(NH3)5X] hydrolysis, and that water is, at most, weakly bound in the transition state. 

Ki is calculated from an electrostatic model. The rate constants, k2, vary by a factor of 5 or less and are close to the rate constant for water exchange. The close agreement 

Moore and R. G. Pearson, Kinetics and Mechanism, 3rd ed., John Wiley Sons, New York, 

The log of the rate constant is plotted against the log of the equilibrium constant for the acid hydrolysis reaction of [Co(NH3)5X] ions. Measurements were made at 25.0°C. Points are designated as follows  

and for F from R. G. Yalman, Inorg. Chem., 1962, 7,16. All other data from 

Limiting Rate Constants for Anation or Water Exchange of tCo(NH3)sW20] + at 45°C. 

The temptation to treat them as distinct physical phenomena should, however, be firmly resisted.  

Induction and field effects are, therefore, yet another example of complementary models that provide useful beginning points for the discussion of organic chemistry but that should not be viewed as pictures of reality. 

It is useful to make a qualitative prediction of the effect of a substituent on one chemical reaction, but we would like to be able to extend this understanding to predict how the same substituent wiU affect some other reaction. Moreover, it would be helpful to have a procedure for using our knowledge of the effect that different substituents have on a particular reaction to help us imderstand the mechanism of that reaction. It is more difficult to treat both steric and electronic effects simultaneously, so we will first consider systems in which the effects of a substituent are primarily induction/field effects or resonance effects but not steric effects. The approach that we will take illustrates the use of a linear free energy relationship (LFER) to relate values of AG or AG from one reaction to another.  

Schematic representation of a model system for studying equilibrium substituent effects. 

Shorter, J. in Zalewski, R. I. Krygowski, T. M. Shorter, J., Eds. Similarity Models in Organic Chemistry, Biochemistry and Related Fields Elsevier Amsterdam, 1991 p. 77. 

In the previous chapters we have considered the effects of physical properties (cohesive forces, polarity, and polarizability) and chemical properties (chiefly acidity and basicity in their various manifestations) on equilibria and rates of reaction. The problem with theoretical approaches to such matters is that these properties never act alone, nor are they independent. For instance, in a series of solvents that are weakly acidic, the acidity, hydrogen-bonding ability, polarity, and solubility parameter may all be expected to increase more or less together. 

Solvent Effects in Chemistry, Second Edition. Erwin Buncel and Robert A. Stairs. 2016 John WQey Sons, Inc. Published 2016 by John WQey Sons, Inc. 

Looking over the array of empirical parameters that have been derived by various authors (see references in Table A.2 and in Reichardt and Welton, 2011, Chapter 7) to correlate effects on reaction rates, equilibria, or spectral frequencies, it appears that there are many effects of the solvent to be considered. The parameters can be divided into two broad categories. First are those that have no sign, that is, they are in principle symmetric in their effect on cationic or anionic species or on molecules that have electron donor or acceptor properties. These are parameters such as cohesive energy density or cohesive pressure (and its square root, the solubility parameter), internal pressure, polarity, polarizability, refractive index, dielectric constant (relative permeability), and a number of empirical parameters based on particular equilibria, rates, or spectral features. An assortment of these parameters is listed in Table A.2a, with an indication of the experimental basis of each. 

The second class contains dual parameters, which occur in pairs of complementary attribntes cationic and anionic charge, Lewis or Brpnsted acidity and basicity (and refinements such as hard or soft acidity and basicity), electrophi-licity and nucleophilicity, and hydrogen-bonding tendency as donor and as acceptor (Table A.2b). A number of the entries in the table are incomplete in that only one of a potential pair of complementary parameters has been investigated. A table of values of most of the listed parameters for selected solvents forms Table A. 3. 

A further distinction that should be kept in mind is between those parameters that clearly pertain to solvents, such as those derived from bulk properties (refractive index, electrical permittivity, rate and equilibrium constants of reactions in the solvent, UV/visible or IR frequency shifts of solutes), and those that relate to molecules in solntion, such as those derived from enthalpies of reactions in dilute solution in an inert solvent. The donor number DN and Drago s E, E, C, Cg acidity and basicity parameters for Lewis acids and bases are examples of the latter kind. 

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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. 

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. °... 

J. Hine, Physical Organic Chemistry, McGraw-Hill, New York, 1962, pp. 95-98 P. R. Wells, Linear Free Energy Relationships, Academic Press, New York, 1968, pp. 35-44 M. Charton, Prog. Phys. Org. Chem. 10 81 (1973) S. Ehrenson, R. T. C. Brownlee, and R. W. Taft, Prog. Phys. Org. Chem. 10 1 (1973). 

Linear Free-Energy Relationships and Substituent Effects... 

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

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