`25ml` of a `0.1(M)` solution of a stable cation of transition metal `z` reacts exactly with `25 ml` of `0.04(M)` acidified `KMnO_(4)` solution. Which of the following is most likely to represent the change in oxidation state of `Z` correctly?

Options

A. `Z^(+)rarrZ^(2+)`
B. `Z^(2+)rarrZ^(3+)`
C. `Z^(3+)rarrZ^(4+)`
D. `Z^(2+)rarrZ^(4+)`

Answer

To solve the problem step by step, we need to analyze the reaction between the stable cation of transition metal \( Z \) and the acidified \( KMnO_4 \) solution.

### Step 1: Understand the Reaction
The reaction involves \( KMnO_4 \) in an acidic medium, where manganese (Mn) is reduced from an oxidation state of +7 in \( KMnO_4 \) to +2 in \( Mn^{2+} \).

### Step 2: Calculate the Equivalent of \( KMnO_4 \)
The equivalent of a substance in a redox reaction can be calculated using the formula:
\[ \text{Equivalent} = \text{Molarity} \times \text{n-factor} \times \text{Volume (L)} \]
For \( KMnO_4 \):
- Molarity (\( M_k \)) = 0.04 M
- n-factor for \( KMnO_4 \) = 5 (since +7 to +2 is a change of 5)
- Volume (\( V_k \)) = 25 mL = 0.025 L

Calculating the equivalent of \( KMnO_4 \):
\[ \text{Equivalent of } KMnO_4 = 0.04 \, \text{mol/L} \times 5 \times 0.025 \, \text{L} = 0.005 \, \text{equivalents} \]

### Step 3: Set Up the Equation for \( Z \)
Since the equivalents of \( Z \) will equal the equivalents of \( KMnO_4 \) in the reaction:
\[ \text{Equivalent of } Z = 0.005 \, \text{equivalents} \]
Let \( n_z \) be the n-factor for \( Z \) and \( M_z \) be the molarity of \( Z \) (which is given as 0.1 M). The volume of \( Z \) is also 25 mL or 0.025 L.

Using the equivalent formula for \( Z \):
\[ 0.005 = 0.1 \, \text{mol/L} \times n_z \times 0.025 \, \text{L} \]

### Step 4: Solve for \( n_z \)
Rearranging the equation to find \( n_z \):
\[ n_z = \frac{0.005}{0.1 \times 0.025} = \frac{0.005}{0.0025} = 2 \]

### Step 5: Determine the Change in Oxidation State
The n-factor of \( Z \) is 2, which indicates that the change in oxidation state of \( Z \) is 2. We need to find which option represents this change correctly.

### Step 6: Analyze the Options
1. \( Z^+ \rightarrow Z^{2+} \) (n-factor = 1) - Incorrect
2. \( Z^{2+} \rightarrow Z^{3+} \) (n-factor = 1) - Incorrect
3. \( Z^{3+} \rightarrow Z^{4+} \) (n-factor = 1) - Incorrect
4. \( Z^{2+} \rightarrow Z^{4+} \) (n-factor = 2) - Correct

### Conclusion
The correct representation of the change in oxidation state of \( Z \) is:
\[ Z^{2+} \rightarrow Z^{4+} \]