Inorganic chemistry, theHammett equation describes a linearfree-energy relationship relatingreaction rates andequilibrium constants for many reactions involvingbenzoic acid derivatives withmeta- and para-substituents to each other with just two parameters: a substituent constant and a reaction constant.^[1][2] This equation was developed and published byLouis Plack Hammett in 1937^[3] as a follow-up to qualitative observations in his 1935 publication.^[4]

The basic idea is that for any two reactions with two aromatic reactants only differing in the type of substituent, the change infree energy of activation is proportional to the change inGibbs free energy.^[5] It is important to understand, that theequilibrium constants in this relationship come fromthermodynamics but thereaction rate constants come fromchemical kinetics , and neither discipline predicts such relationship. Instead, it was introduced by Hammett empirically.^[a] Hammett equation belongs to a larger set of correlations between the rate of a chemical reaction and its driving force.^[6][7][8]

The basic equation is:

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

where

${\displaystyle K_0}$ = Reference constant

${\displaystyle \sigma }$ = Substituent constant

${\displaystyle \rho }$ = Reaction rate constant

relating theequilibrium constant,${\displaystyle K}$, for a given equilibrium reaction with substituent R and thereference constant${\displaystyle K_0}$ when R is a hydrogen atom to thesubstituent constantσ which depends only on the specific substituent R and thereaction rate constantρ which depends only on the type of reaction but not on the substituent used.^[4][3]

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

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

In this equation${\displaystyle k_0}$ is the reference reaction rate of the unsubstituted reactant, and k that of a substituted reactant.

A plot of${\displaystyle \log \frac{K}{K_0}}$ for a given equilibrium versus${\displaystyle \log \frac{k}{k_0}}$ for a given reaction rate with many differently substituted reactants will give a straight line.

The starting point for the collection of the substituent constants is achemical equilibrium for which the substituent constant is arbitrarily set to 0 and the reaction constant is set to 1: the deprotonation ofbenzoic acid orbenzene 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) orp-aminobenzoic acid (R=NH₂, R'=H). These values, combined in the Hammett equation with K₀ and remembering that ρ = 1, give thepara substituent constants compiled in table 1 foramine,methoxy,ethoxy,dimethylamino,methyl,fluorine,bromine,chlorine,iodine,nitro andcyano substituents. Repeating the process with meta-substituents afford themeta substituent constants. This treatment does not includeortho-substituents, which would introducesteric effects.

The σ values displayed in the Table above reveal certain substituent effects. With ρ = 1, the group of substituents with increasing positive values—notablycyano andnitro—cause the equilibrium constant to increase compared to thehydrogen reference, meaning that theacidity 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-withdrawinginductive effect (−I) and also by a negativemesomeric effect (−M).

The next set of substituents are thehalogens, for which the substituent effect is still positive but much more modest. The reason for this is that while theinductive effect is still negative, themesomeric 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 inscheme 3, where, in a para substituted arene1a, oneresonance structure1b is aquinoid 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, likemethoxy andethoxy, can even have opposite signs for the substituent constant as a result of opposing inductive and mesomeric effect. Only alkyl and aryl substituents likemethyl 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.

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(March 2020) (Learn how and when to remove this message)

Because the carbonyl group is unable to serve a source of electrons for −M groups (in contrast to lone pair donors like OH), for reactions involving phenol and aniline starting materials, theσ_p values for electron-withdrawing groups will appear too small. For reactions where resonance effects are expected to have a major impact, a modified parameter, and a modified set ofσ_p⁻ constants may give a better fit. This parameter is defined using the ionization constants ofpara substituted phenols, via a scaling factor to match up the values ofσ_p⁻ with those ofσ_p for "non-anomalous" substituents, so as to maintain comparable ρ values: for ArOH ⇄ ArO⁻ + H⁺, we define${\displaystyle {\sigma }_p^{-}=\frac{1}{2.11}{\log }_{10}\left(\frac{K_{\text{X}}}{K_{\text{H}}}\right)}$.

Likewise, the carbonyl carbon of a benzoic acid is at a nodal position and unable to serve as a sink for +M groups (in contrast to a carbocation at the benzylic position). Thus for reactions involving carbocations at the α-position, theσ_p values for electron-donating groups will appear insufficiently negative. Based on similar considerations, a set ofσ_p⁺ constants give better fit for reactions involving electron-donating groups at thepara position and the formation of a carbocation at the benzylic site. Theσ_p⁺ are based on therate constants of the S_N1 reaction of cumyl chlorides in 90% acetone/water: forArCMe₂Cl + H₂O → ArCMe₂OH + HCl, we define${\displaystyle {\sigma }_p^{+}=-\frac{1}{4.54}{\log }_{10}\left(\frac{k_{\text{X}}}{k_{\text{H}}}\right)}$. Note that the scaling factor is negative, since an electron-donating group speeds up the reaction. For a reaction whose Hammett plot is being constructed, these alternative Hammett constants may need to be tested to see if a better linearity could be obtained.

With knowledge of substituent constants it is now possible to obtain reaction constants for a wide range oforganic reactions. The archetypal reaction is thealkaline hydrolysis ofethyl benzoate (R=R'=H) in a water/ethanol mixture at 30 °C. Measurement of thereaction rate k₀ combined with that of many substituted ethyl benzoates ultimately result in a reaction constant of +2.498.^{[3][needs update][non-primary source needed]}

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 substitutedcinnamic acid ester in ethanol/water (+1.267)
• the ionization of substitutedphenols in water (+2.008)
• the acid catalyzedesterification of substituted benzoic esters inethanol (−0.085)
• the acid catalyzed bromination of substitutedacetophenones (Ketone halogenation) in anacetic acid/water/hydrochloric acid (+0.417)
• the hydrolysis of substitutedbenzyl chlorides inacetone-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 forρ. 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 ofρ 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 differentσ values and kinetic measurements taken. Once these measurements have been made, a Hammett plot can be constructed to determine the value ofρ. 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, orρ 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.^{[10][page needed]}

3 kinds of ground state orstatic 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 apolar or dipolarsubstituent which is transmitted primarily to the reactive group through space (includingsolvent, if any) according to the laws of classicalelectrostatics

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 ofbenzoic andphenylacetic acids can be quantitatively correlated, by assuming only direct electrostatic action of the substituent on the ionizable proton of thecarboxyl 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.^{[11][non-primary source needed][non-primary source needed][12][non-primary source needed][needs update][non-primary source needed]}

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 througharomatic systems.^{[13][non-primary source needed][non-primary source needed]}

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 ofpotential energy and that thesteric andentropy 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’.

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 solventdielectric constant D^e.

This section cites its sourcesbut itspage reference ranges are too broad or are incorrect. Please helpimprove it by specifying more precise page ranges.(June 2015) (Learn how and when to remove this message)

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.^{[14][page needed]}) 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.

This section mayrequirecleanup to meet Wikipedia'squality standards. The specific problem is:the images in the section are too loosely tied to the text, e.g., there is no mention of sulfinate ester or imine hydrolysis, though this is what the schemes present, and there are no sources indicated for the schemes, violating eitherWP:OR orWP:VERIFY. Please helpimprove this section if you can.(June 2015) (Learn how and when to remove this message)

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 therate-determining step orreaction 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 thetransition 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.^{[15][non-primary source needed][non-primary source needed]}

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 anucleophilic 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.^{[16][non-primary source needed][non-primary source needed]}

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.^{[17][non-primary source needed][non-primary source needed]}

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.^{[18][non-primary source needed][non-primary source needed]} However, it has been observed that in some cases of anSN2 reaction that an EWG does not accelerate the reaction as would be expected^{[19][non-primary source needed][non-primary source needed]} 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.^{[18][non-primary source needed]}

For example, the substituent may determine the mechanism to be anSN1 type reaction over aSN2 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.^{[18][non-primary source needed]} 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.

Other equations now exist that refine the original Hammett equation: theSwain–Lupton equation,^[20] theTaft equation,^{[citation needed]} theGrunwald–Winstein equation,^{[citation needed]} and theYukawa–Tsuno equation.^{[citation needed]} An equation that addresses stereochemistry in aliphatic systems has also been developed.^{[vague][21][non-primary source needed][non-primary source needed]}

This sectioncontains an excessive amount of intricatedetail. Please helpimprove it byspinning off orrelocating relevant information and removing excessive detail that goes againstWikipedia's inclusion policy.(June 2015) (Learn how and when to remove this message)

Core-electron binding energy (CEBE) shifts correlate linearly with the Hammett substituent constants (σ) in substitutedbenzene derivatives.^{[22][non-primary source needed]}

Consider para-disubstituted benzene p-F-C₆H₄-Z, where Z is asubstituent 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 parameterκ and a Hammett substituent constant at the para position,σp. The parameterκ is defined by eq.3:

whereρ andρ^* 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 withdensity 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 Hammettσ_o, σ_m andσ_p constants respectively.

• κ value calculated ≈ 1.

Hence the approximate agreement in numerical value and in sign between the CEBE shifts and their corresponding Hammettσ constant.^{[23][non-primary source needed][non-primary source needed]}

• 
  Plot of calculated CEBE shift (eV) against sigma-para
• 
  Table of CEBE shifts (eV) and sigma-para
• 
  Plot of calculated CEBE shift (eV) against sigma-meta
• 
  Table of CEBE shifts (eV) and sigma-meta
• 
  Plot of calculated CEBE shift (eV) against sigma-o
• 
  Table of CEBE shifts (eV) and sigma-ortho

• Bell–Evans–Polanyi principle
• Craig plot
• Free-energy relationship
• pK_a
• Quantitative structure–activity relationship

• Thomas H. Lowry & Kathleen Schueller Richardson, 1987,Mechanism and Theory in Organic Chemistry, 3rd Edn., New York, NY, US: Harper & Row,ISBN 0060440848, see[2], accessed 20 June 2015.
• Francis A. Carey & Richard J. Sundberg, 2006, "Title Advanced Organic Chemistry: Part A: Structure and Mechanisms," 4th Edn., New York, NY, US: Springer Science & Business Media,ISBN 0306468565, see[3], accessed 19 June 2015.
• Michael B. Smith & Jerry March, 2007, "March's Advanced Organic Chemistry: Reactions, Mechanisms, and Structure," 6th Ed., New York, NY, US: Wiley & Sons,ISBN 0470084944, see[4], accessed 19 June 2015.

• L.P. Hammett, 1970,Physical Organic Chemistry, 2nd Edn., New York, NY, US: McGraw-Hill.
• John Shorter, 1982,Correlation Analysis of Organic Reactivity, Chichester 1982.
• Otto Exner, 1988,Correlation Analysis of Chemical Data, New York, NY, US: Plenum.

• Roberto Todeschini, Viviana Consonni, Raimund Mannhold, Hugo Kubinyi & Hendrik Timmerman, 2008, "Entry: Electronic substituent constants (Hammet substituent constants, σ electronic constants)," inHandbook of Molecular Descriptors, Vol. 11 ofMethods and Principles in Medicinal Chemistry (book series), pp. 144–157, New York, NY, US: John Wiley & Sons,ISBN 3527613110, see[5], accessed 22 June 2015.
• N. Chapman, 2012,Correlation Analysis in Chemistry: Recent Advances, New York, NY, US: Springer Science & Business,ISBN 1461588316, see[6], accessed 22 June 2015.

• Roberts, John D. (1996)."The beginnings of physical organic chemistry in the United States"(PDF).Bull. Hist. Chem.19:48–56.
• John Shorter, 2000, "The prehistory of the Hammett equation,"Chem. Listy,94:210-214.
• Frank Westheimer, 1997, "Louis Plack Hammett, 1894—1987: A Biographical Memoir," pp. 136–149, inBiographical Memoirs, Washington, DC, US: National Academies Press, see[7], accessed 22 June 2015.

• Physical organic chemistry
• Equations

• Wikipedia articles needing page number citations from June 2015
• Wikipedia articles in need of updating from June 2015
• All Wikipedia articles in need of updating
• Webarchive template wayback links
• CS1 maint: multiple names: authors list
• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from March 2020
• All articles needing additional references
• All pages needing factual verification
• Wikipedia articles needing factual verification from June 2015
• All articles with unsourced statements
• Articles with unsourced statements from June 2015
• Articles lacking page references from June 2015
• Articles needing cleanup from June 2015
• All pages needing cleanup
• Cleanup tagged articles with a reason field from June 2015
• Wikipedia pages needing cleanup from June 2015
• All Wikipedia articles needing clarification
• Wikipedia articles needing clarification from June 2015
• Wikipedia articles that are excessively detailed from June 2015
• All articles that are excessively detailed
• Wikipedia articles with style issues from June 2015
• All articles with style issues