`E_(Cu^(2+)//Cu)^(@)=0.34V`
`E_(Cu^(+)//Cu)^(@)=0.522V`
`E_(Cu^(2+)//Cu^(+))^(@)=`

Options

A. ` + 0.158 V `
B. ` -0.182 `V
C. ` -0.158 ` V
D. `0.182 ` V

Answer

To find the standard reduction potential for the half-reaction \( \text{Cu}^{2+} + e^- \rightarrow \text{Cu}^+ \), we can use the provided standard reduction potentials for the following half-reactions:

1. \( \text{Cu}^{2+} + 2e^- \rightarrow \text{Cu} \) with \( E^\circ = 0.34 \, \text{V} \)
2. \( \text{Cu}^+ + e^- \rightarrow \text{Cu} \) with \( E^\circ = 0.522 \, \text{V} \)

We can relate these half-reactions using the following equation:

\[ E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}^+} = E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}} + E^\circ_{\text{Cu}^+ \rightarrow \text{Cu}} \]

### Step-by-Step Solution:

1. **Identify the half-reactions and their potentials**:
- For \( \text{Cu}^{2+} + 2e^- \rightarrow \text{Cu} \), \( E^\circ_1 = 0.34 \, \text{V} \)
- For \( \text{Cu}^+ + e^- \rightarrow \text{Cu} \), \( E^\circ_2 = 0.522 \, \text{V} \)

2. **Write the equation for the overall reaction**:
- The overall reaction from \( \text{Cu}^{2+} \) to \( \text{Cu}^+ \) can be derived from the two half-reactions:
\[ \text{Cu}^{2+} + 2e^- \rightarrow \text{Cu} \quad (1) \]
\[ \text{Cu}^+ + e^- \rightarrow \text{Cu} \quad (2) \]

To find \( E^\circ \) for the reaction \( \text{Cu}^{2+} + e^- \rightarrow \text{Cu}^+ \), we can rearrange the reactions.

3. **Use the relationship between the potentials**:
- The potential for the overall reaction can be expressed as:
\[ E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}^+} = E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}} - E^\circ_{\text{Cu} \rightarrow \text{Cu}^+} \]
- Here, \( E^\circ_{\text{Cu} \rightarrow \text{Cu}^+} = -E^\circ_2 \) since we are reversing the second half-reaction.

4. **Substitute the values**:
- Substitute the known values:
\[ E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}^+} = 0.34 \, \text{V} - 0.522 \, \text{V} \]

5. **Calculate the potential**:
- Calculate:
\[ E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}^+} = 0.34 \, \text{V} - 0.522 \, \text{V} = -0.182 \, \text{V} \]

### Final Answer:
\[ E^\circ_{\text{Cu}^{2+} \rightarrow \text{Cu}^+} = -0.182 \, \text{V} \]