`E_(M^(3+)//(M))^(@) = -0.036V` , `E_(M^(2+)//M)^(@)= -0.439V`. The value of standard electrode potential for the change, `M^(3+)(aq) + e^(-)rightarrow M^(2+) (aq)` will be :

Options

A. `(-0.072V)`
B. `(0.385V)`
C. `(0.770V)`
D. `0.270V`

Explanation

To find the standard electrode potential for the reaction \( M^{3+}(aq) + e^- \rightarrow M^{2+}(aq) \), we can use the given standard electrode potentials for the half-reactions involving \( M^{3+} \) and \( M^{2+} \).

### Given Data:
1. \( E^\circ_{M^{3+}/M} = -0.036 \, V \)
2. \( E^\circ_{M^{2+}/M} = -0.439 \, V \)

### Step-by-Step Solution:

1. **Identify the half-reactions**:
- The reduction of \( M^{3+} \) to \( M \):
\[ M^{3+} + 3e^- \rightarrow M \quad (E^\circ = -0.036 \, V) \]
- The reduction of \( M^{2+} \) to \( M \):
\[ M^{2+} + 2e^- \rightarrow M \quad (E^\circ = -0.439 \, V) \]

2. **Write the equation for the desired reaction**:
- We want to find the electrode potential for:
\[ M^{3+} + e^- \rightarrow M^{2+} \]

3. **Use the relationship between the half-reactions**:
- We can express the overall reaction as:
\[ M^{3+} + 3e^- \rightarrow M \quad (1) \]
\[ M^{2+} + 2e^- \rightarrow M \quad (2) \]
- We can rearrange equation (2) to express \( M^{2+} \) in terms of \( M \):
\[ M \rightarrow M^{2+} + 2e^- \quad (reverse of (2)) \]

4. **Combine the equations**:
- Adding the half-reactions:
\[ M^{3+} + 3e^- \rightarrow M \quad \text{(1)} \]
\[ M \rightarrow M^{2+} + 2e^- \quad \text{(reverse of (2))} \]
- The electrons cancel out:
\[ M^{3+} + e^- \rightarrow M^{2+} \]

5. **Calculate the standard electrode potential**:
- The overall potential for the reaction can be calculated using the formula:
\[ E^\circ_{M^{3+}/M^{2+}} = E^\circ_{M^{3+}/M} - E^\circ_{M^{2+}/M} \]
- Substituting the values:
\[ E^\circ_{M^{3+}/M^{2+}} = (-0.036 \, V) - (-0.439 \, V) \]
- Simplifying:
\[ E^\circ_{M^{3+}/M^{2+}} = -0.036 + 0.439 = 0.403 \, V \]

### Final Answer:
The standard electrode potential for the change \( M^{3+}(aq) + e^- \rightarrow M^{2+}(aq) \) is \( 0.403 \, V \).

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