Hammett equation

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 parentheses):

• 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.

Dominating electronic effects

3 kinds of ground state or static electrical influences predominate:

• Resonance
• Inductive effect: electrical influence of a group which is transmitted primarily by polarization of the bonding electrons from one atom to the next
• Direct electrostatic (field) effect: electrical influence of a polar or dipolar substituent which is transmitted primarily to the reactive group through space (including solvent, if any) according to the laws of classical electrostatics

The latter two influences are often treated together as a composite effect, but are treated here separately. Westheimer demonstrated that the electrical effects of π-substituted dipolar groups on the acidities of benzoic and phenylacetic acids can be quantitatively correlated, by assuming only direct electrostatic action of the substituent on the ionizable proton of the carboxyl group. Westheimer's treatment worked well except for those acids with substituents that have unshared electron pairs such as –OH and –OCH3, as these substituents interact strongly with the benzene ring.

Roberts and Moreland studied the reactivities of 4-substituted bicyclo[2.2.2]octane-1-carboxylic acids and esters. In such a molecule, transmission of electrical effects of substituents through the ring by resonance is not possible. Hence, this hints on the role of the π-electrons in the transmission of substituent effects through aromatic systems.

Reactivity of 4-substituted bicyclo[2.2.2]octane-1-carboxylic acids and esters were measured in 3 different processes, each of which had been previously used with the benzoic acid derivatives. A plot of log(k) against log(K_A) showed a linear relationship. Such linear relationships correspond to linear free energy relationships, which strongly imply that the effect of the substituents are exerted through changes of potential energy and that the steric and entropy terms remain almost constant through the series. The linear relationship fit well in the Hammett Equation. For the 4-substituted bicyclo[2.2.2.]octane-1-carboxylic acid derivatives, the substituent and reaction constants are designated σ’ and ρ’.

Comparison of ρ and ρ’

Reactivity data indicate that the effects of substituent groups in determining the reactivities of substituted benzoic and bicyclo[2.2.2.]-octane-1-carboxylic acids are comparable. This implies that the aromatic π-electrons do not play a dominant role in the transmission of electrical effects of dipolar groups to the ionizable carboxyl group. The difference between ρ and ρ’ for the reactions of the acids with diphenylazomethane is probably due to an inverse relation to the solvent dielectric constant D^e.

Comparison of σ and σ’

For meta-directing groups (electron withdrawing group or EWG), σ_meta and σ_para are more positive than σ’. (The superscript, c, in table denotes data from Hammett, 1940.) For ortho-para directing groups (electron donating group or EDG), σ’ more positive than σ_meta and σ_para. The difference between σ_para and σ’ (σ_para − σ’) is greater than that between σ_meta and σ’(σ_meta − σ’). This is expected as electron resonance effects are felt more strongly at the p-positions. The (σ − σ’) values can be taken as a reasonable measurement of the resonance effects.

Nonlinearity

The plot of the Hammett equation is typically seen as being linear, with either a positive or negative slope correlating to the value of rho. However, nonlinearity emerges in the Hammett plot when a substituent affects the rate of reaction or changes the rate-determining step or reaction mechanism of the reaction. For the reason of the former case, new sigma constants have been introduced to accommodate the deviation from linearity otherwise seen resulting from the effect of the substituent. σ+ takes into account positive charge buildup occurring in the transition state of the reaction. Therefore, an electron donating group (EDG) will accelerate the rate of the reaction by resonance stabilization and will give the following sigma plot with a negative rho value.

σ− is designated in the case where negative charge buildup in the transition state occurs, and the rate of the reaction is consequently accelerated by electron withdrawing groups (EWG). The EWG withdraws electron density by resonance and effectively stabilizes the negative charge that is generated. The corresponding plot will show a positive rho value.

In the case of a nucleophilic acyl substitution the effect of the substituent, X, of the non-leaving group can in fact accelerate the rate of the nucleophilic addition reaction when X is an EWG. This is attributed to the resonance contribution of the EWG to withdraw electron density thereby increasing the susceptibility for nucleophilic attack on the carbonyl carbon. A change in rate occurs when X is EDG, as is evidenced when comparing the rates between X = Me and X = OMe, and nonlinearity is observed in the Hammett plot.

The effect of the substituent may change the rate-determining step (rds) in the mechanism of the reaction. A certain electronic effect may accelerate a certain step so that it is no longer the rds.

A change in the mechanism of a reaction also results in nonlinearity in the Hammett plot. Typically, the model used for measuring the changes in rate in this instance is that of the SN2 reaction. However, it has been observed that in some cases of an SN2 reaction that an EWG does not accelerate the reaction as would be expected and that the rate varies with the substituent. In fact, the sign of the charge and degree to which it develops will be affected by the substituent in the case of the benzylic system.

For example, the substituent may determine the mechanism to be an SN1 type reaction over a SN2 type reaction, in which case the resulting Hammett plot will indicate a rate acceleration due to an EDG, thus elucidating the mechanism of the reaction.

Another deviation from the regular Hammett equation is explained by the charge of nucleophile. Despite nonlinearity in benzylic SN2 reactions, electron withdrawing groups could either accelerate or retard the reaction. If the nucleophile is negatively charged (e.g. cyanide) the electron withdrawing group will increase the rate due to stabilization of the extra charge which is put on the carbon in the transition state. On the other hand, if the nucleophile is not charged (e.g. triphenylphosphine), electron withdrawing group is going to slow down the reaction by decreasing the electron density in the anti bonding orbital of leaving group in the transition state.

Hammett modifications

Other equations now exist that refine the original Hammett equation: the Swain–Lupton equation, the Taft equation, the Grunwald–Winstein equation, and the Yukawa–Tsuno equation. An equation that addresses stereochemistry in aliphatic systems has also been developed.

Estimation of Hammett sigma constants

Core-electron binding energy (CEBE) shifts correlate linearly with the Hammett substituent constants (σ) in substituted benzene derivatives.

Consider para-disubstituted benzene p-F-C₆H₄-Z, where Z is a substituent such as NH₂, NO₂, etc. The fluorine atom is para with respect to the substituent Z in the benzene ring. The image on the right shows four distinguished ring carbon atoms, C1(ipso), C2(ortho), C3(meta), C4(para) in p-F-C₆H₄-Z molecule. The carbon with Z is defined as C1(ipso) and fluorinated carbon as C4(para). This definition is followed even for Z = H. The left-hand side of (1) is called CEBE shift or ΔCEBE, and is defined as the difference between the CEBE of the fluorinated carbon atom in p-F-C₆H₄-Z and that of the fluorinated carbon in the reference molecule FC₆H₅.

The right-hand side of Eq. 1 is a product of a parameter κ and a Hammett substituent constant at the para position, σp. The parameter κ is defined by eq. 3:

where ρ and ρ^* are the Hammett reaction constants for the reaction of the neutral molecule and core ionized molecule, respectively. ΔCEBEs of ring carbons in p-F-C₆H₄-Z were calculated with density functional theory to see how they correlate with Hammett σ-constants. Linear plots were obtained when the calculated CEBE shifts at the ortho, meta and para carbon were plotted against Hammett σ_o, σ_m and σ_p constants respectively.

• κ value calculated ≈ 1.

Hence the approximate agreement in numerical value and in sign between the CEBE shifts and their corresponding Hammett σ constant.

Force–work interpretation of the Hammett equation

A computational force–work interpretation of the Hammett equation has been proposed in terms of substituent-induced forces acting on the nuclei of the reacting substrate. This approach interprets the Hammett relationship by relating substituent effects to the mechanical work performed by forces exerted by the substituent along a reaction path. In this framework, a substituent modifies the potential energy surface of the substrate by exerting forces on its nuclei. These substituent-induced forces can be estimated computationally by replacing the substituent by hydrogen at a fixed substrate geometry and calculating the resulting nuclear forces. Their electronic component can be related to the substituent-induced change in electron density through the Hellmann–Feynman theorem.

If the substituent is not directly involved in the reaction coordinate, the change in activation energy or reaction energy can be expressed as the work performed by the average substituent-induced force along the reaction path:

${\displaystyle W_R=-⟨{\mathbf{F}}_R(\mathbf{q})⟩\cdot \mathrm{\Delta }{\mathbf{q}}_{\mathrm{r}\mathrm{x}}}$

where ${\displaystyle ⟨{\mathbf{F}}_R(\mathbf{q})⟩}$ is the average substituent-induced force and ${\displaystyle \mathrm{\Delta }{\mathbf{q}}_{\mathrm{r}\mathrm{x}}}$ is the reaction-path displacement. By comparing this force–work expression with the Hammett equation, the energetic term associated with the substituent effect can be written as:

${\displaystyle 2.303RT\log \left(\frac{k_R}{k_H}\right)=⟨{\mathbf{F}}_R(\mathbf{q})⟩\cdot \mathrm{\Delta }{\mathbf{q}}_{\mathrm{r}\mathrm{x}}=2.303RT\sigma \rho }$

where ${\displaystyle R}$ is the gas constant, ${\displaystyle T}$ is the absolute temperature, and ${\displaystyle k_R}$ and ${\displaystyle k_H}$ are the rate constants for the substituted and unsubstituted reactions, respectively. This expression relates the Hammett product ${\displaystyle \sigma \rho }$ to the scalar product between a substituent-dependent force and a reaction-dependent displacement. In this interpretation, the substituent constant ${\displaystyle \sigma }$ can be interpreted as a relative effective force characteristic of the substituent, while the reaction constant ${\displaystyle \rho }$ can be interpreted as a relative effective reaction-path length. The approach was illustrated for the alkaline hydrolysis of substituted ethyl benzoates, for which substituent-induced forces projected along the path from the reactant to the transition state reproduced density functional theory changes in activation energy.