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

Abstract

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 (T_iso), so that at T = T_iso, 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 T_iso 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 T_iso and T_δ (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.

Graphic abstract