Linear Free-Energy Relationships for the Alkyl Radical Affinities of Nitroxides: A Theoretical Study

High-level <i>ab initio</i> calculations have been used to construct linear free-energy relationships describing the kinetics and thermodynamics of the combination and dissociation reactions between alkyl radicals and nitroxides in terms of easily accessible parameters that quantify the electronic, steric and radical stabilization characteristics of the coreactants. For the gas-phase equilibrium constant (<i>K</i><sub>eq</sub> = <i>k</i><sub>c</sub>/<i>k</i><sub>d</sub>) of the combination reaction at 298 K, the following equation was obtained: log (<i>K</i><sub><i>eq</i></sub>) = −0.10IP − 0.177RSE − 0.130RSE<sub><i>nxd</i></sub> + 38.3. In this equation, IP is the vertical ionization potential of the alkyl radical, RSE is the standard radical stabilization energy the alkyl radical, while RSE<sub><i>nxd</i></sub> is a new descriptor for the nitroxide radical, related to the standard radical stabilization energy, but measuring in this case the flexibility of the nitroxide to the geometric changes associated with formation of an alkoxyamine. The equation was successful for combinations of substituents not included in the original fitting and can thus be used to predict the behavior for larger systems for which direct calculation is impractical. Similar equations were also fitted to available experimental data for <i>k</i><sub><i>c</i></sub>, at 298 K and <i>k</i><sub><i>d</i></sub>, at 393 K, both in <i>tert</i>-butyl benzene, to allow the prediction of rate constants. The equation-determined rate constants, <i>k</i><sub><i>c,eq</i></sub> and <i>k</i><sub><i>d,eq</i></sub> are given by log (<i>k</i><sub><i>c,eq</i></sub>) = −0.408IP − 0.0597RSE − 0.103RSE<sub><i>nxd</i></sub> + 14.5 and log (<i>k</i><sub><i>d,eq</i></sub>) = 0.794IP + 5.68θ + 0.0873RSE + 0.0821RSE<sub><i>nxd</i></sub> − 27.7. For the decomposition rate, an additional parameter, Tolman’s cone angle θ, which measures the steric bulk of the attacking alkyl radical, was found to improve the fit to the data. The equations could in principle be fitted to experimental or calculated rate and equilibrium constants under a variety of reaction conditions. On the basis of our analysis, it appears that the stability of the alkyl radical has the largest effect on the kinetics and thermodynamics of the combination and dissociation reactions, with smaller but significant contributions from the remaining parameters.