In the given typical redox reaction : `M^(X+) + MnO_(4)^(-) to MO_(3)^(-) + Mn^(2+) + 1/2 O_(2)` , If one mole of `MnO_(4)^(-)` oxidises 2.5 moles of `M^(x+)` , then the value of x is :

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the oxidation states
In the reaction, we have the following species:
- \( \text{MnO}_4^{-} \) (permanganate ion)
- \( \text{MO}_3^{-} \) (the product containing metal M)
- \( \text{Mn}^{2+} \) (manganese ion)
- \( \frac{1}{2} \text{O}_2 \) (oxygen gas)

We need to determine the oxidation states of manganese in \( \text{MnO}_4^{-} \) and \( \text{Mn}^{2+} \).

### Step 2: Calculate the oxidation state of manganese in \( \text{MnO}_4^{-} \)
Let the oxidation state of manganese in \( \text{MnO}_4^{-} \) be \( y \). The equation for the oxidation state is:
\[ y + 4(-2) = -1 \]
\[ y - 8 = -1 \implies y = +7 \]
So, the oxidation state of manganese in \( \text{MnO}_4^{-} \) is +7.

### Step 3: Determine the change in oxidation state
In the reaction, manganese is reduced from +7 in \( \text{MnO}_4^{-} \) to +2 in \( \text{Mn}^{2+} \). The change in oxidation state is:
\[ 7 - 2 = 5 \]
This means that 5 electrons are gained by one mole of \( \text{MnO}_4^{-} \).

### Step 4: Determine the n-factor for \( \text{MnO}_4^{-} \)
The n-factor for \( \text{MnO}_4^{-} \) is the number of electrons gained or lost, which we found to be 5.

### Step 5: Set up the equivalence equation
According to the problem, 1 mole of \( \text{MnO}_4^{-} \) oxidizes 2.5 moles of \( \text{M}^{x+} \). The n-factor for \( \text{M}^{x+} \) can be expressed as \( 5 - x \) since it is undergoing oxidation from +5 to +x.

The equivalence of \( \text{MnO}_4^{-} \) and \( \text{M}^{x+} \) can be set up as:
\[ \text{Equivalents of } \text{MnO}_4^{-} = \text{Equivalents of } \text{M}^{x+} \]
This gives us:
\[ 1 \times 5 = 2.5 \times (5 - x) \]

### Step 6: Solve for x
Now, we can solve the equation:
\[ 5 = 2.5(5 - x) \]
Expanding the right side:
\[ 5 = 12.5 - 2.5x \]
Rearranging gives:
\[ 2.5x = 12.5 - 5 \]
\[ 2.5x = 7.5 \]
Dividing both sides by 2.5:
\[ x = \frac{7.5}{2.5} = 3 \]

### Final Answer
The value of \( x \) is **3**.

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