Supplemental Information: Rate Constants

*m* and *n* generally differ by less than an order of magnitude within the range of rate constant values expected for Reaction (1). Thus, fits to time-dependent data in the form of \(y(t) = y_0+\Lambda\mathrm{e}^{-\lambda{}t}+K\mathrm{e}^{-\kappa{}t}\) tend to converge to the form \(y(t) = y_0+\alpha{}\mathrm{e}^{-\beta{}t}\). We take advantage of this to approximate \(\textit{k}_{-1}\) by fitting data for major isomer *B* with a single exponential such that equation (6) may be rearranged as: \[B(t) = B_0[\frac{k_{1}k_{-2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}+[\frac{k_{-1}k_2+k_{-1}k_{-2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}]\mathrm{e}^{-x_1t}]\] Similarly, the integrated rate laws for isomers A and C may be rewritten as: \[A(t) = B_0[\frac{k_{-1}k_{-2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}+[\frac{k_{-1}k_{-2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}]\mathrm{e}^{-x_2t}]\\\] \[C(t) = B_0[\frac{k_{-1}k_{2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}+[\frac{k_{-1}k_{2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}]\mathrm{e}^{-x_3t}]\] Note that in equations (12) and (14), \(x_1\) and \(x_3\) are roughly equivalent to \(k_{-1}\) and \(k_{-2}\), respectively. \(x_2\) does not have an obvious analogue. At equilibrium, for a normalized data set, the following relations also hold: \[1 =[A]+[B]+[C]\\\] \[[B]_e =\frac{k_{1}}{k_{-1}}[A]_e={K_1}[A]_e\\\] \[[C]_e =\frac{k_2}{k_{-2}}[A]_e=K_2[A]_e\] Where \(K_1\) and \(K_2\) are the equilibrium constants for the first and second step of Reaction (1), respectively. Substituting Eqns (16) and (17) into (15) and rearranging yields the following relationship: \[[A]_e = \frac{k_{-1}k_{-2}B_0}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}\] Then, in solving for Eqn (3) at equilibrium, \[[B]_e = \frac{k_{-1}k_{1}k_{-2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}+\frac{k_{-1}k_2+k_{-1}k_{-2}}{k_{-1}k_2+k_{1}k_{-2}+k_{-1}k_{-2}}\mathrm{e}^{-k_{-1}t}\] Thus, by inspection of Eqns (12) and (19), \(x_1\) \(\approx\) \(k_{-1}\). Equation (4) may be similarly rearranged and integrated to solve assuming equilibrium conditions, where \(x_3\) \(\approx\) \(k_{-2}\). Unfortunately, equations (12)-(14) cannot be expressly used to determine exact values for \(k_1\), \(k_{-1}\), \(k_2\), and \(k_{-2}\) explicitly because those variables are coupled due to the simplifying assumption that a three-component exponential fit to the experimental data converges to a two-component exponential fit when values of \(m\) and \(n\) are similar. It is, however, relatively straightforward to determine \(K_1\) and \(K_2\) by fitting the experimental data with a function of the form \(y(t) = y_0+\alpha{}\mathrm{e}^{-\beta{}t}\) to obtain \(y_0\) and \(\beta\) for \(A\), \(B\), and \(C\) at each temperature. It then follows that \[K_1=\frac{y_0(B)}{y_0(A)}\] and \[K_2=\frac{y_0(C)}{y_0(A)}\] Finally, \(K_1\) and \(K_2\) may be used to estimate the remaining two unknowns, \(k_{1}\), \(k_2\), from the estimated values of \(k_{-1}\), and \(k_{-2}\).

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