`M(s)rarrM^(n+)(aq)+n e^(-)`
Choose the correct statement(s).

Options

A. `E_(M|M^(n+))` decrease with increase in `[M^(n+)]`
B. `E_(M^(n+)|M)` increases on increasing temeperature.
C. `E_(M^(n+)|M)` increases on increasing `[M^(n+)]`
D. `E_(M|M^(n+))` increases on increasing temperature.

Explanation

To solve the question regarding the oxidation of a metal \( M \) to \( M^{n+} \) and to choose the correct statements, we will analyze the Nernst equation and its implications step by step.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The given reaction is:
\[ M(s) \rightarrow M^{n+}(aq) + n e^- \]
This indicates that solid metal \( M \) is oxidized to its ion \( M^{n+} \) in aqueous solution, releasing \( n \) electrons.

2. **Writing the Nernst Equation**:
The Nernst equation for the half-reaction can be expressed as:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q \]
where \( Q \) is the reaction quotient. For this reaction, \( Q \) can be defined as:
\[ Q = \frac{1}{[M^{n+}]} \]
Therefore, the Nernst equation becomes:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln \left(\frac{1}{[M^{n+}]}\right) \]

3. **Analyzing the Effect of Concentration**:
- **Option A**: "E of \( M \) to \( M^{n+} \) decreases with increases in concentration of \( M^{n+} \)".
- As the concentration of \( M^{n+} \) increases, \( \ln \left(\frac{1}{[M^{n+}]}\right) \) becomes more negative, which decreases \( E_{\text{cell}} \). Thus, this statement is **correct**.

4. **Analyzing the Effect of Temperature**:
- **Option B**: "E of \( M^{n+} \) to \( M \) increases on increasing temperature".
- When we increase temperature, the term \( -\frac{RT}{nF} \ln [M^{n+}] \) increases, leading to an increase in \( E_{\text{cell}} \). Thus, this statement is **correct**.

5. **Effect of Concentration on Reverse Reaction**:
- **Option C**: "E of \( M^{n+} \) to \( M \) increases on increasing concentration of \( M^{n+} \)".
- As the concentration of \( M^{n+} \) increases, \( E_{\text{cell}} \) indeed increases because the logarithmic term becomes less negative. Thus, this statement is **correct**.

6. **Effect of Temperature on Forward Reaction**:
- **Option D**: "E of \( M \) to \( M^{n+} \) increases on increasing temperature".
- Here, increasing temperature increases the negative term \( -\frac{RT}{nF} \ln [M^{n+}] \), which decreases \( E_{\text{cell}} \). Thus, this statement is **incorrect**.

### Conclusion:
The correct statements are:
- Option A: Correct
- Option B: Correct
- Option C: Correct
- Option D: Incorrect

Thus, the correct options are A, B, and C.