Hammett equation derivation

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)... 

If the perturbations thus caused are relatively slight, the accepted perturbation theory can be used to interpret observed spectral changes (3,10,39). The spectral effect is calculated as the difference of the long-wavelength band positions for the perturbed and the initial dyes. In a general form, the band maximum shift, AX, can be derived from equation 4 analogous to the weU-known Hammett equation. Here p is a characteristic of an unperturbed molecule, eg, the electron density or bond order change on excitation or the difference between the frontier level and the level of the substitution. The other parameter. O, is an estimate of the perturbation. 

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. ... 

Although the application of the Hammett equation to side-chain reactions of disubstituted benzene derivatives (1) is relatively straightforward, the introduction of a heteroatom somewhere in the aromatic... 

Once the concept that the Hammett equation may be applied to heterocyclic systems as well as to simple benzene derivatives is accepted the possibilities of its use are virtually unlimited. Curiously enough, not too many such applications seem to have been reported. 

It seems reasonable to assume that in regard to their formation and decomposition the derivatives of phenylpentazole correspond roughly to the unsubstituted compound (see Table V). The rate data for m-and p-substituted phenylpentazoles conform to the Hammett equation with p= -f-1.01. Finally, the high rate of decomposition and... 

Before we close this section we make reference to an extended form of the Hammett equation in which the substituent constant and the reaction constant are separated into contributions from the field effect (F) and the mesomeric effect (R). This procedure was suggested by Taft in 1957 for 4-substituted benzene derivatives. It is called a dual substituent parameter (DSP) equation (Scheme 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. 

Miller226 applied the Hammett equation to the rate constants for the reaction of 4-substituted l-chloro-2-nitrobenzenes with OMe in methanol at 50°C. a values (denoted ct in accordance with the practice briefly in vogue at that time, 1956) were used for + R substituents, and S02Me conformed well at a a value of 1.04952. Act value of 1.117 for S02Ph was derived from the Hammett plot, intermediate between the values based on phenol and anilinium ionizations by Szmant and Suld88 at about the same time. 

Table IV lists comparative SD and / values for fittings of all of the reactions of Table II and III with each of the gr scales derived in this paper. The comparison includes figures for fittings with F and R values of Swain and Lupton (S L) and fitting with the Hammett equation. We believe the results given in Table IV provide a clear confirmation of the uniqueness and limited generality of the o/2(ba) scale. Very consistently, the fit achieved by the or (BA) scale is shown in Table IV to be superior by significant factors to that achieved by any of the other scales or by the simple op treatment. This fact is clearly reflected in the overall / values and the similarly weighted root-mean-square / values, F = y/lfnif lN, sum taken over all reactions. The value of F is. 067 for the basis sets of Table II (compare with overall / of. 058). The comparable F values are. 140 for Or, . 088 for S L, and. 155 for 0(p) with the data differences as explained in Table IV. For all sets of Table IV, the corresponding figures are. 073 for o (ba). -143 for a%, . 097 for S L, and. 209 for 0(py...

Shen (118) suggested a formal analogy between the Hammett equation and the Price-Alfrey equation. Charton and Capato (119) have derived relationships between e and q values and the Hammett equation as follows. 

Finally, in this account of multiparameter extensions of the Hammett equation, we comment briefly on the origins of the a, scale. This had its beginning around 1956 in the a scale of Roberts and Moreland for substituents X in the reactions of 4-X-bicyclo[2.2.2]octane-l derivatives. However, at that time few values of o were available. A more practical basis for a scale of inductive substituent constants lay in the o values for XCHj groups derived from Taft s analysis of the reactivities of aliphatic esters into polar, steric and resonance effects . For the few o values available it was shown that o for X was related to o for XCHj by the equation o = 0.45 <7. Thereafter the factor 0.45 was used to calculate c, values of X from o values of XCH2 . ... 

The Hammett equation is the best-known and most widely studied of the various linear free energy relations for correlating reaction rate and equilibrium constant data. It was first proposed to correlate the rate constants and equilibrium constants for the side chain reactions of para and meta substituted benzene derivatives. Hammett (37-39) noted that for a large number of reactions of these compounds plots of log k (or log K) for one reaction versus log k (or log K) for a second reaction of the corresponding member of a series of such derivatives was reasonably linear. Figure 7.5 is a plot of this type involving the ionization constants for phenylacetic acid derivatives and for benzoic acid derivatives. The point labeled p-Cl has for its ordinate log Ka for p-chlorophenylacetic acid and for its abscissa log Ka for p-chloroben-zoic acid. The points approximate a straight line, which can be expressed as... 

It is of interest to note that almost fifty years ago physical organic chemists began to derive the corresponding relationships between substituents and chemical reactivity. These are the well known free energy relationships such as the Hammett Equation (6) in which a is a substituent constant, p a reaction constant, kR the rate of the reaction with substituent R present and k0 the rate of reaction with a standard substituent (usually hydrogen) (33). 

One of the oldest and most familiar quantitative relationships for relating the structure of substituted benzene derivatives to both equilibrium constants and rate constants is the "Hammett Equation." See Louis Hammett, Physical Organic Chemistry, 184199. 

Trimm and Irshad 154) have used the influence of the substituents upon the rate of the oxidation of toluene and its derivatives to corresponding aldehydes on a molybdenum oxide catalyst at 460°C for obtaining insight into the mechanism. The rate constants could be correlated by the Hammett equation (series 139, five reactants, 450°C, slope -1). 

It was at Columbia that Hammett carried out his research on the physical properties of organic compounds that made him famous. His first accomplishment was his development of a concept now known as the acidity function, a new interpretation of the behavior of acids in concentrated and dilute solutions. He also derived the Hammett equation and Hammett function for organic substances, accomplishments for which he is perhaps best known today. In 1940, he published Uistextbook Physical-Organic Chemistry, which some chemists have called "one of the great textbooks in the history of chemistry."... 

The Hammett equation traditionally combines 3-substituted and 4-substituted benzene derivatives into a single data set using the composite constants for the former and constants for the latter. This is justifiable only if p3 is equal to P4 (Ap 0). We have shown that this is generally not the case except when the geometry of XGY is comparable to that of the benzoic acids from which the and Op constants were defined. The Yukawa-Tsuno and related equations are based on the assumption that P3 is equal to p4 and are limited to applications in which this is at least approximately correct. They are briefly reviewed elsewhere. ... 

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