Which is the correct Nernst equation at 25^oC involving the reaction Fe³⁺ + e^- → Fe²⁺ ?

E = Eo - 0.059ln\(\frac{[Fe^{2+}]}{[Fe^{3+}]}\)

The correct answer is option 4. \(E = Eo - 0.059 ln \frac{[Fe^{2+}]}{[Fe^{3+}]}\).

The Nernst equation describes the relationship between the standard electrode potential (\( E^o \)) and the actual electrode potential (\( E \)) under non-standard conditions.

For the reaction \( \text{Fe}^{3+} + e^- \rightarrow \text{Fe}^{2+} \), we're dealing with a reduction reaction where one electron is gained by \( \text{Fe}^{3+} \) to form \( \text{Fe}^{2+} \).

The Nernst equation for this reaction is:

\[E = E^o - \frac{0.059}{n} \ln\left( \frac{[\text{Fe}^{2+}]}{[\text{Fe}^{3+}]} \right) \]

Here's what each part of the equation represents:

\( E \) is the cell potential under non-standard conditions.

\( E^o \) is the standard cell potential, which is the potential difference measured under standard conditions (usually 1 M concentration, 1 atm pressure, and 25°C).

\( n \) is the number of moles of electrons transferred in the reaction. In this case, it's 1, because one electron is transferred.

\( 0.059 \) is the value representing the conversion between natural logarithms (ln) and base 10 logarithms (log) in electrochemistry (at 25°C). This value is derived from the gas constant (\( R \)) and Faraday's constant (\( F \)).

\( \ln \) denotes the natural logarithm.

\( [\text{Fe}^{2+}] \) and \( [\text{Fe}^{3+}] \) are the concentrations of \(Fe^{2+}\) and \(Fe^{3+}\) ions, respectively.

For the given reaction \(n = 1\), thus

\[E = E^o - 0.059\ln \left( \frac{[\text{Fe}^{2+}]}{[\text{Fe}^{3+}]} \right) \]

The Nernst equation allows us to calculate the cell potential under non-standard conditions (different concentrations, temperatures, or pressures) by taking into account the concentrations of the ions involved in the redox reaction.