Calculate the standard reduction potentials of galvanic cells in which reaction are
`Fe^(2+) (aq) + Ag^(+) (aq) to Fe^(3+) (aq) + Ag (s)`
Given that `E^(Theta) (Cr^(3+) //Cr) = -0.74 V , E^(Theta) (Cd^(2+) // Cd) = -0.40 V ` , `E^(Theta) (Fe^(3+) // Fe^(2+)) = 0.77 V , E^(Theta) (Ag^(+) // Ag) = 0.80 V `

Explanation

To calculate the standard reduction potentials for the given galvanic cell reaction:

**Reaction:**
\[ \text{Fe}^{2+} (aq) + \text{Ag}^{+} (aq) \rightarrow \text{Fe}^{3+} (aq) + \text{Ag} (s) \]

**Step 1: Identify the oxidation and reduction half-reactions.**
- **Oxidation:** The iron ion \(\text{Fe}^{2+}\) is oxidized to \(\text{Fe}^{3+}\).
- **Reduction:** The silver ion \(\text{Ag}^{+}\) is reduced to silver metal \(\text{Ag}\).

**Step 2: Write the half-reactions.**
- Oxidation half-reaction:
\[ \text{Fe}^{2+} \rightarrow \text{Fe}^{3+} + e^{-} \]
- Reduction half-reaction:
\[ \text{Ag}^{+} + e^{-} \rightarrow \text{Ag} \]

**Step 3: Find the standard reduction potentials for each half-reaction.**
- Given:
- \(E^{\Theta}(\text{Fe}^{3+}/\text{Fe}^{2+}) = 0.77 \, \text{V}\)
- \(E^{\Theta}(\text{Ag}^{+}/\text{Ag}) = 0.80 \, \text{V}\)

**Step 4: Convert the reduction potential of the oxidation half-reaction.**
- The oxidation potential for \(\text{Fe}^{2+} \rightarrow \text{Fe}^{3+}\) is the negative of the reduction potential:
\[ E^{\Theta}(\text{Fe}^{2+}/\text{Fe}^{3+}) = -E^{\Theta}(\text{Fe}^{3+}/\text{Fe}^{2+}) = -0.77 \, \text{V} \]

**Step 5: Calculate the standard cell potential \(E^{\Theta}_{cell}\).**
- The overall cell potential is calculated by adding the oxidation potential of iron and the reduction potential of silver:
\[ E^{\Theta}_{cell} = E^{\Theta}(\text{Ag}^{+}/\text{Ag}) + E^{\Theta}(\text{Fe}^{2+}/\text{Fe}^{3+}) \]
\[ E^{\Theta}_{cell} = 0.80 \, \text{V} + (-0.77 \, \text{V}) \]
\[ E^{\Theta}_{cell} = 0.80 \, \text{V} - 0.77 \, \text{V} \]
\[ E^{\Theta}_{cell} = 0.03 \, \text{V} \]

**Final Answer:**
The standard cell potential \(E^{\Theta}_{cell}\) for the reaction is \(0.03 \, \text{V}\).

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