Question #44823

Consider two outer-sphere single-electron transfer reactions occurring at $300\text{ K}$: Reaction I: $\ce{M_1^{3+} + Red^{2+} -> M_1^{2+} + Red^{3+}}$ Reaction II: $\ce{M_2^{3+} + Red^{2+} -> M_2^{2+} + Red^{3+}}$ The standard reduction potentials are given as: $E^\circ(\ce{M_1^{3+}/M_1^{2+}}) = 1.50\text{ V}$ $E^\circ(\ce{M_2^{3+}/M_2^{2+}}) = 1.14\text{ V}$ $E^\circ(\ce{Red^{3+}/Red^{2+}}) = 0.30\text{ V}$ For these homologous redox reactions, the activation energy ($E_a$) follows the linear free energy relationship: $E_a = E_{a,0} - 0.5 F \Delta E^\circ_{\text{rxn}}$, where $E_{a,0}$ is a constant and $\Delta E^\circ_{\text{rxn}}$ is the standard cell potential of the reaction. The collision frequency factor for Reaction II is 100 times that of Reaction I ($A_{II} = 100 A_I$). Assuming $\frac{2.303 RT}{F} = 0.06\text{ V}$ at $300\text{ K}$, what is the ratio of the rate constants $\frac{k_I}{k_{II}}$?