Interdependence of the Hammett and isokinetic relationships: a numerical simulation approach

Many homologous reaction series present linear correlations between the enthalpy ( $$\Delta H_{{ \ne {\text{,i}}}}^{{\text{o}}}$$ ) and entropy ( $$\Delta S_{{ \ne {\text{,i}}}}^{{{\text{ o}}}}$$ ) of activation (kinetic compensation effect), the slope being the isokinetic temperature of the series (Tiso), so that at T = Tiso, all the reactions of the family share the same value of the rate constant. However, the random errors committed in the laboratory in the determination of $$\Delta H_{{ \ne {\text{,i}}}}^{{\text{o}}}$$ and $$\Delta S_{{ \ne {\text{,i}}}}^{{{\text{ o}}}}$$ are interrelated, and so tend to produce false isokinetic relationships. As a result, the existence of physically meaningful isokinetic relationships is a topic of lasting controversy. Here, it is shown that both the LFER (linear free energy relationships)-type and isokinetic linear correlations are direct consequences of two other correlations, those of $$\Delta H_{{ \ne {\text{,i}}}}^{{\text{o}}}$$ vs. $$\sigma _{{\text{i}}}$$ and $$\Delta S_{{ \ne {\text{,i}}}}^{{{\text{ o}}}}$$ vs. $$\sigma _{{\text{i}}}$$ , where the abscissa is the Hammett (or Taft) substituent parameter. A mathematical model has been developed, according to which Tiso can be interpreted as the temperature at which the reaction constant obtained as the slope of the LFER-type straight line takes a zero value (ρ = 0). Moreover, the numerical simulations performed indicated that the $$\log {\text{ }}k_{{{\text{ T}}}}$$ vs. $$\sigma _{{\text{i}}}$$ and $$\Delta H_{{ \ne {\text{,i}}}}^{{\text{o}}}$$ vs. $$\Delta S_{{ \ne {\text{,i}}}}^{{{\text{ o}}}}$$ linear plots can be visualized as two faces of the same coin, since, if the kinetic data obey the first with a correlation coefficient high enough, the probability of fulfillment of the second will be very high. Finally, it has been found that values of Tiso and Tδ (the slope of the linear correlation between the enthalpy–entropy deviations) very close to the mean working temperature, as well as correlation coefficients of the $$\Delta H_{{ \ne {\text{,i}}}}^{{\text{o}}}$$ vs. $$\Delta S_{{ \ne {\text{,i}}}}^{{{\text{ o}}}}$$ linear plots much higher than those corresponding to the $$\Delta H_{{ \ne {\text{,i}}}}^{{\text{o}}}$$ vs. $$\sigma _{{\text{i}}}$$ and $$\Delta S_{{ \ne {\text{,i}}}}^{{{\text{ o}}}}$$ vs. $$\sigma _{{\text{i}}}$$ plots, are all indicative of false isokinetic relationships, highly contaminated by the statistical correlation between the enthalpy and entropy experimental errors.