Which of the following has longest `C-O` bond length? (Free `C-O` bond length in CO is `1.128Å`).

Options

A. (a) `[Co(CO)_(4)]^(-)`
B. (b) `[Fe(CO)_(4)]^(2-)`
C. (c) `[Mn(CO)_(6)]^(+)`
D. (d) `Ni(CO)_(4)`

Explanation

To determine which of the given compounds has the longest C-O bond length, we need to analyze the back-bonding effect in each compound. The bond length of the C-O bond is influenced by the oxidation state of the central metal atom and the degree of back-bonding that occurs between the metal and the carbonyl group (C=O).

### Step-by-Step Solution:

1. **Identify the Compounds and Their Oxidation States:**
- **Option A: CO (Cobalt in -1 oxidation state)**
- **Option B: Fe(CO)4^2- (Iron in -2 oxidation state)**
- **Option C: Mn(CO)6^+ (Manganese in +1 oxidation state)**
- **Option D: Ni(CO)4 (Nickel in 0 oxidation state)**

2. **Analyze the Back-Bonding:**
- Back-bonding occurs when the metal donates electron density to the π* orbitals of the carbonyl group. The more negative the oxidation state of the metal, the more electron density it has, which leads to increased back-bonding.

3. **Determine the Effect of Back-Bonding on C-O Bond Length:**
- Increased back-bonding results in a higher bond order of the C-O bond, which in turn decreases the C-O bond length. Conversely, decreased back-bonding leads to a lower bond order and a longer C-O bond length.

4. **Evaluate Each Compound:**
- **CO:** The C-O bond length is 1.128 Å (reference value).
- **Fe(CO)4^2-:** Iron has the highest negative oxidation state (-2), indicating the highest electron density and thus the most significant back-bonding. This will result in a shorter C-O bond length compared to CO.
- **Mn(CO)6^+:** Manganese has a +1 oxidation state, which means less electron density and back-bonding compared to Fe(CO)4^2-.
- **Ni(CO)4:** Nickel has a 0 oxidation state, indicating minimal back-bonding, leading to a longer C-O bond length than in the case of Fe(CO)4^2- and Mn(CO)6^+.

5. **Conclusion:**
- Since Fe(CO)4^2- has the highest negative charge and thus the most back-bonding, it will have the shortest C-O bond length. Conversely, the compound with the least back-bonding will have the longest C-O bond length.
- Among the options, Ni(CO)4 will have the longest C-O bond length due to its zero oxidation state and minimal back-bonding.

### Final Answer:
The compound with the longest C-O bond length is **Ni(CO)4**.