How many electrons are present in `t_(2g)` set of d-orbitals of central metal cation in `[Fe(H_(2)O)_(5)(NO)]SO_(4)` brown ring complex?

Explanation

To determine the number of electrons present in the \( t_{2g} \) set of d-orbitals of the central metal cation in the brown ring complex \([Fe(H_2O)_5(NO)]SO_4\), we can follow these steps:

### Step 1: Identify the oxidation state of iron in the complex.
The sulfate ion \((SO_4)^{2-}\) carries a charge of -2. The overall charge of the complex ion must be +2 to balance the -2 charge of the sulfate. Since the nitrosyl \((NO)\) is a neutral ligand, we can deduce that the oxidation state of iron must be +1.

**Hint:** Remember that the sum of the charges in a complex must equal the overall charge of the complex.

### Step 2: Determine the electron configuration of the iron cation.
Iron (Fe) has an atomic number of 26, with an electron configuration of \([Ar] 4s^2 3d^6\). When iron is in the +1 oxidation state, it loses one electron from the 4s subshell, resulting in an electron configuration of \(4s^0 3d^6\).

**Hint:** For transition metals, the 4s electrons are typically lost before the 3d electrons when forming cations.

### Step 3: Analyze the ligands and their field strength.
In the complex, the ligands are five water molecules \((H_2O)\) and one nitrosyl \((NO)\). Both water and nitrosyl are considered weak field ligands, which means they do not cause significant pairing of electrons in the d-orbitals.

**Hint:** The nature of the ligands affects the splitting of d-orbitals and the arrangement of electrons.

### Step 4: Determine the distribution of electrons in the d-orbitals.
With \(3d^6\) configuration and weak field ligands, the electrons will fill the \(t_{2g}\) orbitals first before pairing occurs. The \(3d\) orbitals split into \(t_{2g}\) and \(e_g\) levels, where \(t_{2g}\) consists of three orbitals (dxy, dyz, dzx) and \(e_g\) consists of two orbitals (dx2-y2, dz2).

The filling will occur as follows:
- \(t_{2g}\): 6 electrons fill the three orbitals (1, 2, 3 in \(t_{2g}\) and 4, 5, 6 in \(e_g\)).
- Since we have 6 electrons total, the distribution will be 3 electrons in \(t_{2g}\) and 3 in \(e_g\).

**Hint:** Remember the order of filling and the maximum number of electrons in each set of orbitals.

### Step 5: Count the electrons in the \(t_{2g}\) set.
From the distribution, we find that there are 6 electrons in total, with 3 electrons occupying the \(t_{2g}\) orbitals.

**Final Answer:** The number of electrons present in the \(t_{2g}\) set of d-orbitals is **6**.