Consider the cell `Pt((s),l)H_(2)(g,1atm)|H^(+)(aq,1M)|Fe^(3+)(aq),Fe^(2+)(aq))|Pt(s)` When the potential of the cell is `0.712V` at `298K`,the ratio `[Fe^(2+)]/[Fe^(3+)]` is (Nearest integer)
Given: `Fe^(3+)+e^(-) to Fe^(2+),E^0Fe^(3+), Fe^(2+)|Pt = 0.771`

Answer

To solve the problem, we need to find the ratio \([Fe^{2+}]/[Fe^{3+}]\) given the cell potential and the standard reduction potential for the half-reaction involving iron.

### Step-by-Step Solution:

1. **Identify the Half-Reactions:**
The half-reactions involved in the cell are:
- For hydrogen: \( \frac{1}{2} H_2(g) \rightarrow H^+(aq) + e^- \)
- For iron: \( Fe^{3+}(aq) + e^- \rightarrow Fe^{2+}(aq) \)

2. **Write the Overall Cell Reaction:**
Combining the half-reactions, we get:
\[ \frac{1}{2} H_2(g) + Fe^{3+}(aq) \rightarrow H^+(aq) + Fe^{2+}(aq) \]

3. **Determine the Number of Electrons Transferred (n):**
In this reaction, \( n = 1 \) (one electron is transferred).

4. **Use the Nernst Equation:**
The Nernst equation is given by:
\[ E_{cell} = E^0_{cell} - \frac{0.059}{n} \log \left( \frac{[Products]}{[Reactants]} \right) \]
Here, the products are \( H^+ \) and \( Fe^{2+} \), and the reactants are \( H_2 \) and \( Fe^{3+} \).

5. **Calculate the Standard Cell Potential \(E^0_{cell}\):**
The standard potential for the hydrogen half-reaction is \( 0 \, V \) and for the iron half-reaction is \( E^0_{Fe^{3+}/Fe^{2+}} = 0.771 \, V \).
Thus,
\[ E^0_{cell} = E^0_{Fe^{3+}/Fe^{2+}} - E^0_{H^+/H_2} = 0.771 - 0 = 0.771 \, V \]

6. **Substitute Values into the Nernst Equation:**
Given \( E_{cell} = 0.712 \, V \), we can substitute into the Nernst equation:
\[ 0.712 = 0.771 - \frac{0.059}{1} \log \left( \frac{[Fe^{2+}]}{[Fe^{3+}]} \right) \]

7. **Rearranging the Equation:**
Rearranging gives:
\[ \log \left( \frac{[Fe^{2+}]}{[Fe^{3+}]} \right) = \frac{0.771 - 0.712}{0.059} \]
\[ \log \left( \frac{[Fe^{2+}]}{[Fe^{3+}]} \right) = \frac{0.059}{0.059} = 1 \]

8. **Convert Logarithmic Form to Ratio:**
Since \( \log \left( \frac{[Fe^{2+}]}{[Fe^{3+}]} \right) = 1 \), we can convert this to:
\[ \frac{[Fe^{2+}]}{[Fe^{3+}]} = 10^1 = 10 \]

9. **Final Answer:**
The ratio \([Fe^{2+}]/[Fe^{3+}]\) is \( 10 \).