Which of the electronic configuration according to crystal field theory of the compound is correct `[MnF_(6)]^(4-)` ? .

Options

A. `a. (t_(2g)^(5) e_(g)^(0))`
B. `b. (t_(2g)^3 e_(g)^(2))`
C. `c. (e_(g)^(1) t_(2g)^(4))`
D. `d. (t_(2g)^(1) e_(g)^(4))`

Explanation

To determine the electronic configuration of the compound \([MnF_6]^{4-}\) according to crystal field theory, we can follow these steps:

### Step 1: Determine the oxidation state of manganese (Mn)
- The compound is \([MnF_6]^{4-}\).
- Fluorine (F) has an oxidation state of -1. Since there are 6 fluorine atoms, their total contribution is \(6 \times (-1) = -6\).
- Let the oxidation state of Mn be \(x\). The overall charge of the complex is -4.
- Therefore, we can set up the equation:
\[ x + (-6) = -4 \]
- Solving for \(x\):
\[ x - 6 = -4 \implies x = +2 \]
- Thus, the oxidation state of Mn in \([MnF_6]^{4-}\) is +2.

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

### Step 3: Determine the electronic configuration of Mn in the +2 oxidation state
- In the +2 oxidation state, Mn loses two electrons. The electrons are removed first from the 4s orbital, followed by the 3d orbital.
- Therefore, the electronic configuration of \(Mn^{2+}\) is:
\[ [Ar] 3d^5 \]

### Step 4: Analyze the ligand field strength
- Fluorine (F) is a weak field ligand. According to crystal field theory, weak field ligands lead to high spin complexes.
- In a high spin complex, electrons will occupy the available orbitals singly before pairing occurs.

### Step 5: Split the d-orbitals according to crystal field theory
- The \(d\)-orbitals split into two sets: \(t_{2g}\) and \(e_g\).
- For a \(d^5\) configuration in a weak field, the electrons will fill as follows:
- \(t_{2g}\) orbitals will fill first, followed by \(e_g\) orbitals.

### Step 6: Fill the orbitals
- The filling order for \(Mn^{2+}\) (which has 5 d-electrons) will be:
- 3 electrons will go into the \(t_{2g}\) orbitals and 2 electrons will go into the \(e_g\) orbitals.
- The configuration can be represented as:
\[ t_{2g}^3 e_g^2 \]

### Final Answer
Thus, the electronic configuration of \([MnF_6]^{4-}\) according to crystal field theory is:
\[ t_{2g}^3 e_g^2 \]