Free energy relationships and their linearity: a test example.

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Linear free energy relationships (LFER) are associated with the dawn of physical organic chemistry in the late 1930s and its objectives in understanding chemical reactivity as measured by reaction rates and equilibria.

The Hammett equation is the best known of the LFERs, albeit derived “intuitively”. It is normally applied to the kinetics of aromatic electrophilic substitution reactions and is expressed as;

log KR/K0 = σRρ (for equilibria) and extended to log kR/k0 = σRρ for rates.

The equilibrium constants are normally derived from the ionisation of substituted benzoic acids, with Kbeing that for benzoic acid itself and Kthat of a substituted benzoic acid, with σR being known as the substituent constant and ρ the reaction constant. The concept involved obtaining the substituent constants by measuring the ionisation equilibria. The value of σis then assumed to be transferable to the rates of reaction, where the values can be used to obtain reaction constants for a given reaction. The latter would then be assumed to give insight into the electronic nature of the transition state for that reaction.

The term log kR/k(the ratio of rates of reaction) can be related to ΔΔG = -RT ln kR/kand this latter quantity can be readily obtained from quantum calculations, where ΔΔG is the difference in computed reaction activation free energies for two substituents (of which one might be R=H). The most interesting such Hammett plots are the ones where a discontinuity becomes apparent. The plot comprises two separate linear relationships, but with different slopes. This is normally taken to indicate a change of mechanism, on the assumption that the two mechanisms will have different responses to substituents. 

A test of this is available via the calculated activations energies for acid catalyzed cyclocondensation to give furanochromanes[1] which is a two-step reaction involving two transition states TS1 and TS2, either of which could be rate determining. A change from one to the other would constitute a change in mechanism. In this example, TS1 involves creation of a carbocationic centre which can be stabilized by the substituent on the Ar group; TS2 involves the quenching of the carbocation by a nucleophilic oxygen and hence might be expected to respond differently to the substituents on Ar. As it happens, the reaction coordinate for TS2 is not entirely trivial, since it also includes an accompanying proton transfer which might perturb the mechanism.

Fortunately for this reaction we have available full FAIR data (DOI: 10.14469/hpc/3943), which includes not only the computed free energies for both sets of transition states but also the entropy-free enthalpies for comparison. This allows the table below to be generated. For each substituent, the highest energy point is in bold, indicating the rate limiting step. The span of substituents corresponds to a range of rate constants of almost 1010, which in fact is rarely if ever achievable experimentally.

Highest free energy overall route for HCl catalysed mechanism,

trans stereochemistry

Sub ΔH/ΔG Reactant ΔH/ΔG, TS1 ΔH/ΔG, TS2 RDS
p-NH2 0.2/6.36 0.0/0.0 0.15/4.0 0.2/6.4 TS2/TS2
p-OMe 2.7/8.48 0.0/0.0 2.7/8.45 2.1/8.48 TS1/TS2
p-Me 5.5/10.00 0.0/0.0 5.5/9.9 3.9/10.00 TS1/TS2
p-Cl 7.7/12.28 0.0/0.0 7.7/12.28 5.9/11.84 TS1/TS1
p-H 7.6/13.01 0.0/0.0 7.6/13.01 5.5/11.51 TS1/TS1
p-CN 10.6/18.02 0.0/0.0 10.6 /17.61 10.5/18.02 TS1/TS2
p-NO2 12.4/19.85 0.0/0.0 12.4/18.24 12.0/19.85 TS1/TS2

For the free energies, you can see that TS2 is the rate limiting step for the first two electron donating substituents, and the last two electron withdrawing ones, whilst TS1 represents the rate limiting step for the middle substituents. This represents two changes of rate limiting step over the entire range of substituents. A different picture emerges if only the enthalpies are used. Now TS1 is rate limiting for essentially all the substituents. The difference of course arises because of significant changes to the entropy of the transition states. The Hammett equation, and its use of  σconstants to try to infer the electronic response of a reaction mechanism, does not really factor in entropic responses. Nor is it often if at all applied using a really wide range of substituents. So any linearity or indeed non-linearity in Hammett plots may correspond only very loosely to the underlying mechanisms involved.

Starting in the 1940s and lasting perhaps 40-50 years, thousands of different reaction mechanisms were subjected to the Hammett treatment during the golden era of physical organic chemistry, but very few have been followed up by exploring the computed free energies, as set out above. One wonders how many of the original interpretations will fully withstand such new scrutiny and in general how influential the role of entropy is.


  1. C.D. Nielsen, W.J. Mooij, D. Sale, H.S. Rzepa, J. Burés, and A.C. Spivey, "Reversibility and reactivity in an acid catalyzed cyclocondensation to give furanochromanes – a reaction at the ‘oxonium-Prins’ vs. ‘ortho-quinone methide cycloaddition’ mechanistic nexus", Chemical Science, vol. 10, pp. 406-412, 2019.

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