Question #44877

The kinetics of the outer-sphere electron transfer reaction between tris(2,2'-bipyridine)ruthenium(II) and hexaamminecobalt(III) is given by:

$\ce{[Ru(bpy)3]^{2+} + [Co(NH3)6]^{3+} -> [Ru(bpy)3]^{3+} + [Co(NH3)6]^{2+}}$

The reaction progress is monitored at $300\text{ K}$ by the disappearance of the intense Metal-to-Ligand Charge Transfer (MLCT) band of $\ce{[Ru(bpy)3]^{2+}}$ at $452\text{ nm}$. You are given the following thermodynamic and kinetic parameters: $\bullet$ Standard free energy of reaction, $\Delta G^\circ = -24.0\text{ kJ mol}^{-1}$ $\bullet$ Total reorganization energy, $\lambda = 144.0\text{ kJ mol}^{-1}$ $\bullet$ Theoretical collision frequency, $Z = 2.0 \times 10^{11}\text{ M}^{-1}\text{s}^{-1}$ $\bullet$ Observed bimolecular rate constant, $k_{obs} = 9.0 \times 10^{-2}\text{ M}^{-1}\text{s}^{-1}$ $\bullet$ Universal gas constant, $R = \frac{1}{120}\text{ kJ K}^{-1}\text{mol}^{-1}$

Assume $e^{-10} = 4.5 \times 10^{-5}$. According to Marcus theory, the simple Arrhenius collision-frequency model fails here due to the non-adiabatic nature of the electron transfer (involving a spin-state change in Cobalt). Calculate the transmission coefficient ($\kappa$) and the activation free energy ($\Delta G^\ddagger$ in $\text{kJ mol}^{-1}$) for this reaction.