The equilibrium `Cu^(. .)(aq)+Cu(s) hArr2Cu^(.)` established at `20^(@)C` corresponds to `([Cu^(. .)])/([Cu^(+)])=2.02xx10^(4+)`. The standard potential . `E_(Cu^(. . ).Cu)^(0)=0.33` volt at this temperature . What is the standard potential `E_(Cu//Cu^(+))^(0)`?

Options

A. `-0.457 V`
B. `-0.125V`
C. `-0.66V`
D. `-0.250V`

Explanation

To solve the problem, we need to find the standard potential \( E^\circ_{Cu/C^+} \) given the equilibrium constant and the standard potential \( E^\circ_{Cu^{2+}/Cu} \).

### Step-by-Step Solution:

1. **Identify the Equilibrium Reaction**:
The equilibrium reaction is:
\[ Cu^{2+}(aq) + Cu(s) \rightleftharpoons 2Cu^+(aq) \]

2. **Write the Expression for the Equilibrium Constant (K)**:
The equilibrium constant \( K \) is given by:
\[ K = \frac{[Cu^+]^2}{[Cu^{2+}]} \]
From the problem, we know that \( K = 2.02 \times 10^4 \).

3. **Use the Nernst Equation**:
At equilibrium, the cell potential \( E_{cell} \) is 0. Therefore, we can use the Nernst equation in the form:
\[ E^\circ_{cell} = E^\circ_{Cu^{2+}/Cu} - \frac{0.0591}{n} \log K \]
where \( n \) is the number of electrons transferred in the half-reaction. For this reaction, \( n = 2 \).

4. **Substituting Known Values**:
We know:
- \( E^\circ_{Cu^{2+}/Cu} = 0.33 \, \text{V} \)
- \( K = 2.02 \times 10^4 \)
- \( n = 2 \)

So we can substitute these values into the Nernst equation:
\[ 0 = 0.33 - \frac{0.0591}{2} \log(2.02 \times 10^4) \]

5. **Calculate the Logarithm**:
First, calculate \( \log(2.02 \times 10^4) \):
\[ \log(2.02 \times 10^4) = \log(2.02) + \log(10^4) = 0.305 + 4 = 4.305 \]

6. **Substituting the Logarithm into the Equation**:
Now substitute back into the Nernst equation:
\[ 0 = 0.33 - \frac{0.0591}{2} \times 4.305 \]

7. **Calculate the Right Side**:
\[ \frac{0.0591}{2} \times 4.305 = 0.02545 \times 4.305 \approx 0.109 \]
Thus, the equation becomes:
\[ 0 = 0.33 - 0.109 \]

8. **Solving for \( E^\circ_{Cu/C^+} \)**:
Rearranging gives:
\[ E^\circ_{Cu/C^+} = 0.33 - 0.109 = 0.221 \, \text{V} \]

9. **Final Answer**:
The standard potential \( E^\circ_{Cu/C^+} \) is approximately:
\[ E^\circ_{Cu/C^+} \approx 0.221 \, \text{V} \]