Hammett Equation - Rho Value

Rho Value

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. Here is 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).

The reaction constant, or sensitivity constant, ρ, describes the susceptibility of the reaction to substituents, compared to the ionization of benzoic acid. It is equivalent to the slope of the Hammett plot. Information on the reaction and the associated mechanism can be obtained based on the value obtained for ρ. If the value of:

1. ρ>1, the reaction is more sensitive to substituents than benzoic acid and negative charge is built during the reaction (or positive charge is lost).
2. 0<ρ<1, the reaction is less sensitive to substituents than benzoic acid and negative charge is built (or positive charge is lost).
3. ρ=0, no sensitivity to substituents, and no charge is built or lost.
4. ρ<0, the reaction builds positive charge (or loses negative charge).

These relations can be exploited to elucidate the mechanism of a reaction. As the value of ρ is related to the charge during the rate determining step, mechanisms can be devised based on this information. If the mechanism for the reaction of an aromatic compound is thought to occur through one of two mechanisms, the compound can be modified with substituents with different σ values and kinetic measurements taken. Once these measurements have been made, a Hammett plot can be constructed to determine the value of ρ. If one of these mechanisms involves the formation of charge, this can be verified based on the ρ value. Conversely, if the Hammett plot shows that no charge is developed, i.e. a zero slope, the mechanism involving the building of charge can be discarded.

Hammett plots may not always be perfectly linear. For instance, a curve may show a sudden change in slope, or ρ value. In such a case, it is likely that the mechanism of the reaction changes upon adding a different substituent. Other deviations from linearity may be due to a change in the position of the transition state. In such a situation, certain substituents may cause the transition state to appear earlier (or later) in the reaction mechanism.