The pi acid ligands donate their lone pairs to the metal to form a normal o bond with the latter in addition to it the vacant orbitals accepct electrons from the filled mental orbitals to form a type of pi bonand which suppliments the o bond
Which of the following has lowest `M-C` bond lenght?
`[Ni(CO)_(4)]`
(b) `[Mn(CO)_(6)]^(o+)`
(c) `[Fe(CO)_(4)]^(2-)`
(d) `[Co(CO)_(4)]^(Θ)` .

Answer

To determine which coordination complex has the lowest M-C bond length among the given options, we need to analyze the nature of the metal-ligand bonding, particularly focusing on the synergic bonding involving the carbonyl (CO) ligands.

### Step-by-Step Solution:

1. **Understanding Synergic Bonding**:
- In coordination compounds with CO as a ligand, synergic bonding occurs. This involves the donation of a lone pair of electrons from the CO ligand to the metal, forming a sigma (σ) bond. Simultaneously, the metal can donate electrons to the empty π* (pi star) orbitals of the CO, forming π back-bonding.

2. **Bond Order Concept**:
- The bond order is defined as:
\[ \text{Bond Order} = \frac{\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons}}{2} \]
- An increase in bond order leads to a decrease in bond length, as stronger bonds are shorter.

3. **Analyzing Each Complex**:
- **(a) \([Ni(CO)_{4}]\)**: Nickel is in the zero oxidation state. The metal-ligand bond will have a moderate bond order.
- **(b) \([Mn(CO)_{6}]^{+}\)**: Manganese is in the +1 oxidation state, which may reduce the electron density and thus the bond order compared to nickel.
- **(c) \([Fe(CO)_{4}]^{2-}\)**: Iron is in the -2 oxidation state. The negative charge on the metal increases its electron density, enhancing the π back-bonding and thus increasing the bond order.
- **(d) \([Co(CO)_{4}]^{\Theta}\)**: Cobalt is in a neutral state, which may lead to a bond order similar to nickel.

4. **Comparing the Bond Orders**:
- The bond order is likely highest in \([Fe(CO)_{4}]^{2-}\) due to the increased electron density and enhanced π back-bonding. This means that the M-C bond length will be the shortest in this complex.

5. **Conclusion**:
- Among the given options, \([Fe(CO)_{4}]^{2-}\) has the highest bond order and thus the lowest M-C bond length.

### Final Answer:
The complex with the lowest M-C bond length is **(c) \([Fe(CO)_{4}]^{2-}\)**.