To solve the problem, we will use the relationship between the standard Gibbs free energy change (ΔG°) and the standard electromotive force (E°) of the cell. The formula we will use is:
\[
\Delta G° = -nFE°
\]
where:
- \( n \) = number of moles of electrons transferred in the reaction
- \( F \) = Faraday's constant (96500 C/mol)
- \( E° \) = standard cell potential (2.0 V)
### Step-by-Step Solution:
1. **Identify the Reaction**:
The given reaction is:
\[
\text{Zn} + \text{Cu}^{2+} \rightarrow \text{Zn}^{2+} + \text{Cu}
\]
2. **Determine the Number of Electrons Transferred (n)**:
- In this reaction, zinc (Zn) is oxidized to zinc ions (Zn²⁺), losing 2 electrons:
\[
\text{Zn} \rightarrow \text{Zn}^{2+} + 2e^-
\]
- Copper ions (Cu²⁺) are reduced to copper (Cu), gaining 2 electrons:
\[
\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}
\]
- Therefore, the total number of electrons transferred (n) is 2.
3. **Substitute Values into the Formula**:
- Now we can substitute the values into the formula:
\[
\Delta G° = -nFE°
\]
- Here, \( n = 2 \), \( F = 96500 \, \text{C/mol} \), and \( E° = 2.0 \, \text{V} \).
4. **Calculate ΔG°**:
\[
\Delta G° = -2 \times 96500 \, \text{C/mol} \times 2.0 \, \text{V}
\]
\[
\Delta G° = -386000 \, \text{J/mol}
\]
5. **Convert to Kilojoules**:
- Since the answer needs to be in kilojoules per mole, we convert joules to kilojoules by dividing by 1000:
\[
\Delta G° = -386000 \, \text{J/mol} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = -386 \, \text{kJ/mol}
\]
### Final Answer:
\[
\Delta G° = -386 \, \text{kJ/mol}
\]
