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 can follow these steps:

### Step 1: Identify the central metal and its oxidation state
- The central metal in the complex is cobalt (Co).
- The atomic number of cobalt is 27, and its electronic configuration is \( [Ar] 3d^7 4s^2 \).
- In the complex \([Co(H_2O)_3F_3]\), there are three neutral water ligands (H2O) and three fluoride ions (F^-). Since fluoride has a -1 charge, the overall charge of the complex is 0. Therefore, cobalt must be in the +3 oxidation state to balance the charges.
- The oxidation state of cobalt in this complex is +3, which means it loses 3 electrons (2 from 4s and 1 from 3d). Thus, the electronic configuration of \( Co^{3+} \) is \( [Ar] 3d^6 \).

### Step 2: Determine the geometry and splitting of d-orbitals
- The coordination number of the complex is 6 (3 H2O and 3 F^-), which gives it an octahedral geometry.
- In an octahedral field, the d-orbitals split into two sets: \( t_{2g} \) (lower energy) and \( e_g \) (higher energy).

### Step 3: Identify the nature of the ligands
- Water (H2O) is a neutral ligand and is considered a weak field ligand, while fluoride (F^-) is also a weak field ligand.
- Since both ligands are weak field ligands, they do not cause significant splitting of the d-orbitals, leading to a high-spin configuration.

### Step 4: Fill the d-orbitals according to Hund's rule
- For \( Co^{3+} \) with \( 3d^6 \) configuration in an octahedral field:
- The \( t_{2g} \) set can hold up to 6 electrons, and the \( e_g \) set can hold up to 4 electrons.
- We fill the \( t_{2g} \) orbitals first. According to Hund's rule, we will fill each orbital singly before pairing:
- Fill \( t_{2g} \): 1 electron in each of the three \( t_{2g} \) orbitals (3 electrons) and then pair the next three electrons in the same orbitals (3 more electrons).

Thus, the filling will look like this:
- \( t_{2g} \): ↑ ↑ ↑ (3 unpaired) and then pairing: ↑↓ ↑↓ ↑↓ (total 6 electrons in \( t_{2g} \))

### Step 5: Count the number of unpaired electrons in \( t_{2g} \)
- In the \( t_{2g} \) set, we have 6 electrons filled as follows:
- 3 orbitals are filled with paired electrons.
- Therefore, there are 0 unpaired electrons in the \( t_{2g} \) set.

### Final Answer
The number of unpaired electrons in the \( t_{2g} \) set of d-orbitals in the complex \([Co(H_2O)_3F_3]\) is **0**.