Question

The pair of complex ions that exhibits the slowest outer-sphere electron-exchange reaction at 25 \(^{\circ}\mathrm{C}\) is:

• A. \(\mathrm{[Co(NH_3)_6]^{3+}}\) and \(\mathrm{[Co(NH_3)_6]^{2+}}\)
• B. \(\mathrm{[Fe(H_2O)_6]^{3+}}\) and \(\mathrm{[Fe(H_2O)_6]^{2+}}\)
• C. \(\mathrm{[Ru(NH_3)_6]^{3+}}\) and \(\mathrm{[Ru(NH_3)_6]^{2+}}\)
• D. \(\mathrm{[Mn(CN)_6]^{3-}}\) and \(\mathrm{[Mn(CN)_6]^{4-}}\)

Hint:

Outer-sphere self-exchange rate drops when the metal-ligand bond length (and spin state) changes a lot between the two oxidation states; check which pair has the biggest jump in \(e_g\) electron occupancy.

Answer 1

The Correct Option is A

Step 1: Recall what controls outer-sphere self-exchange rate.
Marcus theory says the rate of an outer-sphere electron-transfer self-exchange (same metal, two oxidation states) depends on the reorganization energy \(\lambda\): the energy needed to distort the metal-ligand bond lengths and angles of both partners to a common geometry before the electron jumps, since electron transfer is much faster than nuclear motion (Franck-Condon principle). A larger \(\lambda\) means a higher barrier and a slower rate.

Step 2: Identify what makes \(\lambda\) large.
\(\lambda\) is large when the metal-ligand bond length changes a lot between the two oxidation states, and it gets even larger if the spin state also changes, since electrons must then reorganize between orbitals of very different bonding character.

Step 3: Analyze the cobalt ammine pair.
\(\mathrm{[Co(NH_3)_6]^{3+}}\) is \(\mathrm{Co(III)}\), \(d^6\), low spin, \(t_{2g}^6e_g^0\). \(\mathrm{[Co(NH_3)_6]^{2+}}\) is \(\mathrm{Co(II)}\), \(d^7\), high spin, \(t_{2g}^5e_g^2\), with two electrons now in the strongly \(\sigma^*\)-antibonding \(e_g\) set. Populating \(e_g\) stretches the \(\mathrm{Co-N}\) bonds by a large amount, and the spin state itself changes, so \(\lambda\) becomes very large.

Step 4: Compare with the other pairs.
\(\mathrm{[Fe(H_2O)_6]^{3+/2+}}\) are both high spin with a smaller bond-length change and no spin-state change. \(\mathrm{[Ru(NH_3)_6]^{3+/2+}}\) are both low spin (the heavier 4d metal has a stronger field), with only a tiny bond-length change, giving one of the fastest known outer-sphere rates. \(\mathrm{[Mn(CN)_6]^{3-/4-}}\) are both low spin with strong-field \(\mathrm{CN^-}\), also a small bond-length change.

Step 5: Conclude.
The cobalt ammine pair has by far the largest reorganization energy (large bond-length change plus a spin-state change), so it has the slowest self-exchange rate. This is a classic example: \(\mathrm{[Co(NH_3)_6]^{3+/2+}}\) self-exchange is famously slow, around \(10^{-6}\ \mathrm{M^{-1}s^{-1}}\), against \(\mathrm{[Ru(NH_3)_6]^{3+/2+}}\) at around \(10^{2}\text{-}10^{3}\ \mathrm{M^{-1}s^{-1}}\).

Final Answer:
The slowest pair is \(\mathrm{[Co(NH_3)_6]^{3+}}\) and \(\mathrm{[Co(NH_3)_6]^{2+}}\), option (A). \[ \boxed{\text{(A)}} \]