Which of the following reactions are disproportionation reaction ?

Options

A. (a) `2Cu^(+)toCu^(2+)+Cu^(0)`
B. (b) `3KMnO_(4)^(2-) + 4H^(+)to2MnO_(4)^(-)+MnO_(2)+2H_(2)O`
C. (c) `2KMnO_(4)overset(triangle)toK_(2)MnO_(4)+MnO_(2)+O_(2)`
D. (d) `2MnO_(4)^(-)+3Mn^(2+)+2H_(2)Oto5MnO_(2)+4H^(+)`

Explanation

To determine which of the given reactions are disproportionation reactions, we will analyze each reaction step by step. A disproportionation reaction is a specific type of redox reaction where a single species is simultaneously oxidized and reduced.

### Step-by-Step Solution:

1. **Identify the Reactions**: We have four reactions to analyze:
- Reaction A: \( 2 \text{Cu}^+ \rightarrow \text{Cu}^{2+} + \text{Cu}^0 \)
- Reaction B: \( 3 \text{KMnO}_4 + 4 \text{H}^+ \rightarrow 2 \text{MnO}_4^{-} + \text{MnO}_2 + 2 \text{H}_2O \)
- Reaction C: \( 2 \text{KMnO}_4 \rightarrow \text{K}_2\text{MnO}_4 + \text{MnO}_2 + \text{O}_2 \)
- Reaction D: \( 2 \text{MnO}_4^{-} + 3 \text{Mn}^{2+} + 2 \text{H}_2O \rightarrow 5 \text{MnO}_2 + 4 \text{H}^+ \)

2. **Analyze Reaction A**:
- In \( 2 \text{Cu}^+ \rightarrow \text{Cu}^{2+} + \text{Cu}^0 \):
- The oxidation state of Cu changes from +1 to +2 (oxidation).
- The oxidation state of Cu changes from +1 to 0 (reduction).
- Since the same element (Cu) is both oxidized and reduced, this is a disproportionation reaction.

3. **Analyze Reaction B**:
- In \( 3 \text{KMnO}_4 + 4 \text{H}^+ \rightarrow 2 \text{MnO}_4^{-} + \text{MnO}_2 + 2 \text{H}_2O \):
- The oxidation state of Mn in KMnO4 is +7 and in MnO4^- it is +6, while in MnO2 it is +4.
- Here, Mn is being reduced from +7 to +4 (reduction) and also being oxidized from +7 to +6 (oxidation).
- Therefore, this is also a disproportionation reaction.

4. **Analyze Reaction C**:
- In \( 2 \text{KMnO}_4 \rightarrow \text{K}_2\text{MnO}_4 + \text{MnO}_2 + \text{O}_2 \):
- The oxidation state of Mn changes from +7 in KMnO4 to +6 in K2MnO4 and +4 in MnO2.
- Here, Mn is only being reduced (from +7 to +6 and +4).
- Since there is no oxidation occurring, this is not a disproportionation reaction.

5. **Analyze Reaction D**:
- In \( 2 \text{MnO}_4^{-} + 3 \text{Mn}^{2+} + 2 \text{H}_2O \rightarrow 5 \text{MnO}_2 + 4 \text{H}^+ \):
- The Mn in MnO4^- is +7 and in MnO2 it is +4, while Mn2+ is +2.
- Here, MnO4^- is being reduced and Mn2+ is being oxidized, involving different species.
- Therefore, this is not a disproportionation reaction.

### Conclusion:
The reactions that are disproportionation reactions are:
- **Reaction A**: \( 2 \text{Cu}^+ \rightarrow \text{Cu}^{2+} + \text{Cu}^0 \)
- **Reaction B**: \( 3 \text{KMnO}_4 + 4 \text{H}^+ \rightarrow 2 \text{MnO}_4^{-} + \text{MnO}_2 + 2 \text{H}_2O \)

### Final Answer:
The correct options are **A and B**, which are disproportionation reactions.