The C-O bond length is the shortest in:

Options

A. `[Cr(CO)_(6)]`
B. `[Mo(CO)_(6)]`
C. `[Mn(CO)_(6)]^(+)`
D. `[V(CO)_(6)]^(-)`

Answer

To determine which compound has the shortest C-O bond length, we need to analyze the bond order associated with each option. The bond order is directly related to bond length: a higher bond order corresponds to a shorter bond length.

### Step-by-Step Solution:

1. **Understand Bond Order and Bond Length Relationship**:
- Bond order is the number of chemical bonds between a pair of atoms. Higher bond order means stronger bonds and shorter bond lengths.
- Conversely, lower bond order results in weaker bonds and longer bond lengths.

2. **Analyze the Charge on the Central Metal**:
- The charge on the central metal ion affects the extent of back-bonding, which influences bond order.
- A higher positive charge on the metal ion generally leads to a higher bond order due to increased back-bonding, while a higher negative charge leads to lower bond order.

3. **Identify the Charges on the Central Metals**:
- Cobalt (Co): 0 charge (neutral)
- Chromium (Cr): 0 charge (neutral)
- Manganese (Mn): +1 charge
- Vanadium (V): -1 charge

4. **Compare the Bond Orders**:
- For neutral cobalt and chromium, the bond order will be moderate.
- Manganese, with a +1 charge, will have a higher bond order compared to neutral metals, resulting in a shorter bond length.
- Vanadium, with a -1 charge, will have a lower bond order due to decreased back-bonding, leading to a longer bond length.

5. **Conclusion**:
- Between manganese and vanadium, manganese has a higher bond order (due to +1 charge) than vanadium (due to -1 charge).
- Therefore, the C-O bond length is shortest in the compound with manganese.

### Final Answer:
The C-O bond length is shortest in the compound with manganese.