Hammett equation examples

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. 

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... 

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 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. 

It is the purpose of the present review to examine in what wa the Hammett equation can be applied to heterocyclic systems, to give examples of such applications, and to examine the special problems which arise in the process. In view of the tremendous difficulties involved in systematically searching the literature for the type of data required, no attempt wiU be made at an exhaustive coverage of all available information. The different possible applications will be discussed and, where feasible, illustrated. If an unjustified number of such illustrations are taken from the authors work, this should be... 

One interesting problem frequently recurring in heterocyclic chemistry, particularly with respect to nitrogen heterocycles, is tautomeric equilibria. Too many methods are available for the elucidation of equilibrium positions and tautomeric equilibrium constants (Kj) to adequately review the whole question here. However, the Hammett equation provides one independent method this method has the advantage that it can be used to predict the equilibrium position and to estimate the equilibrium constant, even in cases where the equilibrium position is so far to one side or the other that experimental determination of the concentration of the minor component is impossible. The entire method will be illustrated using nicotinic acid as an example but is, of course, completely general. 

The dediazoniation kinetics of 3- and 4-substituted benzenediazonium ions are probably the best known example of a failure of the classical Hammett equation (Scheme 7-1, see discussion in Sec. 7.2). 

The Hammett equation is the best-known example of a linear free-energy relationship (LFER), that is, an equation which implies a linear relationship between free energies of reaction or activation for two related processes48. It describes the influence of polar meta-or para-substituents on reactivity for side-chain reactions of benzene derivatives. 

Jaffe (1953)52 showed that while many rate or equilibrium data conform well to the Hammett equation (as indicated by the correlation coefficient), many such data are outside the scope of the equation in its original form and mode of application. Deviations are commonly shown by para-substituents with considerable + Rot — R effect53. Hammett himself found that p-NOz (+ R) showed deviations in the correlation of reactions of anilines or phenols. The deviations were systematic in that a a value of ca. 1.27 seemed to apply, compared with 0.78 based on the ionization of p-nitrobenzoic acid. Other examples were soon discovered and it became conventional to treat them similarly in terms of a duality of substituent constants . 

In order to gain an insight into the mechanism on the basis of the slope of a Type A correlation requires a more complicated procedure. Consider the Hammett equation. The usual statement that electrophilic reactions exhibit negative slopes and nucleophilic ones positive slopes may not be true, especially when the values of the slopes are low. The correct interpretation has to take the reference process into account, for example, the dissociation equilibrium of substituted benzoic acids at 25°C in water for which the slope was taken, by definition, as unity (p = 1). The precise characterization of the process under study is therefore that it is more or less nucleophilic than the reference process. However, one also must consider the possible influence of temperature on the value of the slope when the catalytic reaction has been studied under elevated temperatures there is disagreement in the literature over the extent of this influence (cf. 20,39). The sign and value of the slope also depend on the solvent. The situation is similar or a little more complex with the Taft equation, in which the separation of the molecule into the substituent, link, and reaction center may be arbitrary and may strongly influence the values of the slopes obtained. This problem has been discussed by Criado (33) with respect to catalytic reactions. 

Estimation of Acidity Constants The Hammett Correlation Illustrative Example 8.2 Estimating Acidity Constants of Aromatic Acids and Bases Using the Hammett Equation... 

We have stated earlier that because of proximity effects, no generally applicable aj values may be derived for ortho substitution. Nevertheless, one can determine a set of apparent 0)ortho values for a specific type of reaction, as for example, for the dissociation of substituted phenols. Table 8.7 gives such apparent O)ortho constants for estimating pKa values of substituted phenols and anilines. Of course, in cases of multiple substitution, substituents may interact with one another, thereby resulting in larger deviations of experimental from predicted pKa values. Some example calculations using the Hammett equation are given in Illustrative Example 8.2. 

While the examples chosen here concern only dissociation of protons, the Hammett equation has a much broader application. Equilibria for other types of reactions can be treated. Furthermore, since rates of reactions are related to Gibbs energies of activation, many rate constants can be correlated. For these purposes the Hammett equation can be written in the more general form in which k may be either an equilibrium constant or a rate constant.13 The subscript / denotes the reaction under consideration and i the substituent influencing the reaction. 

In fact this is a specific example of a general contention116 that in all heterocyclic systems Eq. (3) rather than the straightforward Hammett equation is the more correct. Equation (15), however, gives no indication that electron acceptors influence pK values of pyridinium ions by induction alone, although this circumstance has been carefully authenticated,104,109 but it does give voice to the contention that a correlation with o+ would be more realistic, to allow for the presence of canonical 19 in y-substituted pyridines for groups of +R character. Evidence... 

The Hammett equation (26-6) states that the relative reactivity (expressed in logarithmic form) of a substituted benzene derivative is proportional to the substituent constant a. For a given reaction, a plot of log (/c//c0) or of log (K/Ka) versus cr should be linear with slope p. Some idea of the validity of the Hammett equation can be gained from Figure 26-4, which shows plots of log kjko) or of log (KjKa) against a for several different reactions. For the examples given, the fit to the Hammett equation is fair. A number of p values (slopes) are listed separately in Table 26-7. It can be seen that p values vary... 

The Hammett equation also fails for open-chain aliphatic derivatives. For example, there is no simple linear relationship between log K for a scries of substituted ethanoic acids (RCH2C02H) and log k for the hydrolysis rates of similarly substituted ethyl ethanoates (RCH2C02C2H5). The freedom of motion available to a flexible open-chain compound permits a much wider range of variations in steric effects than for meta- and para-substituted aromatic compounds. 

Equations 2.11 and 2.12 express a linear relationship of free energies known as the Hammett a-p rejationshjp. or simply as the Hammett equation. It can be applied to reaction rales of substituted aromatic compounds as well as to equilibrium constants, and we shall find that it is a very useful tool for obtaining information about reaction mechanisms. (See, for example, Problem 2.1.)... 

Then the differences in rate caused by the electronic effect of the substituent are correlated by the Hammett equation log(kz/kH) = poz, where kz is the rate constant obtained for a compound with a particular meta or para substituent, ku is the rate constant for the unsubstituted phenyl group, and crz is the substituent constant for each substituent used. The proportionality constant p relates the substituent constant (electron donating or wididrawing) and the substituent s effect on rate. It gives information about the type and extent of charge development in the activated complex. It is determined by plotting log(kz/kQ) versus ov for a series of substituents. The slope of the linear plot is p and is termed the reaction constant. For example, the reaction shown above is an elimination reaction in which a proton and the nosy late group are eliminated and a C-N n bond is formed in their place. The reaction is second order overall, first order in substrate, and first order in base. The rate constants were measured for several substituted compounds ... 

Organic chemists have studied the influence of substituents on various reactions for the better part of a century. Linear free energy relationships have played an important role in this pursuit by correlating equilibrium and rate processes. One of the earliest examples is now known as the Hammett equation. It emerged from the observation that the acidities of benzoic acids correlated with the rates at which ethyl esters of benzoic acids hydrolyzed. The relationship was expressed as follows in which K represents an equilibrium constant and k is a rate constant. The proportionality constant, m, is the slope of the log-log data plot for the two processes. 

There are principally two different approaches of correlating experimental rate data of electrophilic substitution with reactivity indices (1) Correlating the index with the rate data of a given reaction, e.g. bromination. For example, a satisfying correlation of Dewar reactivity numbers with the log of rate constants of the bromination of benzene, naphthalene (1- and 2-position), biphenyl (4-position), phenanthrene (9-position), and anthracene (9-position) has been observed [55]. In correlations of this type the reactivity index corresponds to the reactivity constant in the Hammett equation while the slope of the linear correlation corresponds to the reaction constant (see also Sect. 3) (2) correlating the index with experimental a values. 

A vast generalization beyond the Bronsted—Lowry acids and bases concepts is the concept of a Lewis29 base (an electron pair donor) and a Lewis acid (an electron pair acceptor). This concept has been used extensively in all branches of chemistry. In physical organic chemistry, quantities of the type pA = —logio[A] have used extensively to study reactivities—for example, in the Hammett equation. 

It is assumed that all the explanatory variables are independent of each other and truly additive as well as relevant to the problem under study [144], MRA has been widely used to establish linear Gibbs energy (LGE) relationships [144, 149, 150], The Hammett equation is an example of the simplest form of MRA, namely bivariate statistical analysis. For applications of MRA to solvent effects on chemical reactions, see Chapter 7.7. 

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