Hammett equation

Free energy relationship in organic chemistry

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. 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". |
! Substituent ! ! para effect ! ! meta effect
|
| Dimethylamino || −0.83 || −0.211
|
| Amino || −0.66 || −0.161
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| Butylamino || −0.51 || −0.34
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| Hydroxy || −0.37 || +0.12
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| Methoxy || −0.268 || +0.115
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| Ethoxy || −0.25 || +0.015
|
| Methyl || −0.170 || −0.069
|
| Trimethylsilyl || −0.07 || −0.04
|
| None || 0.000 || 0.000
|
| Fluoro || +0.062 || +0.337
|
| Chloro || +0.227 || +0.373
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| Bromo || +0.232 || +0.393
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| Iodo || +0.276 || +0.353
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| Ethoxycarbonyl || +0.45 || +0.37
|
| Trifluoromethyl || +0.54 || +0.43
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| Cyano || +0.66 || +0.56
|
| Nitro || +0.778 || +0.710
|}

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, or 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.

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 σp 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 σp– 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 σp– with those of σp for "non anomalous" substituents, so as to maintain comparable ρ values: for ArOH ⇄ ArO– + H+, we define
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 σp values for electron donating groups will appear insufficiently negative. Based on similar considerations, a set of σp+ 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 σp+ are based on the rate constants of the SN1 reaction of cumyl chlorides in 90% acetone/water: for ArCMe2Cl + H2O → ArCMe2OH + HCl, we define 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 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 ρ' ρ De 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 De

Substituent σ’ σparac σmetac σpara − σ’ σmeta − σ’ H 0 0 0 0 0 OH 0.283 −0.341 0.014 −0.624 −0.269 CO2C2H5 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 (electron withdrawing group or EWG), σmeta and σpara 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 σmeta and σpara. The difference between σpara and σ’ (σpara – σ’) is greater than that between σmeta and σ’(σmeta − 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.

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.

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 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. is a product of a parameter κ and a Hammett substituent constant at the para position, σp. The parameter κ is defined by eq. :

where ρ and 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 σo, σm and σp 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.

Thomas H. Lowry & Kathleen Schueller Richardson, 1987, Mechanism and Theory in Organic Chemistry, 3rd Edn., New York, NY, US: Harper & Row, , see , 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, , see , 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, , see , 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)," in Handbook of Molecular Descriptors, Vol. 11 of Methods and Principles in Medicinal Chemistry (book series), pp. 144–157, New York, NY, US: John Wiley & Sons, , see , accessed 22 June 2015. N. Chapman, 2012, Correlation Analysis in Chemistry: Recent Advances, New York, NY, US: Springer Science & Business, , see , accessed 22 June 2015.

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, in Biographical Memoirs, Washington, DC, US: National Academies Press, see , accessed 22 June 2015.