Which of the following statements is correct with respect to the metal carbonyls of `I^(st)` transition series?

Options

A. As `M-Cpi` bonding increase, the `C-O` bond length increases.
B. As positive charge on the central metal atom increases, the `C-O` bond length increases.
C. As electron density on the central metal atom increases, the `C-O` bond length increases.
D. (1) and (3) both

Explanation

To determine which statement is correct regarding the metal carbonyls of the first transition series, we need to analyze the bonding characteristics of metal carbonyls, particularly focusing on the interactions between the metal and the carbon monoxide (CO) ligands.

### Step-by-Step Solution:

1. **Understanding Metal Carbonyls**:
- Metal carbonyls consist of a metal center bonded to carbon monoxide (CO) ligands. CO acts as a ligand that donates electron density to the metal.

**Hint**: Recall that CO is a strong field ligand and participates in both sigma donation and pi back-donation.

2. **Bonding in Metal Carbonyls**:
- The bonding in metal carbonyls involves two types of interactions:
- **Sigma Bonding**: CO donates a pair of electrons from its lone pair to an empty orbital of the metal, forming a sigma bond.
- **Pi Back-Bonding**: The metal, having gained electron density from CO, can donate some of its electron density back to the empty pi* (pi-star) orbitals of CO.

**Hint**: Remember that the interaction is bidirectional: CO donates to the metal, and the metal donates back to CO.

3. **Effect of Back-Bonding**:
- The back-bonding leads to a decrease in the bond order of the C-O bond in CO. As the bond order decreases, the bond length increases. This is because the increased electron density on the metal reduces the electron density on the carbon atom of CO, weakening the C-O bond.

**Hint**: Consider how bond order and bond length are related: lower bond order typically means longer bond length.

4. **Analyzing the Statements**:
- **Statement A**: "Metal carbon pi bonding increases, the CO bond length increases."
- This statement is correct because as pi back-bonding increases, the bond order of C-O decreases, leading to an increase in bond length.

- **Statement B**: "As positive charge on a metal increases, the CO bond length increases."
- This statement is incorrect. An increase in positive charge on the metal typically leads to stronger back-bonding, which would decrease the bond length of CO.

- **Statement C**: "As the electron density on a central metal atom increases, the CO bond length increases."
- This statement is also correct. Increased electron density on the metal enhances back-bonding, which decreases the bond order and increases the bond length of C-O.

5. **Conclusion**:
- The correct statements are A and C. Therefore, the correct answer is that options A and C are true regarding the metal carbonyls of the first transition series.

### Final Answer:
The correct statements with respect to the metal carbonyls of the first transition series are:
- A: Metal carbon pi bonding increases, the CO bond length increases.
- C: As the electron density on a central metal atom increases, the CO bond length increases.