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{{Short description|Free-energy relationship in organic chemistry}}

In [[organic chemistry]], the '''Hammett equation''' describes a linear [[free-energy relationship]] relating [[reaction rate]]s and [[equilibrium constant]]s for many reactions involving [[benzoic acid]] derivatives with [[Arene substitution patterns|meta- and para-]][[substituent]]s to each other with just two parameters: a substituent constant and a reaction constant.<ref name=goldbook>{{GoldBookRef | title = Hammett equation (Hammett relation) | file = H02732}}</ref><ref>{{cite journal | last1 = Keenan | first1 = Sheue L. | last2 = Peterson | first2 = Karl P. | last3 = Peterson | first3 = Kelly | last4 = Jacobson | first4 = Kyle | title = Determination of Hammett Equation Rho Constant for the Hydrolysis of p-Nitrophenyl Benzoate Esters | journal = [[J. Chem. Educ.]] | volume = 85 | pages = 558 | year = 2008 | doi = 10.1021/ed085p558 | issue = 4|bibcode = 2008JChEd..85..558K }}</ref> This equation was developed and published by [[Louis Plack Hammett]] in 1937<ref name=Hammett1937>{{cite journal | last1 = Hammett | first1 = Louis P. | journal = [[J. Am. Chem. Soc.]] | volume = 59 | pages = 96–103 | year = 1937 | doi = 10.1021/ja01280a022 | title=The Effect of Structure upon the Reactions of Organic Compounds. Benzene Derivatives | issue=1}}</ref> as a follow-up to qualitative observations in his 1935 publication.<ref name=Hammett1935>{{cite journal | title = Some relations between Reaction Rates and Equilibrium Constants | author = Louis P. Hammett | journal =  [[Chem. Rev.]] | year = 1935 | volume = 17 | issue = 1 | pages = 125–136 | doi = 10.1021/cr60056a010}}</ref>

The basic idea is that for any two reactions with two aromatic reactants only differing in the type of substituent, the change in [[activation energy|free energy of activation]] is proportional to the change in [[Gibbs free energy]].<ref name="Carey1983">{{cite book |title=Advanced Organic Chemistry Part A, 2nd edition |first1=Francis A. |last1=Carey |first2=Richard J. |last2=Sundberg |year=1983 |publisher=Plenum Press |url=https://books.google.com/books?id=uWQaAQAAMAAJ |oclc=1064985344 |isbn=9780306410871}}</ref> This notion does not follow from elemental [[thermochemistry]] or [[chemical kinetics]] and was introduced by Hammett intuitively.{{efn|The opening line in his 1935 publication reads: "The idea that there is some sort of relationship between the rate of a reaction and the equilibrium constant is one of the most persistently held and at the same time most emphatically denied concepts in chemical theory".<ref name=Hammett1935 />}}

The basic equation is:

:<math>\log \frac{K}{K_0} = \sigma\rho </math>
where
:<math>{K}_0</math> = Reference constant
:<math>\sigma</math> = Substituent constant
:<math>\rho</math> = [[Reaction rate constant]]

relating the [[equilibrium constant]], <math>{K}</math>, for a given equilibrium reaction with substituent R and the '''reference constant''' <math>{K}_0</math> when R is a hydrogen atom to the '''substituent constant''' [[sigma|{{mvar|σ}}]] which depends only on the specific substituent R and the '''reaction rate constant''' [[rho|ρ]] which depends only on the type of reaction but not on the substituent used.<ref name=Hammett1935 /><ref name=Hammett1937 />

The equation also holds for [[reaction rate]]s k of a series of reactions with substituted benzene derivatives:

:<math>\log \frac{k}{k_0} = \sigma\rho</math>

In this equation <math>{k}_0</math> is the reference reaction rate of the unsubstituted reactant, and k that of a substituted reactant.

A plot of <math>\log \frac{K}{K_0}</math> for a given equilibrium versus <math>\log \frac{k}{k_0}</math> 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&nbsp;°C.

[[File:BenzoicAcidDissociation.svg|center|600px|Scheme 1. Dissociation of benzoic acids]]

{| class="wikitable sortable" style="float:right;"
|+ Substituent constants: ''para'' and ''meta'' substituted benzene rings.<ref name="Hammett1937"/>{{update after|2015|6|23}}<ref>Table values are this original 1937 publication, and differ from values appearing in subsequent publications. For more standard values, see: {{cite journal |author1=C. Hansch |author2=A. Leo |author3=R. W. Taft | title = A survey of Hammett substituent constants and resonance and field parameters | journal = [[Chemical Reviews|Chem. Rev.]] | volume = 91 | year = 1991 | pages = 165–195 | doi = 10.1021/cr00002a004 | issue = 2|s2cid=97583278 }}</ref>
|-
! Substituent !! ''para-'' effect !! ''meta-'' effect 
|-
| Dimethylamino || &minus;0.83 || &minus;0.211
|-
| Amino  || &minus;0.66 || &minus;0.161
|-
| Butylamino || &minus;0.51 || &minus;0.34
|-
| Hydroxy || &minus;0.37 || +0.12
|-
| Methoxy || &minus;0.268 || +0.115
|-
| Ethoxy || &minus;0.25 || +0.015
|-
| Methyl || &minus;0.170 || &minus;0.069
|-
| Trimethylsilyl || &minus;0.07 || &minus;0.04
|-
| None || 0.000 || 0.000
|-
| Fluoro || +0.062 || +0.337
|-
| Chloro || +0.227 || +0.373
|-
| Bromo || +0.232 || +0.393
|-
| Iodo || +0.276 || +0.353
|-
| Ethoxycarbonyl || +0.45 || +0.37
|-
| Trifluoromethyl || +0.54 || +0.43
|-
| Cyano || +0.66 || +0.56
|-
| Nitro || +0.778 || +0.710
|}

Having obtained a value for K<sub>0</sub>, a series of equilibrium constants (K) are now determined based on the same process, but now with variation of the para substituent—for instance, {{nobr|[[p-hydroxybenzoic acid]] (R{{=}}OH, R'{{=}}H)}} or {{nobr|[[4-aminobenzoic acid|p-aminobenzoic acid]] (R{{=}}NH<sub>2</sub>, R'{{=}}H)}}. These values, combined in the Hammett equation with K<sub>0</sub> 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 compound|nitro]] and [[cyano]] substituents. Repeating the process with meta-substituents afford the '''meta substituent constants'''. This treatment does not include [[Arene substitution patterns|ortho-]]substituents, which would introduce [[steric effect]]s.

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 compound|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 [[halogen]]s, 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.
  
[[Image:HammettInductiveMesomericEffects.png|center|600px|Scheme 3. Hammett Inductive Mesomeric Effects]]

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 ''σ<sub>p</sub>''<sup>−</sup> and ''σ<sub>p</sub>''<sup>+</sup> constants ===
{{Unreferenced section|reason=No literature is cited to support this section.|date=March 2020}}

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, the ''σ<sub>p</sub>'' 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 of ''σ<sub>p</sub>''<sup>−</sup> constants may give a better fit.  This parameter is defined using the ionization constants of ''para'' substituted phenols, via a scaling factor to match up the values of ''σ<sub>p</sub>''<sup>−</sup> with those of ''σ<sub>p</sub>'' for "non-anomalous" substituents, so as to maintain comparable ρ values: for ArOH ⇄ ArO<sup>−</sup> + H<sup>+</sup>, we define <math chem>\sigma_p^- = \frac{1}{2.11}\log_{10}\left(\frac{K_\ce{X}}{K_\ce{H}}\right)</math>.

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, the ''σ<sub>p</sub>'' values for electron-donating groups will appear insufficiently negative.  Based on similar considerations, a set of ''σ<sub>p</sub><sup>+</sup>'' constants give better fit for reactions involving electron-donating groups at the ''para'' position and the formation of a carbocation at the benzylic site.  The ''σ<sub>p</sub>''<sup>+</sup> are based on the ''rate'' constants of the S<sub>N</sub>1 reaction of cumyl chlorides in 90% acetone/water: for {{nowrap|ArCMe<sub>2</sub>Cl + H<sub>2</sub>O → ArCMe<sub>2</sub>OH + HCl}}, we define <math chem>\sigma_p^+ = -\frac{1}{4.54}\log_{10}\left(\frac{k_\ce{X}}{k_\ce{H}}\right)</math>.  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.

==Rho value==
With knowledge of substituent constants it is now possible to obtain reaction constants for a wide range of [[organic reaction]]s. The archetypal reaction is the [[alkaline hydrolysis]] of [[ethyl benzoate]] (R=R'=H) in a water/ethanol mixture at 30&nbsp;°C. Measurement of the [[reaction rate]] k<sub>0</sub> combined with that of many substituted ethyl benzoates ultimately result in a reaction constant of +2.498.<ref name="Hammett1937"/>{{update after|2015|6|23}}{{primary source inline|date=June 2015}}

[[Image:BenzoateEsterHydrolysis.png|center|600px|Scheme 2. Hydrolysis of benzoic acid esters]]

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 [[acetophenone]]s ([[Ketone halogenation]]) in an [[acetic acid]]/water/[[hydrochloric acid]] (+0.417)
* the hydrolysis of substituted [[benzyl chloride]]s in [[acetone]]-water at 69.8&nbsp;°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:

# ''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).
# ''0<ρ<1'', the reaction is less sensitive to substituents than benzoic acid and negative charge is built (or positive charge is lost).
# ''ρ=0'', no sensitivity to substituents, and no charge is built or lost.
# ''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.<ref>E.V. Anslyn & D.A. Dougherty, ''Modern Physical Organic Chemistry,'' pp. TBD, Sausalito, CA, US: University Science Books, {{ISBN|1891389319}}.{{page needed|date=June 2015}}</ref>{{page needed|date=June 2015}}

==Dominating electronic effects==
3 kinds of ground state or ''static'' electrical influences predominate:
* [[Resonance effect|Resonance]] (mesomeric) effect
* [[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 [[chemical polarity|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 acid|benzoic]] and [[phenylacetic acid]]s 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.<ref>{{cite journal | doi = 10.1021/ja01877a012 | author = Westheimer F.H. | title = The Electrostatic effect of substituents on the dissociation constants of organic acids. IV. Aromatic acids | journal = [[J. Am. Chem. Soc.]] | volume = 61 | pages = 1977–1980 | year = 1939 | issue = 8}}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}<ref>{{cite journal | doi = 10.1063/1.1750302 |author1=Kirkwood J.G. |author2=Westheimer F.H. | title = The electrostatic influence of substituents on the dissociation constants of organic acids. I [Missing Subtitle]| journal = [[J. Chem. Phys.]] | volume = 6 | pages = 506 | year = 1938 | issue = 9|bibcode = 1938JChPh...6..506K }}{{update after|2015|6|23}}</ref>{{primary source inline|date=June 2015}}{{update after|2015|6|23}} {{primary source inline|date=June 2015}}

[[Image:4-substituted bicyclo(2.2.2)octane-1-carboxylic ester.svg|right|100px|4-substituted bicyclo-2.2.2.-octane-1-carboxylic acid]] 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 hydrocarbon|aromatic systems]].<ref>{{cite journal | doi = 10.1021/ja01105a045 |author1=Roberts J.D. |author2=Moreland Jr. W.T. | title = Electrical Effects of Substituent Groups in Saturated Systems. Reactivities of 4-Substituted Bicyclo [2.2.2] octane-1-carboxylic acids | journal = [[J. Am. Chem. Soc.]] | year = 1953 | volume = 75 | pages = 2167–2173 | issue = 9}}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}

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<sub>A</sub>) 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 [[chemical potential|potential energy]] and that the [[steric]] and [[entropy|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’===
{| class="wikitable" style="float:right; margin: 1em"
|-
! Reaction{{citation needed|date=June 2015}} !! ρ' !! ρ !! D<sup>e</sup>
|-
| Ionization of acids || 1.464 || 1.464 || 54
|-
| Alkaline hydrolysis of ethyl esters || 2.24 || 2.494 || 28
|-
| Acids with diphenyldiazomethane || 0.698 || 0.937 || 24
|}

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 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<sup>e</sup>

===Comparison of σ and σ’===
{{Page numbers improve|section|date=June 2015}}
{| class=wikitable style="float: left; margin: 1em"
|-
! Substituent !! σ’ {{citation needed|date=June 2015}} !! σ<sub>para</sub><sup>c</sup> !! σ<sub>meta</sub><sup>c</sup> !! σ<sub>para</sub> − σ’ {{citation needed|date=June 2015}} !! σ<sub>meta</sub> − σ’ {{citation needed|date=June 2015}}
|-
| H || 0 || 0 || 0 || 0 || 0
|-
| OH || 0.283 || −0.341 || 0.014 || −0.624 || −0.269
|-
| CO<sub>2</sub>C<sub>2</sub>H<sub>5</sub> || 0.297 || 0.402 || 0.334 || 0.105 || 0.037
|-
| Br || 0.454 || 0.232 || 0.391 || −0.222 || −0.063
|-
| CN || 0.579 || 0.656 || 0.608 || 0.077 || 0.029
|}

For meta-directing groups ([[Polar effect|electron withdrawing group or EWG]]), σ<sub>meta</sub> and σ<sub>para</sub> are more positive than σ’. (The superscript, c, in table denotes data from Hammett, 1940.<ref>L.P.Hammett, 1940, "Chapter III," "Chapter IV," and "Chapter VII," in ''Physical Organic Chemistry'', New York, NY, US: McGraw-Hill.{{page needed|date=June 2015}}</ref>{{page needed|date=June 2015}}) For ortho-para directing groups ([[Polar effect|electron donating group or EDG]]), σ’ more positive than σ<sub>meta</sub> and σ<sub>para</sub>. The difference between σ<sub>para</sub> and σ’ (σ<sub>para</sub> − σ’) is greater than that between σ<sub>meta</sub> and σ’(σ<sub>meta</sub> − 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.
{{clear}}

== Nonlinearity ==
{{Cleanup|section|reason=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 either [[WP:OR]] or [[WP:VERIFY]]|date=June 2015}}
[[Image:Rate acceleration EDG.png|right|300px|Rate acceleration EDG]]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.<ref>Y. Yukawa & Y. Tsuno, 1959, "Resonance Effect in Hammett Relationship. II. Sigma Constants in Electrophilic Reactions and their Intercorrelation," ''Bull. Chem. Soc. Jpn.'' '''32''':965-971, see [https://www.jstage.jst.go.jp/article/bcsj1926/32/9/32_9_965/_pdf], accessed 22 June 2015.</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}

[[Image:Rate acceleration EWG.png|300px|right|Rate acceleration EWG]] 
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.<ref>{{cite journal | doi = 10.1021/jo035854r | pmid = 15049643 | last1 = Um | first1 = Ik-Hwan | title = Curved Hammett plot in alkaline hydrolysis of ''O''-aryl thionobenzoates: Change in rate-determining step versus ground-state stabilization | journal = [[Journal of Organic Chemistry|J. Org. Chem.]] | year = 2004 | volume = 69 | pages = 2436–2441 | last2 = Lee | first2 = Ji-Youn | last3 = Kim | first3 = Han-Tae | last4 = Bae | first4 = Sun-Kun | issue = 7}}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}

:[[Image:Nuc mech.png|500px|Nucleophilic acyl substitution]]

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.<ref>{{cite journal | doi = 10.1021/ja00986a018 | author = Hart, H. | journal = [[Journal of the American Chemical Society|J. Am. Chem. Soc.]] | year = 1967 | volume = 89 | pages = 2342 | last2 = Sedor | first2 = Edward A. | issue = 10 | title=Mechanism of cyclodehydration of 2-phenyltriarylcarbinols}}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}

:[[Image:Change in rds.png|600px|change in rate determining step]]

[[Image:Change in rds graph.png|right|300px|change in rate determining step]]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.<ref name=SteinJOC80>{{cite journal | doi = 10.1021/jo01305a045 | last1 = Stein | first1 = Allan R. | title = Nonlinearity of Hammett .sigma..rho. correlations for benzylic systems: activation parameters and their mechanistic implications | journal = [[J. Org. Chem.]] | year = 1980 | volume = 45 | pages = 3539–3540 | last2 = Tencer | first2 = Michal | last3 = Moffatt | first3 = Elizabeth A. | last4 = Dawe | first4 = Robert | last5 = Sweet | first5 = James | issue = 17}}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}  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<ref>{{cite journal | doi = 10.1021/ja00506a025 | last1 = Young | first1 = P. R. | title = Separation of polar and resonance substituent effects in the reactions of acetophenones with bisulfite and of benzyl halides with nucleophiles | journal = [[Journal of the American Chemical Society|J. Am. Chem. Soc.]]  | year = 1979 | volume = 101 | pages = 3288 | last2 = Jencks | first2 = W. P. | issue = 12}}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}} 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.<ref name=SteinJOC80/>{{primary source inline|date=June 2015}}

[[Image:Change in mech graph.png|right|300px|Change in emchanism]]For example, the substituent may determine the mechanism to be an [[SN1 reaction|SN1]] type reaction over a [[SN2 reaction|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.
:[[Image:Change in mech.png|600px]]

Another deviation from the regular Hammett equation is explained by the charge of nucleophile.<ref name=SteinJOC80/>{{primary source inline|date=June 2015}} 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]],{{citation needed|date=June 2015}} the [[Taft equation]],{{citation needed|date=June 2015}} the [[Grunwald–Winstein equation]],{{citation needed|date=June 2015}} and the [[Yukawa–Tsuno equation]].{{citation needed|date=June 2015}} An equation that addresses stereochemistry in aliphatic systems has also been developed.{{vague|date=June 2015}}<ref>{{cite journal | last1 = Bols | first1 = Mikael | last2 = Liang | first2 = Xifu | last3 = Jensen | first3 = Henrik H. | title = Equatorial contra axial polar substituents. The relation of a chemical reaction to stereochemical substituent constants | journal = [[J. Org. Chem.]] | volume = 67 | pages = 8970–4 | year = 2002 | doi = 10.1021/jo0205356 | issue = 25| pmid = 12467416 }}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}

==Estimation of Hammett sigma constants==
{{overly detailed|section|date=June 2015}}
[[Image:Carbon positions.png|right|75px]]

Core-electron binding energy (CEBE) shifts correlate linearly with the Hammett substituent constants (''σ'') in substituted [[benzene]] derivatives.<ref>{{cite journal | doi = 10.1016/0009-2614(76)85053-1 |author1=Linderberg, B. |author2=Svensson, S. |author3=Malmquist, P.A. |author4=Basilier, E. |author5=Gelius, U. |author6=Siegbahn, K. | title = Correlation of ESCA shifts and Hammett substituent constants in substituted benzene derivatives | journal = [[Chem. Phys. Lett.]] | year = 1976 | volume = 40 | pages = 175 | issue = 2|bibcode = 1976CPL....40..175L }}</ref>{{primary source inline|date=June 2015}}

{{NumBlk|:| {{math|ΔCEBE ≈ ''κσ<sub>p</sub>''}} |{{EquationRef|1}}}}

Consider para-disubstituted benzene p-F-C<sub>6</sub>H<sub>4</sub>-Z, where Z is a [[substituent]] such as NH<sub>2</sub>, NO<sub>2</sub>, 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([[Arene substitution patterns|ipso]]), C2([[Arene substitution patterns|ortho]]), C3([[Arene substitution patterns|meta]]), C4([[Arene substitution patterns|para]]) in p-F-C<sub>6</sub>H<sub>4</sub>-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 ({{EquationNote|1}}) is called CEBE shift or ΔCEBE, and is defined as the difference between the CEBE of the fluorinated carbon atom in p-F-C<sub>6</sub>H<sub>4</sub>-Z and that of the fluorinated carbon in the reference molecule FC<sub>6</sub>H<sub>5</sub>.

{{NumBlk|:| ΔCEBE ≡ CEBE(C4 in p-F-C<sub>6</sub>H<sub>4</sub>-Z) − CEBE(C4 in p-F-C<sub>6</sub>H<sub>5</sub>) |{{EquationRef|2}}}}

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

{{NumBlk|:|{{math|''κ'' {{=}} 2.3''kT''(''ρ'' − ''ρ''{{sup|*}})}} |{{EquationRef|3}}}}

where {{mvar|ρ}} and {{math|''ρ''{{sup|*}}}} are the Hammett reaction constants for the reaction of the neutral molecule and core ionized molecule, respectively.  ΔCEBEs of ring carbons in p-F-C<sub>6</sub>H<sub>4</sub>-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<sub>o</sub>, σ<sub>m</sub>'' and ''σ<sub>p</sub>'' constants respectively.

* {{mvar|κ}} value calculated ≈ 1. 
Hence the approximate agreement in numerical value and in sign between the CEBE shifts and their corresponding Hammett ''σ'' constant.<ref>{{cite journal | doi = 10.1002/qua.20533 |author1=Takahata Y. |author2=Chong D.P. | title = Estimation of Hammett sigma constants of substituted benzenes through accurate density-functional calculation of core-electron binding energy shifts | journal = [[International Journal of Quantum Chemistry]] | year = 2005 | volume = 103 | pages = 509–515 | issue = 5|bibcode = 2005IJQC..103..509T }}</ref>{{primary source inline|date=June 2015}}{{primary source inline|date=June 2015}}

<gallery>
Image:Plot of CEBE shift against sigma-p.jpg|Plot of calculated CEBE shift (eV) against sigma-para
Image:CEBE shift and sigma-p table.jpg|Table of CEBE shifts (eV) and sigma-para
Image:CEBE shift against sigma-m plot.jpg|Plot of calculated CEBE shift (eV) against sigma-meta
Image:CEBE shift and sigma-m table.jpg|Table of CEBE shifts (eV) and sigma-meta
Image:CEBE shift against sigma-o graph.jpg|Plot of calculated CEBE shift (eV) against sigma-o
Image:CEBE shift and sigma-o table.jpg|Table of CEBE shifts (eV) and sigma-ortho
</gallery>

==See also==
* [[Bell–Evans–Polanyi principle]]
* [[Craig plot]]
* [[Free-energy relationship]]
* [[Acid dissociation constant|p''K''<sub>a</sub>]]
* [[Quantitative structure–activity relationship]]

== Notes ==
{{notelist}}
==References==
{{reflist|30em}}

==Further reading==

===General===
* Thomas H. Lowry & Kathleen Schueller Richardson, 1987, ''Mechanism and Theory in Organic Chemistry,'' 3rd Edn., New York, NY, US: Harper & Row, {{ISBN|0060440848}}, see [https://books.google.com/books?isbn=0060440848], 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 [https://books.google.com/books?isbn=0306468565], 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 [https://books.google.com/books?isbn=0470084944], accessed 19 June 2015.

===Theory===
* 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.

===Surveys of descriptors===
* Roberto Todeschini, Viviana Consonni, Raimund Mannhold, Hugo Kubinyi & Hendrik Timmerman, 2008, "Entry: Electronic substituent constants (Hammet substituent constants, σ electronic constants)," in ''Handbook of Molecular Descriptors,'' Vol. 11 of ''Methods and Principles in Medicinal Chemistry'' (book series), pp.&nbsp;144–157, New York, NY, US: John Wiley & Sons, {{ISBN|3527613110}}, see [https://books.google.com/books?isbn=3527613110], accessed 22 June 2015.
* N. Chapman, 2012, ''Correlation Analysis in Chemistry: Recent Advances,'' New York, NY, US: Springer Science & Business, {{ISBN|1461588316}}, see [https://books.google.com/books?isbn=1461588316], accessed 22 June 2015.

===History===
* {{cite journal|first = John D.|last = Roberts|year = 1996|title = The beginnings of physical organic chemistry in the United States|journal = [[Bull. Hist. Chem.]]|volume = 19|pages = 48–56|url = http://www.scs.illinois.edu/~mainzv/HIST/bulletin_open_access/num19/num19%20p48-56.pdf}}
* 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.&nbsp;136–149, in ''Biographical Memoirs,'' Washington, DC, US: National Academies Press, see [http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/hammett-louis-p.pdf], accessed 22 June 2015.

{{DEFAULTSORT:Hammett Equation}}
[[Category:Physical organic chemistry]]
[[Category:Equations]]