Determining Lowest CO Stretching Frequency

Core Principle: Metal Oxidation State and Back-bonding

The stretching frequency of the Carbonyl ($ CO $) ligand ($ \nu_{CO} $) is sensitive to the electronic environment of the metal center in coordination complexes. Specifically, it depends on the strength of the metal-to-ligand π-back bonding.

• Stronger π-back bonding leads to increased electron density in the CO π* antibonding orbitals.
• This weakens the C-O bond, causing a lower $ \nu_{CO} $.
• π-back bonding is favored by a lower oxidation state (more electron-rich metal) and electron-donating ligands.

Analysis of Metal Oxidation States

Let's determine the oxidation state for the metal in each complex:

• $ [Mn(CO)_6]^+ $: $ Mn = +1 $
• $ [V(CO)_6]^- $: $ V = -1 $
• $ [Cr(CO)_5] $: $ Cr = 0 $
• $ [Cr(dien)(CO)_3] $: $ Cr = 0 $ (dien is a neutral ligand)

Identifying the Lowest Frequency

The complex with the lowest metal oxidation state will exhibit the strongest π-back bonding and thus the lowest $ \nu_{CO} $.

• $ V $ in $ [V(CO)_6]^- $ has the lowest oxidation state ($ -1 $).
• $ Cr $ in $ [Cr(CO)_5] $ and $ [Cr(dien)(CO)_3] $ have an oxidation state of $ 0 $.
• $ Mn $ in $ [Mn(CO)_6]^+ $ has the highest oxidation state ($ +1 $).

Therefore, $ [V(CO)_6]^- $ has the most electron-rich metal center, leading to the strongest back-bonding and the lowest CO stretching frequency.