Give the number of unpaired electrons in `t_(2g)` set of d-orbitals in `[Co(H_(2)O)_(3)F_(3)]` complex .

Explanation

To determine the number of unpaired electrons in the \( t_{2g} \) set of d-orbitals in the complex \([Co(H_2O)_3F_3]\), we will follow these steps:

### Step 1: Determine the oxidation state of cobalt
Let the oxidation state of cobalt be \( x \). The aqua ligands \( (H_2O) \) are neutral, and the fluoride ligands \( (F) \) are -1 charged. The overall charge of the complex is 0.

The equation for the oxidation state can be set up as:
\[ x + 3(0) + 3(-1) = 0 \]
Solving this gives:
\[ x - 3 = 0 \implies x = +3 \]
Thus, cobalt is in the +3 oxidation state.

### Step 2: Write the electron configuration for cobalt
Cobalt (Co) has an atomic number of 27. The electron configuration for neutral cobalt is:
\[ [Ar] 4s^2 3d^7 \]
For cobalt in the +3 oxidation state, we remove 3 electrons (2 from the 4s and 1 from the 3d):
\[ [Ar] 3d^6 \]

### Step 3: Determine the nature of the ligands
In the complex \([Co(H_2O)_3F_3]\), both \( H_2O \) and \( F^- \) are considered weak field ligands. Weak field ligands do not cause significant splitting of the d-orbitals, which means that the pairing energy is greater than the splitting energy.

### Step 4: Fill the d-orbitals according to Hund's rule
For a \( d^6 \) configuration in a weak field, the electrons will fill the \( t_{2g} \) and \( e_g \) orbitals as follows:
1. Fill each orbital singly before pairing (Hund's rule).
2. The \( t_{2g} \) set can hold 6 electrons.

The filling of the orbitals will be:
- \( t_{2g} \): 6 electrons will occupy the orbitals as follows:
- \( \uparrow \) \( \uparrow \) \( \uparrow \) (3 unpaired electrons in \( t_{2g} \))
- Then we start pairing:
- \( \uparrow\downarrow \) \( \uparrow\downarrow \) \( \uparrow \) (2 paired and 1 unpaired)

### Step 5: Count the unpaired electrons in the \( t_{2g} \) set
In the \( t_{2g} \) set, we have:
- 4 electrons in total: 2 are paired and 2 are unpaired.
Thus, the number of unpaired electrons in the \( t_{2g} \) set is 2.

### Final Answer
The number of unpaired electrons in the \( t_{2g} \) set of d-orbitals in the complex \([Co(H_2O)_3F_3]\) is **2**.

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