Electron Transfer Rates in Coordination Complexes

The question asks for the reason behind the significant difference in the self-exchange electron transfer rates between the [Ru(NH$_3$)$_6$]$^{2+}$/[Ru(NH$_3$)$_6$]$^{3+}$ system and the [Co(NH$_3$)$_6$]$^{2+}$/[Co(NH$_3$)$_6$]$^{3+}$ system. The rate constant for the Ru system is $9.2 \times 10^2$ M$^{-1}$ sec$^{-1}$, while for the Co system, it is $\le 10^{-9}$ M$^{-1}$ sec$^{-1}$. The Co system is vastly slower.

Comparing Ru and Co Systems

We need to examine the electron configurations of the metal centers in the initial and final oxidation states for both systems when coordinated with ammonia ligands (which act as $\sigma$-donors and relatively strong field ligands, especially for 3d metal ions in higher oxidation states and 4d/5d metal ions).

• Ru System: [Ru(NH$_3$)$_6$]$^{2+}$/[Ru(NH$_3$)$_6$]$^{3+}$
• Ru(II) is 4d$^6$. With strong field NH$_3$ ligands, it forms a low spin complex. The electron configuration is $t_{2g}^6 e_g^0$.
• Ru(III) is 4d$^5$. With strong field NH$_3$ ligands, it forms a low spin complex. The electron configuration is $t_{2g}^5 e_g^0$.
• The electron transfer is from Ru(II) to Ru(III), effectively transferring an electron within the $t_{2g}$ orbitals: $t_{2g}^6 e_g^0 \rightleftharpoons t_{2g}^5 e_g^0$. Notice there is no change in the number of $e_g$ electrons.

• Co System: [Co(NH$_3$)$_6$]$^{2+}$/[Co(NH$_3$)$_6$]$^{3+}$
• Co(II) is 3d$^7$. With NH$_3$, it typically forms a high spin complex. The electron configuration is $t_{2g}^5 e_g^2$.
• Co(III) is 3d$^6$. With strong field NH$_3$ ligands, it forms a low spin complex. The electron configuration is $t_{2g}^6 e_g^0$.
• The electron transfer is from Co(II) to Co(III), effectively transferring electrons from $t_{2g}$ and $e_g$ to fill the $t_{2g}$ set and empty the $e_g$ set: $t_{2g}^5 e_g^2 \rightleftharpoons t_{2g}^6 e_g^0$. Notice there is a significant change in the number of $e_g$ electrons, from 2 in Co(II) to 0 in Co(III).

Inner-Sphere Reorganization Energy

Outer-sphere electron transfer reactions involve minimal structural change in the coordination sphere during the electron hop itself. However, for the electron transfer to be efficient (according to the Frank-Condon principle), the nuclei must be in a configuration that is momentarily stable for both the reactant and product electronic states. The energy required to distort the reactant and product complexes to this transition state geometry is called the inner-sphere reorganization energy ($\lambda_i$).

Effect of Electron Configuration Change on Rate

The $e_g$ orbitals in octahedral complexes are $\sigma^*$-antibonding orbitals. Electrons in these orbitals contribute to repelling the ligands and thus increasing the metal-ligand bond length. Electrons in $t_{2g}$ orbitals are non-bonding or weakly $\pi$-bonding/antibonding and have less effect on the metal-ligand $\sigma$ bond length.

• In the Ru system ($t_{2g}^6 e_g^0 \rightleftharpoons t_{2g}^5 e_g^0$), the change in electron count occurs within the $t_{2g}$ orbitals. There is no change in the number of $e_g$ electrons. While the overall oxidation state change (Ru(II) to Ru(III)) does cause some contraction of the Ru-N bonds (because the higher charge pulls ligands closer), the lack of change in the antibonding $e_g$ electron count means the change in bond lengths and associated $\lambda_i$ is relatively small. This leads to a fast electron transfer rate.
• In the Co system ($t_{2g}^5 e_g^2 \rightleftharpoons t_{2g}^6 e_g^0$), the number of $e_g$ electrons changes from 2 to 0. Removing two electrons from the $\sigma^*$-antibonding $e_g$ orbitals causes a significant shortening of the Co-N bond lengths. The change in geometry between the high-spin Co(II) complex (longer bonds due to two $e_g$ electrons) and the low-spin Co(III) complex (shorter bonds due to zero $e_g$ electrons) is substantial. This large difference in equilibrium bond lengths requires a large amount of energy to reorganize the inner coordination sphere ($\lambda_i$) to reach the transition state geometry. A large $\lambda_i$ results in a high activation barrier and consequently a very slow electron transfer rate.

Therefore, the vast difference in rates is primarily due to the large inner-sphere reorganization energy in the Co system, which is caused by the significant change in the number of $\sigma^*$-antibonding ($e_g$) electrons upon electron transfer.

Based on this analysis, the correct rationale is related to the change in the number of $\sigma^*$-electrons in the Co(II)/Co(III) system.