How many unpaired electrons are present in `e_(g)` orbital of `MnO_(4) Θ` .

Answer

To determine how many unpaired electrons are present in the `e_(g)` orbital of `MnO₄⁻`, we can follow these steps:

### Step 1: Determine the Oxidation State of Manganese
- The formula for the oxidation state can be set up as follows:
\[ \text{Oxidation state of Mn} + 4 \times (\text{Oxidation state of O}) = \text{Overall charge} \]
- Let the oxidation state of Mn be \( x \).
- Oxygen has an oxidation state of \(-2\), and there are 4 oxygen atoms:
\[ x + 4(-2) = -1 \]
- Simplifying this gives:
\[ x - 8 = -1 \implies x = +7 \]

**Hint:** Remember that the overall charge of the complex is equal to the sum of the oxidation states of the elements involved.

### Step 2: Write the Electron Configuration of Manganese
- The atomic number of manganese (Mn) is 25. Its ground state electron configuration is:
\[ \text{Mn: } [Ar] 4s^2 3d^5 \]

**Hint:** The ground state configuration shows how many electrons are in each subshell.

### Step 3: Adjust the Electron Configuration for the Oxidation State
- Since manganese is in the +7 oxidation state, we remove 7 electrons from its configuration:
- Remove 2 electrons from the 4s subshell and 5 electrons from the 3d subshell:
\[ 4s^2 \rightarrow 4s^0 \quad \text{and} \quad 3d^5 \rightarrow 3d^0 \]
- Thus, the electron configuration for \( \text{Mn}^{7+} \) is:
\[ \text{Mn}^{7+}: [Ar] 4s^0 3d^0 \]

**Hint:** When calculating the oxidation state, remember to remove electrons starting from the outermost shell.

### Step 4: Analyze the d-Orbitals
- The d-orbitals split into two sets: \( t_{2g} \) and \( e_g \).
- Since \( \text{Mn}^{7+} \) has no electrons left in the 3d subshell, both the \( t_{2g} \) and \( e_g \) orbitals are empty:
\[ t_{2g} = 0, \quad e_g = 0 \]

**Hint:** Understand the splitting of d-orbitals in octahedral complexes and how the presence or absence of electrons affects their occupancy.

### Conclusion
- Therefore, the number of unpaired electrons in the \( e_g \) orbital of \( \text{MnO}_4^- \) is:
\[ \text{Number of unpaired electrons in } e_g = 0 \]

**Final Answer:** 0 unpaired electrons in the \( e_g \) orbital of \( \text{MnO}_4^- \).