Arrange the oxides of manganese according to increasing acidic strength.

Options

A. `MnO lt Mn_3O_4 lt Mn_2O_3 lt MnO_2 lt Mn_2O_7`
B. `Mn_2O_7 lt MnO_2 lt Mn_2O_3 lt Mn_3O_4 lt MnO`
C. `MnO_2 lt Mn_2O_7 lt Mn_3O_4 lt Mn_2O_3 lt MnO`
D. `Mn_3O_4 lt Mn_2O_3 lt Mn_2O_7 lt MnO_2 lt MnO`

Explanation

To arrange the oxides of manganese according to increasing acidic strength, we will first identify the oxidation states of manganese in each oxide and then determine their acidic strength based on these oxidation states.

### Step-by-Step Solution:

1. **Identify the Oxides of Manganese**:
The oxides of manganese we need to consider are:
- MnO
- Mn3O4
- Mn2O3
- MnO2
- Mn2O7

2. **Determine the Oxidation States**:
- For **MnO**:
Let the oxidation state of Mn be \( x \). The oxidation state of O is \(-2\).
\[ x + (-2) = 0 \implies x = +2 \]
- For **Mn3O4**:
Let the oxidation state of Mn be \( x \). There are 3 Mn and 4 O.
\[ 3x + 4(-2) = 0 \implies 3x - 8 = 0 \implies 3x = 8 \implies x = \frac{8}{3} \approx +2.67 \]
- For **Mn2O3**:
Let the oxidation state of Mn be \( x \). There are 2 Mn and 3 O.
\[ 2x + 3(-2) = 0 \implies 2x - 6 = 0 \implies 2x = 6 \implies x = +3 \]
- For **MnO2**:
Let the oxidation state of Mn be \( x \). There are 1 Mn and 2 O.
\[ x + 2(-2) = 0 \implies x - 4 = 0 \implies x = +4 \]
- For **Mn2O7**:
Let the oxidation state of Mn be \( x \). There are 2 Mn and 7 O.
\[ 2x + 7(-2) = 0 \implies 2x - 14 = 0 \implies 2x = 14 \implies x = +7 \]

3. **List the Oxides with Their Oxidation States**:
- MnO: +2
- Mn3O4: +2.67
- Mn2O3: +3
- MnO2: +4
- Mn2O7: +7

4. **Determine the Acidic Strength**:
The acidic strength of the oxides generally increases with the increase in oxidation state of manganese. Therefore, we can arrange them in order of increasing acidic strength:
- MnO (least acidic, +2)
- Mn3O4 (+2.67)
- Mn2O3 (+3)
- MnO2 (+4)
- Mn2O7 (most acidic, +7)

5. **Final Arrangement**:
The final arrangement of manganese oxides according to increasing acidic strength is:
\[ \text{MnO} < \text{Mn3O4} < \text{Mn2O3} < \text{MnO2} < \text{Mn2O7} \]