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 find the standard potential \( E^\circ_{Cu/C^+} \) for the half-reaction \( Cu^+ + e^- \leftrightarrow Cu \), we can follow these steps:

### Step 1: Understand the given equilibrium expression
The equilibrium established is:
\[ Cu^{2+}(aq) + Cu(s) \rightleftharpoons 2Cu^{+}(aq) \]
The equilibrium constant \( K \) is given by:
\[ K = \frac{[Cu^{+}]^2}{[Cu^{2+}]} \]
At equilibrium, this is equal to \( 2.02 \times 10^4 \).

### Step 2: Use the Nernst equation
At equilibrium, the cell potential \( E_{cell} \) is 0. Therefore, we can use the Nernst equation:
\[ E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log K \]
Since \( E_{cell} = 0 \), we have:
\[ 0 = E^\circ_{cell} - \frac{0.0591}{n} \log K \]
Thus,
\[ E^\circ_{cell} = \frac{0.0591}{n} \log K \]

### Step 3: Determine \( n \)
In the half-reaction \( Cu^{2+} + 2e^- \leftrightarrow 2Cu^{+} \), the transfer of electrons \( n \) is 2.

### Step 4: Calculate \( E^\circ_{cell} \)
Substituting \( n = 2 \) and \( K = 2.02 \times 10^4 \):
\[ E^\circ_{cell} = \frac{0.0591}{2} \log(2.02 \times 10^4) \]
Calculating \( \log(2.02 \times 10^4) \):
\[ \log(2.02 \times 10^4) = \log(2.02) + \log(10^4) = 0.305 + 4 = 4.305 \]
Now substituting this value:
\[ E^\circ_{cell} = \frac{0.0591}{2} \times 4.305 \approx 0.1275 \text{ volts} \]

### Step 5: Relate \( E^\circ_{cell} \) to the standard potentials
We know:
\[ E^\circ_{cell} = E^\circ_{Cu^{2+}/Cu^{+}} - E^\circ_{Cu^{+}/Cu} \]
Given \( E^\circ_{Cu^{2+}/Cu} = 0.33 \) volts, we can rearrange:
\[ E^\circ_{Cu^{+}/Cu} = E^\circ_{Cu^{2+}/Cu^{+}} - E^\circ_{cell} \]
Substituting the known values:
\[ E^\circ_{Cu^{+}/Cu} = 0.33 - 0.1275 \approx 0.2025 \text{ volts} \]

### Step 6: Final answer
Thus, the standard potential \( E^\circ_{Cu/C^+} \) is approximately \( 0.2025 \) volts.

---