What is the oxidation number of `Mn` is `KMnO_(4)" "K_(2)MNO_(4)" "MnSO_(4)" "MnO_(2)`, and `Mn_(3)O_(4)`?

Explanation

To find the oxidation number of manganese (Mn) in the given compounds, we will analyze each compound one by one.

### Step-by-Step Solution:

1. **For KMnO4:**
- Let the oxidation number of Mn be \( x \).
- The oxidation number of K (potassium) is \( +1 \).
- The oxidation number of O (oxygen) is \( -2 \).
- The equation for the compound is:
\[ +1 + x + 4(-2) = 0 \]
- Simplifying the equation:
\[ +1 + x - 8 = 0 \implies x - 7 = 0 \implies x = +7 \]
- **Oxidation number of Mn in KMnO4 is +7.**

2. **For K2MnO4:**
- Let the oxidation number of Mn be \( x \).
- The oxidation number of K is \( +1 \) (and there are 2 K atoms).
- The oxidation number of O is \( -2 \).
- The equation for the compound is:
\[ 2(+1) + x + 4(-2) = 0 \]
- Simplifying the equation:
\[ 2 + x - 8 = 0 \implies x - 6 = 0 \implies x = +6 \]
- **Oxidation number of Mn in K2MnO4 is +6.**

3. **For MnSO4:**
- Let the oxidation number of Mn be \( x \).
- The oxidation number of S (sulfur) is \( +6 \).
- The oxidation number of O is \( -2 \) (and there are 4 O atoms).
- The equation for the compound is:
\[ x + 6 + 4(-2) = 0 \]
- Simplifying the equation:
\[ x + 6 - 8 = 0 \implies x - 2 = 0 \implies x = +2 \]
- **Oxidation number of Mn in MnSO4 is +2.**

4. **For MnO2:**
- Let the oxidation number of Mn be \( x \).
- The oxidation number of O is \( -2 \) (and there are 2 O atoms).
- The equation for the compound is:
\[ x + 2(-2) = 0 \]
- Simplifying the equation:
\[ x - 4 = 0 \implies x = +4 \]
- **Oxidation number of Mn in MnO2 is +4.**

5. **For Mn3O4:**
- Let the oxidation number of Mn be \( x \).
- The oxidation number of O is \( -2 \) (and there are 4 O atoms).
- The equation for the compound is:
\[ 3x + 4(-2) = 0 \]
- Simplifying the equation:
\[ 3x - 8 = 0 \implies 3x = 8 \implies x = \frac{8}{3} \]
- **Oxidation number of Mn in Mn3O4 is \( \frac{8}{3} \) or approximately +2.67.**

### Summary of Oxidation Numbers:
- KMnO4: +7
- K2MnO4: +6
- MnSO4: +2
- MnO2: +4
- Mn3O4: \( \frac{8}{3} \) or approximately +2.67