Most of the metal carbonyls obey inert gas rule which states the the compounds in which the central metal atom appears to have attained the configuration of a noble gas either by the sharing or by the transfer of electrons tend to be more stable
Which of the following has highest `C-O` bond length ?
(a) `[mn(CO)_(6)]^(o+)`
(b) `[Co(CO)_(4)]^(Θ)`
(c) `[Fe(CO)_(4))]^(2-)`
(d) `[Ni(CO)_(4)]` .

Answer

To determine which of the given metal carbonyl complexes has the highest C-O bond length, we need to analyze the nature of the bonding in each complex, particularly focusing on the concept of synergic bonding and the oxidation states of the central metal ions.

### Step-by-Step Solution:

1. **Understanding Synergic Bonding**:
- In coordination compounds, ligands like CO donate electron pairs to the central metal ion. Simultaneously, the metal can donate electrons back to the anti-bonding orbitals (π* orbitals) of the CO ligand. This interaction is known as synergic bonding.
- The presence of synergic bonding affects the bond order of the C-O bond.

2. **Bond Order Concept**:
- Bond order is calculated using the formula:
\[ \text{Bond Order} = \frac{(\text{Number of Bonding Electrons} - \text{Number of Anti-bonding Electrons})}{2} \]
- A decrease in bond order leads to an increase in bond length, as bond length is inversely proportional to bond order.

3. **Analyzing Each Complex**:
- **(a) \([Mn(CO)_6]^{o+}\)**:
- Mn is in the +1 oxidation state. The central metal ion has a moderate tendency to donate electrons.
- **(b) \([Co(CO)_4]^{Θ}\)**:
- Co is in the 0 oxidation state. The central metal ion has a balanced tendency to donate and accept electrons.
- **(c) \([Fe(CO)_4]^{2-}\)**:
- Fe is in the -2 oxidation state. The central metal ion has a high tendency to donate electrons, leading to significant synergic bonding.
- **(d) \([Ni(CO)_4]\)**:
- Ni is in the 0 oxidation state. Similar to Co, it has a balanced tendency to donate and accept electrons.

4. **Comparing the Tendency to Donate Electrons**:
- The more negative the oxidation state of the metal, the greater the tendency to donate electrons. Thus, the order of electron donation tendency is:
- \([Fe(CO)_4]^{2-} > [Mn(CO)_6]^{o+} > [Co(CO)_4]^{Θ} = [Ni(CO)_4]\)

5. **Conclusion**:
- Since \([Fe(CO)_4]^{2-}\) has the highest tendency to donate electrons, it will exhibit the most significant synergic bonding. This results in the lowest bond order for the C-O bond, leading to the highest C-O bond length.
- Therefore, the complex with the highest C-O bond length is **(c) \([Fe(CO)_4]^{2-}\)**.

### Final Answer:
(c) \([Fe(CO)_4]^{2-}\)