Question

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 \(\mathrm{Fe}^{3+}(a q) .\left(\Delta G_{\mathrm{f}}^{\circ} \text { for } \mathrm{e}^{-}=0 .\right)\)

Short answer

The standard Gibbs free energies of formation for \(\mathrm{Fe}^{2+}(\mathrm{aq})\) and \(\mathrm{Fe}^{3+}(\mathrm{aq})\) ions are approximately \(8.48 \times 10^4\, \mathrm{J/mol}\) and \(1.19 \times 10^4\, \mathrm{J/mol}\) respectively.

Step-by-step solution

Write the half-reactions for the formation of iron ions

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: \[\mathrm{Fe}(s) \rightarrow \mathrm{Fe}^{3+}(\mathrm{aq}) + 3\mathrm{e}^-\]

Find the number of electrons exchanged for each half-reaction

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: \(n = 3\)

Look up the standard reduction potentials for each half-reaction

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: \(\mathscr{E}^\circ (\mathrm{Fe}^{3+}) = -0.041\,\mathrm{V}\)

Calculate the standard Gibbs free energy of formation for each iron ion

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.

Key concepts

Half-Reactions

Half-reactions are simplified representations of redox processes. They exhibit the change occurring at one electrode during the redox reaction. They isolate oxidation or reduction in the chemical equations. Here, we are dealing with the formation of iron ions:

• For the formation of Fe²⁺: Fe(s) loses two electrons to form Fe²⁺(aq).
• For the formation of Fe³⁺: Fe(s) loses three electrons to form Fe³⁺(aq).

Understanding the number of electrons lost or gained in these half-reactions helps in determining how many electrons are involved in each process. It's crucial because these values of n (number of electrons) are needed when applying them to the formula involving Gibbs free energy.

Standard Reduction Potentials

Standard reduction potentials, symbolized as \(\mathscr{E}^\circ\), are measured under standard conditions (25°C, 1 M concentration, and 1 atm pressure). They indicate how likely a species is to gain electrons (be reduced). These potentials are useful to predict the direction of redox reactions.For the iron ions:

• The standard reduction potential for Fe²⁺ is \(-0.44\, \mathrm{V}\).
• The standard reduction potential for Fe³⁺ is \(-0.041\, \mathrm{V}\).

The negative values indicate that both reactions are non-spontaneous as written, and energy is required for the reduction of iron cations. This information is vital to estimate the change in standard Gibbs free energy of formation using the formula \(\Delta G^{\circ} = -nF \mathscr{E}^\circ\).

Faraday’s Constant

Faraday's constant is a critical number in electrochemistry. It represents the charge of one mole of electrons, which is approximately \(96485 \, \mathrm{C/mol}\). This constant allows conversion between moles of electrons and their corresponding electric charge. It's used in calculating Gibbs free energy changes in redox reactions.In the formula \(\Delta G^{\circ} = -nF \mathscr{E}^\circ\), Faraday's constant helps to convert the product of electron moles (n) and voltage (\(\mathscr{E}^\circ\)) into an energy quantity in Joules, providing a measure of the energy change in the formation of ions.

Formation of Iron Ions

The formation of iron ions involves converting metallic iron into its ionic form. This redox reaction involves:

• Fe turning into Fe²⁺, involving the loss of two electrons.
• Fe turning into Fe³⁺, involving the loss of three electrons.

Calculating the Gibbs free energy for these conversions helps to understand the energy change associated with each reaction.Using the determined values for \(n\), \(\mathscr{E}^\circ\), and Faraday's constant, you can calculate the standard Gibbs free energy of formation:

• For Fe²⁺: \(\Delta G_{\mathrm{f}}^{\circ} \approx 8.48 \times 10^4 \, \mathrm{J/mol}\)
• For Fe³⁺: \(\Delta G_{\mathrm{f}}^{\circ} \approx 1.19 \times 10^4 \, \mathrm{J/mol}\)

These values represent the energy required to form the ions under standard conditions, providing insight into the feasibility and energy requirements of the reactions.