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 `C-O` bond length in the following molecules .
(I) `[Mo(CO)_(3)(PF_(3))_(3)]` (II) `[Mo(CO)_(3)(PCI_(3))_(3)]`
(III) `[Mo(CO)_(3)(P(Me)_(3)_(3)]`
(a) `IgtIIgtIII`
(b) `IIIgtIgtII`
(c) `IIgtIIIgtI`
(d) `IltIIltIII`

Explanation

To solve the given question, we need to analyze the impact of the ligands on the C-O bond length in the mixed carbonyl complexes. The key points to consider are the pi-accepting abilities of the ligands and how they affect the back donation from the metal to the CO ligands.

### Step-by-Step Solution:

1. **Understanding the Ligands**:
- We have three complexes:
- (I) \[ Mo(CO)_{3}(PF_{3})_{3} \]
- (II) \[ Mo(CO)_{3}(PCl_{3})_{3} \]
- (III) \[ Mo(CO)_{3}(P(Me)_{3})_{3} \]
- The ligands PF3, PCl3, and P(Me)3 have different abilities to accept pi-electrons.

2. **Pi-Acceptor Strength**:
- PF3 is a better pi-acceptor than PCl3 due to the higher electronegativity of fluorine, which stabilizes the empty p-orbitals.
- PCl3 is a better pi-acceptor than P(Me)3 because chlorine is more electronegative than the methyl group, which has a weak electron-donating effect.

3. **Order of Pi-Accepting Ability**:
- The order of pi-accepting ability from highest to lowest is:
- PF3 > PCl3 > P(Me)3

4. **Effect on C-O Bond Order**:
- The stronger the pi-acceptor, the more back donation occurs from the metal to the CO ligands.
- Higher back donation leads to a higher bond order for the C-O bond, which results in a shorter bond length.

5. **Bond Order and Bond Length Relationship**:
- Bond order is inversely related to bond length. Therefore, if we have the order of pi-accepting ability, we can deduce the order of bond lengths:
- Higher pi-accepting ability (higher bond order) results in shorter C-O bond lengths.

6. **Determining the C-O Bond Length Order**:
- From the pi-accepting ability:
- PF3 (highest pi-acceptor) will have the shortest C-O bond length.
- PCl3 will have a longer C-O bond length than PF3 but shorter than P(Me)3.
- P(Me)3 (lowest pi-acceptor) will have the longest C-O bond length.
- Therefore, the order of C-O bond lengths is:
- C-O bond length: P(Me)3 > PCl3 > PF3

7. **Final Order**:
- Thus, the correct order of C-O bond lengths in the given complexes is:
- (III) > (II) > (I)

### Conclusion:
The correct answer is option (d) \( I < II < III \).