What is the correct electronic configuration of the central atom in `K_(4)[Fe(CN)_(6)]` based on crystal field theory
(A) `e^(4)t_(2)^(2)`
(B) `t_(2g)^(4)e_(g)^(2)`
(C) `t_(2g)^(6)e_(g)^(0)`
(D) `e^(3)t_(2)^(3)`

Options

A. `e^(4)t_(2)^(2)`
B. `t_(2g)^(4)e_(g)^(2)`
C. `t_(2g)^(6)e_(g)^(0)`
D. `e^(3)t_(2)^(3)`

Explanation

To determine the correct electronic configuration of the central atom in the coordination compound \( K_4[Fe(CN)_6] \) based on crystal field theory, we will follow these steps:

### Step 1: Identify the oxidation state of iron (Fe)
In the complex \( [Fe(CN)_6]^{4-} \), we need to find the oxidation state of iron. Let the oxidation state of iron be \( x \).

The cyanide ion (CN) has a charge of -1, and since there are 6 cyanide ions, their total contribution is \( 6 \times (-1) = -6 \).

Setting up the equation:
\[ x + (-6) = -4 \]
Solving for \( x \):
\[ x - 6 = -4 \implies x = +2 \]

### Step 2: Determine the electronic configuration of \( Fe^{2+} \)
The atomic number of iron (Fe) is 26. The ground state electronic configuration of iron is:
\[ [Ar] 3d^6 4s^2 \]

When iron is in the +2 oxidation state, it loses two electrons, which are removed from the 4s orbital first:
\[ Fe^{2+} : [Ar] 3d^6 4s^0 \]

### Step 3: Analyze the effect of the ligands
In the complex \( [Fe(CN)_6]^{4-} \), the cyanide ion (CN) is a strong field ligand. Strong field ligands cause splitting of the d-orbitals into two sets: \( t_{2g} \) (lower energy) and \( e_g \) (higher energy).

### Step 4: Fill the d-orbitals according to crystal field theory
With 6 electrons in the \( 3d \) subshell, we will fill the \( t_{2g} \) and \( e_g \) orbitals:
- The first three electrons will fill the \( t_{2g} \) orbitals singly.
- The next three electrons will pair up in the \( t_{2g} \) orbitals due to the strong field nature of the cyanide ligands.

The filling will look like this:
- \( t_{2g}^6 \) (6 electrons in the lower energy orbitals)
- \( e_g^0 \) (0 electrons in the higher energy orbitals)

### Step 5: Write the final electronic configuration
Thus, the electronic configuration of the central atom (iron) in \( K_4[Fe(CN)_6] \) based on crystal field theory is:
\[ t_{2g}^6 e_g^0 \]

### Conclusion
The correct answer is option (C) \( t_{2g}^6 e_g^0 \).

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