In organic chemistry, the Hammett equation describes a linear 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 his 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. It is important to understand, that the equilibrium constants in this relationship come from thermodynamics but the reaction rate constants come from chemical kinetics, and neither discipline predicts such relationship. Instead, it was introduced by Hammett empirically. Hammett equation belongs to a larger set of correlations between the rate of a chemical reaction and its driving force. The basic equation is:
log K K 0 = σ ρ {\displaystyle \log {\frac {K}{K_{0}}}=\sigma \rho }
where
K 0 {\displaystyle {K}_{0}} = Reference constant
σ {\displaystyle \sigma } = Substituent constant
ρ {\displaystyle \rho } = Reaction constant relating the equilibrium constant, K {\displaystyle {K}} , for a given equilibrium reaction with substituent R and the reference constant K 0 {\displaystyle {K}_{0}} when R is a hydrogen atom to the substituent constant σ which depends only on the specific substituent R and the reaction rate 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 k k 0 = σ ρ {\displaystyle \log {\frac {k}{k_{0}}}=\sigma \rho }
In this equation k 0 {\displaystyle {k}_{0}} is the reference reaction rate of the unsubstituted reactant, and k that of a substituted reactant. A plot of log K K 0 {\displaystyle \log {\frac {K}{K_{0}}}} for a given equilibrium versus log k k 0 {\displaystyle \log {\frac {k}{k_{0}}}} 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 the substituent constant is arbitrarily set to 0 and the reaction constant is set to 1: the deprotonation of benzoic acid or benzene carboxylic 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 p-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 σ values displayed in the Table above 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 carboxylic acid (depicted on the left 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 negative, the mesomeric effect is positive, causing partial cancellation. The data also show that for these substituents, the meta effect is much larger than the para effect, due to the fact that the mesomeric effect is greatly reduced in a meta substituent. With meta substituents a carbon atom bearing the negative charge is further away from the carboxylic acid group (structure 2b). 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 a 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.
The σp− and σp+ constants
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