To calculate the standard free energy change (ΔG°) for the reaction \( \text{Cu}^{2+} + \text{Sn}(s) \rightarrow \text{Cu}(s) + \text{Sn}^{2+} \), we can use the relationship between Gibbs free energy and the standard cell potential (E°) given by the formula:
\[
\Delta G^\circ = -nFE^\circ
\]
Where:
- \( n \) = number of moles of electrons transferred in the reaction
- \( F \) = Faraday's constant (approximately \( 96500 \, \text{C/mol} \))
- \( E^\circ \) = standard cell potential (given as \( 0.48 \, \text{V} \))
### Step 1: Determine the number of electrons transferred (n)
In the given reaction, copper ions (\( \text{Cu}^{2+} \)) are reduced to copper metal (\( \text{Cu} \)), and tin metal (\( \text{Sn} \)) is oxidized to tin ions (\( \text{Sn}^{2+} \)).
- The reduction half-reaction for copper is:
\[
\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}
\]
- The oxidation half-reaction for tin is:
\[
\text{Sn} \rightarrow \text{Sn}^{2+} + 2e^-
\]
From both half-reactions, we see that 2 electrons are transferred in total. Therefore, \( n = 2 \).
### Step 2: Substitute values into the formula
Now, we can substitute the values into the Gibbs free energy equation:
\[
\Delta G^\circ = -nFE^\circ
\]
Substituting the known values:
- \( n = 2 \)
- \( F = 96500 \, \text{C/mol} \)
- \( E^\circ = 0.48 \, \text{V} \)
\[
\Delta G^\circ = -2 \times 96500 \, \text{C/mol} \times 0.48 \, \text{V}
\]
### Step 3: Calculate ΔG°
Now, we perform the calculation:
\[
\Delta G^\circ = -2 \times 96500 \times 0.48
\]
Calculating this step-by-step:
1. Calculate \( 96500 \times 0.48 = 46320 \)
2. Now multiply by 2: \( 2 \times 46320 = 92640 \)
3. Apply the negative sign: \( \Delta G^\circ = -92640 \, \text{J/mol} \)
### Step 4: Convert to kilojoules
Since the answer is typically expressed in kilojoules per mole, we convert joules to kilojoules:
\[
\Delta G^\circ = -92640 \, \text{J/mol} \div 1000 = -92.64 \, \text{kJ/mol}
\]
### Final Answer
Thus, the standard free energy change for the reaction is:
\[
\Delta G^\circ = -92.64 \, \text{kJ/mol}
\]
