Arrange the following in the increasing order of oxidation state of Mn:
(i) `Mn^(2+)` (ii) `MnO_(2)` (iii) `KMnO_(4)` (iv) `K_(2)MnO_(4)`

Explanation

To arrange the given compounds in the increasing order of oxidation state of manganese (Mn), we will first calculate the oxidation state of Mn in each compound step by step.

### Step 1: Determine the oxidation state of Mn in `Mn^(2+)`
- The oxidation state of Mn in `Mn^(2+)` is simply +2.

### Step 2: Determine the oxidation state of Mn in `MnO_(2)`
- Let the oxidation state of Mn be \( x \).
- The oxidation state of oxygen is -2. Since there are 2 oxygen atoms, the total contribution from oxygen is \( 2 \times (-2) = -4 \).
- The compound is neutral, so:
\[ x + (-4) = 0 \]
\[ x = +4 \]
- Therefore, the oxidation state of Mn in `MnO_(2)` is +4.

### Step 3: Determine the oxidation state of Mn in `KMnO_(4)`
- Let the oxidation state of Mn be \( x \).
- The oxidation state of potassium (K) is +1, and the oxidation state of oxygen is -2. There are 4 oxygen atoms, so their total contribution is \( 4 \times (-2) = -8 \).
- The compound is neutral, so:
\[ +1 + x + (-8) = 0 \]
\[ x - 7 = 0 \]
\[ x = +7 \]
- Therefore, the oxidation state of Mn in `KMnO_(4)` is +7.

### Step 4: Determine the oxidation state of Mn in `K_(2)MnO_(4)`
- Let the oxidation state of Mn be \( x \).
- The oxidation state of potassium (K) is +1, and there are 2 potassium atoms, contributing +2. The oxidation state of oxygen is -2, and there are 4 oxygen atoms, contributing \( 4 \times (-2) = -8 \).
- The compound is neutral, so:
\[ +2 + x + (-8) = 0 \]
\[ x - 6 = 0 \]
\[ x = +6 \]
- Therefore, the oxidation state of Mn in `K_(2)MnO_(4)` is +6.

### Summary of oxidation states:
1. `Mn^(2+)`: +2
2. `MnO_(2)`: +4
3. `K_(2)MnO_(4)`: +6
4. `KMnO_(4)`: +7

### Step 5: Arrange in increasing order
Now, we can arrange the oxidation states in increasing order:
- +2 (Mn^(2+)) < +4 (MnO2) < +6 (K2MnO4) < +7 (KMnO4)

### Final Answer:
The increasing order of oxidation state of Mn is:
`Mn^(2+) < MnO_(2) < K_(2)MnO_(4) < KMnO_(4)`

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