Hammett equation

The Hammett equation in organic chemistry describes a free-energy relationship relating reaction rates and equilibrium constants for many reactions involving benzoic acid derivatives with meta- and para-substituents to each other with just two parameters: a substituent constant and a reaction constant. This equation was developed and published by Louis Plack Hammett in 1937 as a follow up to qualitative observations in a 1935 publication.

The basic idea is that for any two reactions with two aromatic reactants only differing in the type of substituent the change in free energy of activation is proportional to the change in Gibbs free energy. This notion does not follow from elemental thermochemistry or chemical kinetics and was introduced by Hammett intuitively

Hammett equation
The basic equation is:

$$log \frac{K}{K_0} = \sigma\rho $$

relating the equilibrium constant K for a given equilibrium reaction with substituent R and the reference K0 constant with R=H to the substituent constant σ which depends only on the specific substituent R and the reaction constant ρ which depends only on the type of reaction but not on the substituent used.

The equation also holds for reaction rates k of a series of reactions with substituted benzene derivatives:

$$log \frac{k}{k_0} = \sigma\rho $$

In this equation k0 is the reference reaction rate of the unsubstituted reactant and k that of a substituted reactant.

A plot of log(K/K0) for a given equilibrium versus log(k/k0) for a given reaction rate with many differently substituted reactants will give a straight line.

Substituent constants
The starting point for the collection of the substituent constants is a chemical equilibrium for which both the substituent constant and the reaction constant are arbitrarily set to 1: the ionization of benzoic acid (R and R' both H) in water at 25°C.



Having obtained a value for K0, a series of equilibrium constants (K) are now determined based on the same process but now with variation of the para substituent for instance p-Hydroxybenzoic acid (R=OH, R'=H) or 4-aminobenzoic acid (R=NH2, R'=H). These values combined in the Hammett equation with K0 and remembering that ρ = 1 give the para substituent constants compiled in table 1 for amine, methoxy, ethoxy, dimethylamino, methyl, fluorine, bromine, chlorine, iodine, nitro and cyano substituents. Repeating the process with meta-substituents afford the meta substituent constants. This treatment does not include ortho-substituents which would introduce steric effects.

The δ values displayed in table 1 reveal certain substituent effects. With ρ = 1 the group of substituents with increasing positive values, notably cyano and nitro cause the equilibrium constant to increase compared to the hydrogen reference meaning that the acidity of the carboxylate anion (depicted on the right of the equation) has increased. These substituents stabilize the negative charge on the carboxylate oxygen atom by an electron-withdrawing inductive effect (-I) and also by a negative mesomeric effect (-M).

The next set of substituents are the halogens for which the substituent effect is still positive but much more modest. The reason for this is that while the inductive effect is still positive, the mesomeric effect is negative causing partial cancellation. The data also that for these substituents the meta effect is much larger than the para effect and this is due to the fact that the mesomeric effect is cancelled in a meta substituent.

This effect is depicted in scheme 3 where in a para substituted arene 1a, one resonance structure 1b is a quinoid with positive charge on the X substituent releasing electrons and thus destabilizing the Y substituent. This destabilizing effect is not possible when X has a meta orientation.

Other substituents like methoxy and ethoxy can even have opposite signs for the substituent constant as result of opposing inductive and mesomeric effect. Only alkyl and aryl substituents like methyl are electron-releasing in both respects.

Of course when the sign for the reaction constant is negative (next section) only substituents with a likewise negative substituent constant will increase equilibrium constants.

Reaction constants
With knowledge of substituent constants it is now possible to obtain reaction constants for a wide range of organic reactions. The archetypal reaction is the alkaline hydrolysis of ethyl benzoate (R=R'=H) in a water/ethanol mixture at 30°C. Measurement of the reaction rate k0 combined with that of many substituted ethyl benzoates ultimately result in a reaction constant of +2.498.



Reaction constants are known for many other reactions and equilibria, a selection of those provided by Hammett himself (with their values in parenthesis):
 * the hydrolysis of substituted cinnamic acid ester in ethanol/water (+1.267)
 * the ionization of substituted phenols in water (+2.008)
 * the acid catalyzed esterification of substituted benzoic esters in ethanol (-0.085)
 * the acid catalyzed bromination of substituted acetophenones (Ketone halogenation) in an acetic acid/water/hydrochloric acid (+0.417)
 * the hydrolysis of substituted benzyl chlorides in acetone-water at 69.8°C (-1.875).

Hammett modifications
Other equations now exist that refine the original Hammett equation: the Swain-Lupton equation, the Taft equation and the Yukawa-Tsuno equation. An equation that address stereochemistry in aliphatic systems is also known.