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 Δ₀ (Delta naught) for the given compounds, we need to analyze the crystal field splitting energy (CFSE) based on the ligands and the oxidation state of the central metal ion.

### Step-by-Step Solution:

1. **Identify the oxidation states of manganese in each complex:**
- For `[Mn(H₂O)₆]²⁺`:
- The charge on the complex is +2. Since H₂O is a neutral ligand, the oxidation state of Mn is +2.
- For `[Mn(CN)₂(H₂O)₄]`:
- The charge on the complex is +2. CN⁻ is a -1 ligand, and H₂O is neutral.
- Let the oxidation state of Mn be x:
\[ x + 2(-1) + 4(0) = +2 \implies x - 2 = 2 \implies x = +2 \]
- For `[Mn(CN)₄(H₂O)₂]²⁻`:
- The charge on the complex is -2.
- Let the oxidation state of Mn be x:
\[ x + 4(-1) + 2(0) = -2 \implies x - 4 = -2 \implies x = +2 \]

All three complexes have Mn in the +2 oxidation state.

2. **Determine the nature of the ligands:**
- H₂O is a weak field ligand.
- CN⁻ is a strong field ligand.

3. **Assess the ligand field strength and its effect on Δ₀:**
- The first complex `[Mn(H₂O)₆]²⁺` has 6 weak field ligands (H₂O), leading to the least splitting and thus the smallest Δ₀.
- The second complex `[Mn(CN)₂(H₂O)₄]` has 2 strong field ligands (CN⁻) and 4 weak field ligands (H₂O). The presence of CN⁻ will increase Δ₀ compared to the first complex.
- The third complex `[Mn(CN)₄(H₂O)₂]²⁻` has 4 strong field ligands (CN⁻) and 2 weak field ligands (H₂O). The higher number of CN⁻ ligands will result in the largest Δ₀ among the three complexes.

4. **Order the complexes based on Δ₀:**
- From the analysis:
- **Δ₀ for `[Mn(H₂O)₆]²⁺`** < **Δ₀ for `[Mn(CN)₂(H₂O)₄]`** < **Δ₀ for `[Mn(CN)₄(H₂O)₂]²⁻`**
- Thus, the order of Δ₀ is:
\[ [Mn(H₂O)₆]²⁺ < [Mn(CN)₂(H₂O)₄] < [Mn(CN)₄(H₂O)₂]²⁻ \]

### Final Answer:
The order of Δ₀ for the given compounds is:
\[ [Mn(H₂O)₆]²⁺ < [Mn(CN)₂(H₂O)₄] < [Mn(CN)₄(H₂O)₂]²⁻ \]