Predict the order of `Delta_(@)` for the following compounds
I `[Mn(H_(2)O)_(6)]^(2+)`
II `[Mn(CN)_(2)(H_(2)O)_(4)]`
III `[Mn(CN)_(4)(H_(2)O)_(2)]^(2-)`

Options

A. `Delta_(@) (I) lt Delta_(@)(II) lt Delta_(@) (III)`
B. `Delta_(@)(II) lt Delta_(@)(I) lt Delta_@)(III)`
C. `Delta_(@)(III) lt Delta_(@) (II) lt Delta_(@)(I)`
D. `Delta_(@)(I) lt Delta_(@)(III) lt Delta_(@)(II)`

Answer

To predict the order of \( \Delta_0 \) (the crystal field splitting energy) for the given compounds, we need to consider the nature of the ligands and the oxidation state of the central metal ion. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the oxidation state of manganese in each complex.
- **For I**: \( [Mn(H_2O)_6]^{2+} \)
- Manganese is in the +2 oxidation state.

- **For II**: \( [Mn(CN)_2(H_2O)_4] \)
- The oxidation state can be calculated as follows:
- Let the oxidation state of Mn be \( x \).
- The charge of CN is -1 (2 CN gives -2) and H2O is neutral.
- Thus, \( x - 2 + 0 = 0 \) → \( x = +2 \).

- **For III**: \( [Mn(CN)_4(H_2O)_2]^{2-} \)
- Similarly, let the oxidation state of Mn be \( x \).
- The charge of CN is -1 (4 CN gives -4) and H2O is neutral.
- Thus, \( x - 4 + 0 = -2 \) → \( x = +2 \).

### Step 2: Analyze the ligands present in each complex.
- **I**: Contains only \( H_2O \) (a weak field ligand).
- **II**: Contains \( CN^- \) (a strong field ligand) and \( H_2O \).
- **III**: Contains more \( CN^- \) ligands (4 CN and 2 H2O), which are strong field ligands.

### Step 3: Determine the strength of the ligands.
- According to the spectrochemical series:
- \( CN^- \) is a stronger field ligand than \( H_2O \).
- The more strong field ligands present, the larger the crystal field splitting energy \( \Delta_0 \).

### Step 4: Compare the number of strong field ligands in each complex.
- **I**: 0 strong field ligands → \( \Delta_0 \) is the lowest.
- **II**: 2 strong field ligands (CN) and 4 weak field ligands (H2O) → moderate \( \Delta_0 \).
- **III**: 4 strong field ligands (CN) and 2 weak field ligands (H2O) → highest \( \Delta_0 \).

### Step 5: Order the complexes based on \( \Delta_0 \).
- From highest to lowest \( \Delta_0 \):
- III > II > I

### Final Order:
The order of \( \Delta_0 \) for the compounds is:
\[ [Mn(CN)_4(H_2O)_2]^{2-} > [Mn(CN)_2(H_2O)_4] > [Mn(H_2O)_6]^{2+} \]

### Answer:
The correct order is: **III > II > I**.