According to crystal field theory the electronic configuration of the compound `[Mn(CN)_(4)]^(2-)` is `(DeltagtP)` :

Options

A. `4(e^(4)t_(2)^(1))`
B. `(e^(2) t_(2)^(3))`
C. `4(e^(2)t_(2)^(3))`
D. `3(e^(4).t_(2)^(1))` .

Explanation

To determine the electronic configuration of the manganese ion in the complex \([Mn(CN)_{4}]^{2-}\) according to crystal field theory, we will follow these steps:

### Step 1: Determine the oxidation state of manganese (Mn)
1. The overall charge of the complex is \(-2\).
2. Each cyanide ligand (CN) has a charge of \(-1\). Since there are 4 cyanide ligands, their total contribution is \(-4\).
3. Let the oxidation state of Mn be \(x\). Therefore, we have:
\[ x + 4(-1) = -2 \]
Simplifying this gives:
\[ x - 4 = -2 \implies x = +2 \]
Thus, the oxidation state of Mn in \([Mn(CN)_{4}]^{2-}\) is \(+2\).

### Step 2: Write the electronic configuration of Mn in its elemental state
1. The atomic number of manganese (Mn) is 25.
2. The electronic configuration of neutral manganese is:
\[ [Ar] 3d^5 4s^2 \]

### Step 3: Determine the electronic configuration of Mn in the +2 oxidation state
1. When manganese loses two electrons to form \(Mn^{2+}\), it loses the two 4s electrons first:
\[ Mn^{2+}: [Ar] 3d^5 \]

### Step 4: Identify the type of ligands and their effect on electron pairing
1. Cyanide (CN\(^-\)) is a strong field ligand.
2. Strong field ligands cause a large crystal field splitting energy (\(\Delta\)) and favor pairing of electrons in the lower energy orbitals (t2g).

### Step 5: Apply crystal field theory to fill the d-orbitals
1. According to crystal field theory, the \(d\) orbitals split into two sets: \(t_{2g}\) (lower energy) and \(e_g\) (higher energy).
2. For \(Mn^{2+}\) with a \(3d^5\) configuration and a strong field ligand, we fill the orbitals to maximize pairing:
- Fill the \(t_{2g}\) orbitals first before moving to \(e_g\).
- Since we have 5 electrons, they will fill as follows:
- \(t_{2g}\): 5 electrons (fully filled)
- \(e_g\): 0 electrons

### Final Electronic Configuration
Thus, the electronic configuration of \(Mn^{2+}\) in the complex \([Mn(CN)_{4}]^{2-}\) is:
\[ t_{2g}^5 \, e_g^0 \]

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