Given standard `E^(c-):`
`Fe^(3+)+3e^(-)rarrFe," "E^(c-)=-0.036`
Fe^(2+)+2e^(-)rarr Fe," "E^(c-)=-0.440V`
The `E^(c-)` of `Fe^(3+)+e^(-) rarr Fe^(2+) ` is

Options

A. `-0.476V`
B. `-0.404V`
C. `0.404V`
D. `0.772V`

Explanation

To find the standard reduction potential \( E^{\circ} \) for the reaction \( \text{Fe}^{3+} + e^- \rightarrow \text{Fe}^{2+} \), we can use the given standard reduction potentials for the reactions involving iron.

### Step-by-Step Solution:

1. **Write down the given reactions and their standard potentials**:
- Reaction 1: \( \text{Fe}^{3+} + 3e^- \rightarrow \text{Fe} \) with \( E^{\circ} = -0.036 \, \text{V} \)
- Reaction 2: \( \text{Fe}^{2+} + 2e^- \rightarrow \text{Fe} \) with \( E^{\circ} = -0.440 \, \text{V} \)

2. **Identify the target reaction**:
- We need to find \( E^{\circ} \) for the reaction:
\[ \text{Fe}^{3+} + e^- \rightarrow \text{Fe}^{2+} \]

3. **Manipulate the given reactions**:
- We can derive the target reaction by subtracting Reaction 2 from Reaction 1:
\[ \text{Fe}^{3+} + 3e^- \rightarrow \text{Fe} \quad \text{(Reaction 1)} \]
\[ \text{Fe}^{2+} + 2e^- \rightarrow \text{Fe} \quad \text{(Reaction 2)} \]
- Rearranging Reaction 2 gives:
\[ \text{Fe} \rightarrow \text{Fe}^{2+} + 2e^- \quad \text{(Reverse Reaction 2)} \]

4. **Combine the reactions**:
- Adding Reaction 1 and the reversed Reaction 2:
\[ \text{Fe}^{3+} + 3e^- + \text{Fe} \rightarrow \text{Fe} + \text{Fe}^{2+} + 2e^- \]
- Simplifying this gives:
\[ \text{Fe}^{3+} + e^- \rightarrow \text{Fe}^{2+} \]

5. **Calculate the standard potential for the target reaction**:
- Using the formula for the standard potential:
\[ E^{\circ}_{\text{target}} = \frac{n_1 E^{\circ}_1 - n_2 E^{\circ}_2}{n_3} \]
- Where:
- \( n_1 = 3 \) (from Reaction 1)
- \( E^{\circ}_1 = -0.036 \, \text{V} \)
- \( n_2 = 2 \) (from Reaction 2)
- \( E^{\circ}_2 = -0.440 \, \text{V} \)
- \( n_3 = 1 \) (for the target reaction)
- Plugging in the values:
\[ E^{\circ}_{\text{target}} = \frac{3(-0.036) - 2(-0.440)}{1} \]
\[ = \frac{-0.108 + 0.880}{1} = 0.772 \, \text{V} \]

6. **Conclusion**:
- The standard reduction potential \( E^{\circ} \) for the reaction \( \text{Fe}^{3+} + e^- \rightarrow \text{Fe}^{2+} \) is \( 0.772 \, \text{V} \).