If in the mixed carbonyl the other ligand is also `pi` acceptor it would compete with the ligand `CO` for gaining the metal `d_(pi)` electron charge The higher is the extent of back donation in `CO` , the lesser will be the stretching vibration frequency for `C-O` bond If `PP_(3)` is better pi-acceptor than `CO` then answer the following
Select the correct order of `M-C` bond order in the following molecule and ions .
(I) `[Ni(CO)_(4)]`
(II) `[Co(CO)_(4)]^(Θ)`
(III) `[Fe(CO)_(4)]^(2-)`
(a) `IgtIIgtIII`
(b) `I=II=III`
(c) `IIgtIIIgtI`
(d) `IltIIltIII` .

Explanation

To solve the problem, we need to analyze the bond order of the M-C bond in the given coordination compounds based on the extent of back donation from the metal to the ligands, particularly focusing on the carbon monoxide (CO) ligand and the other ligand, which is a π-acceptor.

### Step-by-Step Solution:

1. **Understanding Back Donation**:
- In coordination compounds, back donation occurs when the metal donates electron density from its d-orbitals to the empty π* orbitals of the ligands. This is particularly significant for ligands like CO, which can accept back-donated electrons.
- The extent of back donation is influenced by the oxidation state of the metal and the nature of the ligands.

2. **Analyzing the Given Complexes**:
- **(I) [Ni(CO)₄]**: Nickel in this complex is in the zero oxidation state (Ni^0). CO is a strong π-acceptor, but since Ni is not in a high oxidation state, the back donation is moderate.
- **(II) [Co(CO)₄]⁰**: Cobalt is also in the zero oxidation state here. Similar to Ni, the back donation is moderate, but Co has a slightly higher effective nuclear charge than Ni, which can enhance back donation slightly.
- **(III) [Fe(CO)₄]²⁻**: Iron in this complex has a -2 oxidation state (Fe²⁻). The higher negative charge on the metal increases the ability to donate electron density to CO, leading to stronger back donation and thus a higher bond order.

3. **Comparing Bond Orders**:
- The bond order of M-C bonds can be inferred from the extent of back donation:
- For [Fe(CO)₄]²⁻, the bond order will be the highest due to the -2 charge enhancing back donation.
- For [Co(CO)₄]⁰, the bond order will be moderate.
- For [Ni(CO)₄], the bond order will be the lowest due to the zero oxidation state of Ni.

4. **Conclusion on Bond Order**:
- Based on the analysis, the order of M-C bond order from highest to lowest is:
- [Fe(CO)₄]²⁻ > [Co(CO)₄]⁰ > [Ni(CO)₄]
- Thus, the correct order is **III > II > I**, which corresponds to option (d) **I < II < III**.

### Final Answer:
The correct order of M-C bond order in the given complexes is **(d) I < II < III**.