The Hammett Equation

The relative ability of substituents in an aromatic ring to donate or withdraw electrons is indicated qualitatively by Hammett substituent constants. It was observed that a plot of the logarithms of the rate constants (k) for the alkaline hydrolysis of esters of benzoic acid against the values of the corresponding acids, XC H4C02H, was linear, Le. 

Log k and p. Ta are related to free energies of activation and ionization, respectively, and hence a linear free energy relationship exists between the rates of ester hydrolysis and acid strengths. 

Similar correlations between rate and equilibrium constants exist for various other side-chain reactions of benzene derivatives. The magnitude of p, which is called the reaction constant, is the slope of the line and varies with the reaction. The sign of p can be positive or negative according to whether the reaction rate is increased or decreased by the withdrawal of electrons. 

The term (p aO P a) is given the symbol a (sigma) and is constant for given substituents. Equation 2.3 thus simplifies to  

The data for the ionization of benzoic acid and its derivatives in water at 25 °C are extensive and accurate and this was chosen as the standard reaction to which all other reactions would be compared. The value of p for the standard reaction is 1.00. 

The classic example, and still the most useful one, of a LFER is the Hammett equation, which correlates rates and equilibria of many side-chain reactions of meta- and para-substituted aromatic compounds. The standard reaction is the aqueous ionization equilibrium at 25°C of meta- and para-substituted benzoic acids. 

The reference compound is benzoic acid. The substituent parameter, which is called the Hammett substituent constant cr, is defined by 

The form of Eq. (7-24) suggests, by comparison with Eq. (7-19), that a single interaction mechanism is operative between the substituent R and the reaction site X. That this is not necessarily true is shown as follows. Suppose two interaction mechanisms control the substituent effect. Then a Hammett relationship may be written as in 

If the two interaction mechanisms maintain the same relative importance throughout the reaction series, that is, if the ratio pj/pi is constant, Eq. (7-26) becomes identical in form with Eq. (7-24). This in fact appears to be the usual situation in Hammett correlations, PP- and the two interaction mechanisms are commonly discussed in terms of the inductive and resonance effects of electronic displacement. 

The other point of view is that the a values should be based on the original defining relation of Hammett, Eq. (7-22). Besides the advantage of providing 

Linear correlations between the logarithms of rate constants and pK values of benzoic acids were discovered independently by Hammett and Burkhardt in the 1930 s [31,32]. This approach has since been used extensively as a means of estimating polarity of substituents with the aim of eliminating structural aspects of the model. The measure of polarity is simply the effect of substituents on the dissociation constant of benzoic acid. For the simple hydrolysis reaction (Eqn. 40) the ionisation is clearly a good model and a plot of log /cq versus a (Eqn. 41) gives a good straight line (Fig. 8). 

Hammett relationships are possible for equilibria and Fig. 9 illustrates the linear dependence of pAT for substituted phenylacetic acids (Eqn. 42). 

The slope of the Hammett plot (p) does not bear a simple relationship with the transition-state structure (relative to ground state) because the model and the reaction or equilibrium in question are usually entirely dissimilar. Even the phenylacetic acid dissociation constants are not strictly comparable with those of benzoic acids because the polar effect is transmitted via different routes. 

Comparison of the Hammett p values for phenylacetic acid ( + 0.47) and benzoic acid (-I-1.00) dissociation constants provides us with a measure of the transmission of the methylene group (0.47) compared with direct coupling. The ratio (0.47) is remarkably constant for attenuation of the polar effect down a methylene side chain and essentially the same value is obtained for the reaction of phenylacetic and 

The only information that we may obtain a priori from p the value for hydrolysis of ethyl benzoates (Eqn. 43) is that electron-withdrawing substituents (X) increase 

A different, and basically empirical method of correlating aromatic reactivity with aromatic structures is represented by Hammett s equation42  

1 (a) Ingold, C. K. A, Rep. Chem. Soc. 23 (1926) 129 b) Robinson, R. Two Lectures on an Outline of an Electrochemical Electronic) Theory of the Course of Organic Reactions, London (Institute of Chemistry) 1932 c) Ingold, G. K. Chem. Rev. 15 (1934) 225 d) idem, Structure and Mechanism in Organic Chemistry, London (Bell) 1953 e) ibid. p. 255 

28 Mason, S. F. The Chemistry and Biology of Purines (Giba Foundation Symposium) 1957, p. 72 

32 Eyring, H. Chem. Rev. 10 (1932) 103 Glasstone, S., Laidler, K. J. and Eyring, H. The Theory of Rate Processes, New York (McGraw-Hill) 1941 

34 Wheland, G. W. J. Am. chem. Soc. 64 (1942) 900 Resonance in Organic Chemistry, Chapter 8, New York (Wiley) 1955 

Suppose we have a series of compounds S TR, where is a reaction center (i.e., a group of atoms that undergoes some reaction independently of the backbone T to which it is attached) and S, is a substituent. If the substituent and reaction center are attached to fixed points in T, r in equation (4.133) will be constant. We can then write equation (4.133) in the form 

Experimental values for equilibrium constants give only values for the product ap. Each is therefore uncertain by a numerical factor. We can multiply all the p s by a common factor A if we divide all the as by the same factor. In order to resolve this ambiguity, we have to assume the value of some specific p or a. Hammett did this in the case of his benzene derivatives by assuming that p = 1 for ionization of carboxylic acids in water. If then the pK values for benzoic acid and for a substituted benzoic acid are (pX )o 

Our derivation of the Hammett equation rested on the assumption that the sole mode of interaction between substituent S and reaction center R is by direct electrostatic interaction across space. This is true if the substrate T is saturated, but not if it is conjugated. If T is conjugated, there may be mutual conjugation between R and S in RTS. Even if there is not, n interactions between S and T may lead to changes in the latter that in turn can influence R by the field effect. If these alternative processes are important, the Hammett equation will fail if we use in it a constants based on the field effect alone. 

Substituent effects due to mutual conjugation between -f E and — E groups are described as the mesomeric effect and those due to mutual conjugation between a -f 7 substituent and — E reaction center as the n-inductive effect or inductoelectromeric effect The stabilizing effects of + and -f 7 substituents on an odd AH anion are both proportional to the formal negative charge at the position to which these are attached. If a substituent exerts both -f and + 7 effects, the stabilizations due to them will then be in fixed proportion to one another for all odd AH anions. It follows that mutual 

The other complication is illustrated by the ionization of m-amino-benzoic acid, 

We have seen numerous examples of substituent effects on rates and equilibria of organic reactions and have developed a qualitative feel for various groups as electron-donating or electron-withdrawing. Beginning in the 1930s, Lewis P. Hammett of Columbia University developed a quantitative treatment of substituent effects represented in the equations  

Source O. Exner, Correlation Analysis in Chemistry, N.B. Chapman and J. Shorter, eds. Plenum Press, New York, 1978, Chapter 10. 

Which compound undergoes this reaction at the fastest rate  

60 p for the reaction of a series of benzylic chlorides with potassium iodide in acetone is +0.8. 

Icq and Kq are, respectively, the rate and equilibrium constants for the unsubstituted parent compound that is, the substituent is H. The standard substituent H is assigned a cr value of 0. The standard reaction, assigned a value of p = 1.0, is the ionization of substituted benzoic acids. 

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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. 

The application of the Taft equation, derived from the Hammett equation (262)... 

Many equilibrium and rate processes can be systematized when the influence of each substituent on the reactivity of substrates is assigned a characteristic constant cr and the reaction parameter p is known or can be calculated. The Hammett equation... 

Linear Free Energy—Linear Solvation Energy Relationships. Linear free energy (LFER) and linear solvation energy (LSER) relationships are used to develop correlations between selected properties of similar compounds. These are fundamentally a collection of techniques whereby properties can be predicted from other properties for which linear dependency has been observed. Linear relationships include not only simple y = rax + b relationships, but also more compHcated expressions such as the Hammett equation (254) which correlates equiUbrium constants for ben2enes,... 

The numerical values of the terms a and p are defined by specifying the ionization of benzoic acids as the standard reaction to which the reaction constant p = 1 is assigned. The substituent constant, a, can then be determined for a series of substituent groups by measurement of the acid dissociation constant of the substituted benzoic acids. The a values so defined are used in the correlation of other reaction series, and the p values of the reactions are thus determined. The relationship between Eqs. (4.12) and (4.14) is evident when the Hammett equation is expressed in terms of fiee energy. For the standard reaction, o%K/Kq = ap. Thus,... 

The Hammett equation in the form of Eq. (4.14) or Eq. (4.15) is free of complications due to steric effects, since it is applied only to meta and para substituents. The geometry of the benzene ring ensures that groups in these positions cannot interact stoically with the site of reaction. Tables of a values for many substituents have been collected some values are given in Table 4.5, but substituent constants are available for a much wider range of... 

One approach is to correct for the added resonance interaction. This is done in a modification of the Hammett equation known as the Yukawa-Tsuno equation. ... 

Not all reactions can be fitted by the Hammett equations or the multiparameter variants. There can be several reasons for this. The most common is that the mechanism of the reaction depends on the nature of the substituent. In a multistep reaction, for example, one step may be rate-determining in the case of electron-withdrawing substituents, but a different step may become rate-limiting when the substituent is electron-releasing. The rate of semicarbazone formation of benzaldehydes, for example, shows a nonlinear Hammett... 

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

The rates of substituted benzaldehydes are correlated by the Hammett equation with p = +3.76. 

The chlorination of a series of compounds having electron-withdrawing substituents has been studied. The relative rates of chlorination and the isomer distributions are known. The data give a satisfactory correlation with the Hammett equation using (t, but no rate measurement for benzene under precisely comparable conditions is possible. How could you estimate f,f y and fp for chlorination from the available data ... 

Nevertheless, many free-radical processes respond to introduction of polar substituents, just as do heterolytic processes that involve polar or ionic intermediates. The substituent effects on toluene bromination, for example, are correlated by the Hammett equation, which gives a p value of — 1.4, indicating that the benzene ring acts as an electron donor in the transition state. Other radicals, for example the t-butyl radical, show a positive p for hydrogen abstraction reactions involving toluene. ... 

The Hammett equation is said to be followed when a plot of log k against a is linear. Most workers take as the criterion of linearity the correlation coefficient r, which is required to be at least 0.95 and preferably above 0.98. A weakness of r as a statistical measure of goodness of fit is that r is a function of the slope p if the slope is zero, the correlation coefficient is zero. A slope of zero in an LEER is a chemically informative result, for it demonstrates an absence of a substituent... 

Table 7-2 lists 15 reactions whose rates are correlated by the Hammett equation. Besides the reaction constant p, the table gives a value for k° (from the fitted line), which provides all the information needed to estimate k for any member of the series, if the corresponding o is available, by means of Eq. (7-24). Note that kP in Table 7-2 is generally not identical to the experimental value of k for the ct = 0 member of the series, because this experimental point may deviate from the regression line. 

The discussion in Section 7.1 should prepare us to expect deviations from such a simple relationship as the Hammett equation if the reaction being correlated differs greatly from the standard reaction. When this happens we have two choices (within this extrathermodynamic approach) We can select a different standard reaction, or we can increase the number of parameters. 

The Hammett equation and LFER in general added no new concepts to the qualitative picture that had been built up of electronic effects in organic reactions, but they did provide a quantitative measure that had been lacking and that has been found very useful. Here we will describe the further development of ideas concerning the substituent constant. 

Table 7-8 lists these ortho E% constants. An LFER can now be written by analogy with the Hammett equation. 

The large success of the Hammett equation in relating and systematizing the side-chain reactivities of meta- and poro-substituted benzenes has stimulated similar studies in the thiophene series. 

The aforementioned exception and the rather limited experimental material available do not allow any conclusions about the general applicability of the Hammett equation, using the same a- and p-values as for benzenes, to be drawn with certainty. The present author has pointed out that large deviations should be expected with strong - -M-substituents, as is also indicated from the rates of alkaline hydrolysis of methyl 5-amino- and 5-acylamino-2-thenoates. From the chemical shifts in the NMR spectra of thiophenes and benzenes it appears that another set of cr-values should be used in the thiophenes series which seems plausible since the transmission of the sub-... 

Hydroxypyridine 1-oxide is insoluble in chloroform and other suitable solvents, and, although the solid-state infrared spectrum indicates that strong intermolecular hydrogen bonding occurs, no additional structural conclusions could be reached. Jaffe has attempted to deduce the structure of 4-hydroxypyridine 1-oxide using the Hammett equation and molecular orbital calculations. This tautomeric compound reacts with diazomethane to give both the 1- and 4-methoxy derivatives, " and the relation of its structure to other chemical reactions has been discussed by Hayashi. ... 

Earlier studies of 4-aminopyridine 1-oxide were less conclusive. The solid-state infrared spectrum could be interpreted to indicate the existence of both the imino structure and/or, more probably, the amino structure. Comparison of the actual pKa value of 4-aminopyridine 1-oxide wdth the value calculated using the Hammett equation was considered to indicate that the compound existed as such or as an equilibrium mixture with l-hydroxypyrid-4-onimine, the latter possibility being considered the less likely on the basis of resonance and bond energies/ Resonance energy and ultraviolet spectral considerations have been advanced to support the 4-aminopyridine 1-oxide structure/ The presence of an infrared absorption band at... 

The ultraviolet spectra of the pyridinecarboxylic acids (334) were initially interpreted assuming that the proportion of the zwitterion structure 335 was not appreciable,and the early pK work was inconclusive. However, Jaffe s calculations based on the Hammett equation indicated that about 95% of nicotinic and isonicotinic acids existed in the zwitterion form, and ultraviolet spectral data showed that the actual percentages of picolinic, nicotinic, and isonieotinie acids existing in the zwitterion form in aqueous solution are 94, 91, and 96%, respectively.This was later confirmed by Stephenson and Sponer, who further demonstrated that the proportion of the zwitterion form decreases in solvents of low dielectric constant, becoming very low in ethanol. Dipole moment data indicate that isonicotinic acid exists as such in dioxane, and 6-hydroxypyridine-3-carboxyiic acid has been shown to exist in form 336 u.sing pK data. ... 

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