The correct bond order for carbon-oxygen bond in following metal carbonyls $[Cr(CO)_6], [V(CO)_6]^-, [Mn(CO)_6]^+$ is:

$[V(CO)_6]^-<[Cr(CO)_6] < [Mn (CO)_6]^+$

The correct answer is Option 1. $[V(CO)_6]^-<[Cr(CO)_6] < [Mn (CO)_6]^+$.

The bond order of carbon-oxygen bonds in metal carbonyls is primarily influenced by two factors: the electron configuration of the metal and the extent of \(\pi \)-backbonding between the metal and the carbon monoxide ligands \((CO)\).

Key Concepts:

Bond Order in \(CO\):

In a free \(CO\) molecule, the bond order is 3, as \(CO\) has a triple bond (one sigma bond and two \(pi \)-bonds). In metal carbonyl complexes, the carbon monoxide ligands interact with the metal center in two key ways:

\(\sigma \) donation: The \(CO\) molecule donates electron density from the lone pair on carbon to the metal (σ-donation).

\(\pi \) back-donation: The metal can donate electron density from its filled d-orbitals into the empty \(\pi ^*\) (anti-bonding) orbitals of \(CO\) (\(pi \)-backbonding).

Effect of \(\pi \)-Backbonding:

Increased back-donation: When the metal donates more electron density into the \(\pi ^*\) orbitals of \(CO\), the bond between carbon and oxygen weakens (because the anti-bonding orbitals are filled), reducing the bond order of the \(C-O\) bond.

Decreased back-donation: When the metal is less able to donate electron density, the \(C-O\) bond remains stronger, and the bond order is higher.

The Metal Carbonyl Complexes:

\([Cr(CO)_6]\) (Hexacarbonylchromium(0)):

Chromium in this complex is in the 0 oxidation state.

Electron configuration: \( 3d^5 4s^1 \).

This neutral metal center can engage in moderate back-donation to the \(CO\) ligands.

Since this is a neutral complex, the back-donation is balanced, resulting in an intermediate \(C-O\) bond order

\([V(CO)_6]^{-}\) (Hexacarbonylvanadate(-I)):

Vanadium is in the -1 oxidation state, meaning it has gained an additional electron compared to the neutral complex.

Electron configuration: \( 3d^4 4s^2 \) (extra electron in the d-orbitals).

The extra electron enhances the metal's ability to back-donate to the \(CO\) ligands.

Increased back-donation means more electron density is transferred to the \(CO\) \(\pi ^*\) orbitals, weakening the \(C-O\) bond and lowering its bond order.

\([Mn(CO)_6]^{+}\) (Hexacarbonylmanganese(+I)):

Manganese is in the +1 oxidation state, meaning it has lost an electron compared to the neutral complex.

Electron configuration: \( 3d^5 4s^0 \). Since Mn has fewer electrons available for back-donation, it cannot effectively transfer electron density to the \(CO\) \(\pi ^*\) orbitals.

This leads to a stronger \(C-O\) bond and a higher bond order because the \(\pi ^*\) orbitals are less filled.

Comparison and Trend:

\([V(CO)_6]^{-}\): Extra electron increases back-donation → \(C-O\) bond order is the lowest.

\([Cr(CO)_6]\): Neutral complex with moderate back-donation → \(C-O\) bond order is intermediate.

\([Mn(CO)_6]^{+}\): Fewer electrons for back-donation → \(C-O\) bond order is the highest.

Conclusion:

The bond order trend is driven by the extent of \(\pi \)-backbonding. The more electron-rich the metal center (i.e., the greater the electron density available for back-donation), the more weakened the \(C-O\) bond becomes, leading to a lower bond order.

Thus, the correct bond order trend for the carbon-oxygen bonds in the given metal carbonyls is:

\([V(CO)_6]^- < [Cr(CO)_6] < [Mn(CO)_6]^+\)