Kinetic Analysis of TEMPOL-H Reduction

The least-square fits of theoretical models, described below, were applied to the experimental concentrations of TEMPOL-H in time, [TEMPOL-H] (t), obtained from the integral intensities of TEMPOL-H methyl resonance in the real-time pseudo-2D ¹H NMR spectra, and were performed in MATLAB by the function lsqcurvefit.
For large excesses of AA over TEMPOL, k₁ was estimated by fitting the solution of Equation (2): $$\left[\mathrm{TEMPOL}\text{-}\mathrm{H}\right]\left(t\right)={\left[\mathrm{TEMPOL}\text{-}\mathrm{H}\right]}_{\infty }$$
Besides k₁, the optimized parameters were the final TEMPOL-H concentration [TEMPOL-H]_∞ and the origin of the reaction Δt before the start of the measurements.
More complex fits needed for the series of backward and forward reactions, including (1) and (3), and more as described in Supporting Information, involved sets of differential equations describing first and second order kinetics, analogous to Equation (2). A vector of the time derivatives of the concentrations of individual compounds was calculated and the set of ordinary differential equations was solved by the MATLAB function ode45 (Supporting Information). In this way, [TEMPOL-H] (t) was calculated from initial concentrations and rate constants, compared to the experimental values, and fitted by varying selected parameters as described in Supporting Information.
All of the errors in rate constants are estimated as 5%, coming mainly from uncertainties in sample preparation.