The equation \(\Delta G^{\circ}=-\mathrm{nF} \mathscr{E}^{\circ}\) also can be applied to half-reactions. Use standard reduction potentials to estimate \(\Delta G_{\mathrm{f}}^{\circ}\) for \(\mathrm{Fe}^{2+}(a q)\) and $\mathrm{Fe}^{3+}(a q) .\left(\Delta G_{\mathrm{f}}^{\circ} \text { for } \mathrm{e}^{-}=0 .\right)$

We need to find the half-reactions for the formation of \(\mathrm{Fe}^{2+}(\mathrm{aq})\) and \(\mathrm{Fe}^{3+}(\mathrm{aq})\) ions. The half-reactions are: For \(\mathrm{Fe}^{2+}(\mathrm{aq})\) formation: \[\mathrm{Fe}(s) \rightarrow \mathrm{Fe}^{2+}(\mathrm{aq}) + 2\mathrm{e}^-\] For \(\mathrm{Fe}^{3+}(\mathrm{aq})\) formation: \[\mathrm{Fe}(s) \rightarrow \mathrm{Fe}^{3+}(\mathrm{aq}) + 3\mathrm{e}^-\]

The number of electrons exchanged for each half-reaction is determined by the number of electrons released during the oxidation process. From the half-reactions above, we can see that: For \(\mathrm{Fe}^{2+}(\mathrm{aq})\) formation: \(n = 2\) For \(\mathrm{Fe}^{3+}(\mathrm{aq})\) formation: \(n = 3\)

We will consult a standard reduction potential table to find the standard reduction potentials (\(\mathscr{E}^\circ\)) for both half-reactions: For \(\mathrm{Fe}^{2+}\) formation: \(\mathscr{E}^\circ (\mathrm{Fe}^{2+}) = -0.44 \,\mathrm{V}\) For \(\mathrm{Fe}^{3+}\) formation: \(\mathscr{E}^\circ (\mathrm{Fe}^{3+}) = -0.041\,\mathrm{V}\)

Now, we will use the given equation, \(\Delta G^\circ = -nF\mathscr{E}^\circ\), and plug in the values determined in Steps 1-3 to calculate the standard Gibbs free energy of formation (\(\Delta G_{\mathrm{f}}^\circ\)) for each iron ion: For \(\mathrm{Fe}^{2+}(\mathrm{aq})\): \[\Delta G_{\mathrm{f}}^\circ(\mathrm{Fe}^{2+}) = -2 \times \mathrm{F} \times (-0.44\,\mathrm{V})\] For \(\mathrm{Fe}^{3+}(\mathrm{aq})\): \[\Delta G_{\mathrm{f}}^\circ(\mathrm{Fe}^{3+}) = -3 \times \mathrm{F} \times (-0.041\,\mathrm{V})\] Where Faraday's constant F is approximately \(96485\, \mathrm{C/mol}\). Calculating the \(\Delta G_{\mathrm{f}}^\circ\) values: \[\Delta G_{\mathrm{f}}^\circ(\mathrm{Fe}^{2+}) \approx 8.48 \times 10^4 \, \mathrm{J/mol}\] \[\Delta G_{\mathrm{f}}^\circ(\mathrm{Fe}^{3+}) \approx 1.19 \times 10^4 \, \mathrm{J/mol}\] Therefore, the standard Gibbs free energies of formation for \(\mathrm{Fe}^{2+}(\mathrm{aq})\) and \(\mathrm{Fe}^{3+}(\mathrm{aq})\) are approximately \(8.48 \times 10^4\, \mathrm{J/mol}\) and \(1.19 \times 10^4\, \mathrm{J/mol}\) respectively.