Consider the addition of the following half-reactions: (1) Fe^3+(a q)+3 e^-⟶Fe(s) E1^∘=-0

step 1 There are two half -cells reaction in the first reaction. On the first reaction, ferric ion can

Consider the addition of the following half-reactions: ...

Consider the addition of the following half-reactions: $$\begin{array}{ll}{\text { (1) } \mathrm{Fe}^{3+}(a q)+3 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s)} & {E_{1}^{\circ}=-0.04 \mathrm{V}} \\ {\text { (2) } \mathrm{Fe}(s) \longrightarrow \mathrm{Fe}^{2+}(a q)+2 \mathrm{e}^{-}} & {E_{2}^{\circ}=0.45 \mathrm{V}} \\ {\text { (3) } \mathrm{Fe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{Fe}^{2+}(a q)} & {E_{3}^{\circ}=?}\end{array}$$ (a) Starting with the relationship between $\Delta G^{\circ}$ and $E^{\circ},$ derive a general equation that relates the $E^{\circ}$ values for half-reactions $(1),(2),$ and $(3) .$ (b) Calculate the value of $E_{3}^{\circ}$ for the $\mathrm{Fe}^{3+} / \mathrm{Fe}^{2+} / \mathrm{Fe}^{2+}$ half-reaction. (c) Explain why the $E^{\circ}$ values would be additive $\left(E_{3}^{\circ}=E_{1}^{\circ}+E_{2}^{\circ}\right)$ if reaction $(3)$ were an overall cell reaction rather than a half-reaction.

Consider the addition of the following half-reactions:
$$\begin{array}{ll}{\text { (1) } \mathrm{Fe}^{3+}(a q)+3 \mathrm{e}^{-} \longrightarrow \mathrm{Fe}(s)} & {E_{1}^{\circ}=-0.04 \mathrm{V}} \\ {\text { (2) } \mathrm{Fe}(s) \longrightarrow \mathrm{Fe}^{2+}(a q)+2 \mathrm{e}^{-}} & {E_{2}^{\circ}=0.45 \mathrm{V}} \\ {\text { (3) } \mathrm{Fe}^{3+}(a q)+\mathrm{e}^{-} \longrightarrow \mathrm{Fe}^{2+}(a q)} & {E_{3}^{\circ}=?}\end{array}$$
(a) Starting with the relationship between $\Delta G^{\circ}$ and $E^{\circ},$ derive a general equation that relates the $E^{\circ}$ values for half-reactions $(1),(2),$ and $(3) .$
(b) Calculate the value of $E_{3}^{\circ}$ for the $\mathrm{Fe}^{3+} / \mathrm{Fe}^{2+} / \mathrm{Fe}^{2+}$ half-reaction.
(c) Explain why the $E^{\circ}$ values would be additive $\left(E_{3}^{\circ}=E_{1}^{\circ}+E_{2}^{\circ}\right)$ if reaction $(3)$ were an overall cell reaction rather than a half-reaction.

Key Concepts

Standard Electrode Potentials

Standard electrode potentials (E°) are a measure of the tendency of a chemical species to be reduced, measured under standard conditions. They provide a benchmark to compare the intrinsic redox properties of different half-reactions and are related to the Gibbs free energy change of the reaction via the equation ?G° = -nFE°. This concept is fundamental in understanding the spontaneity of redox reactions and is used to predict the direction of electron flow in electrochemical cells.

Relationship between Gibbs Free Energy and Electrode Potential

The relationship ?G° = -nFE° connects thermodynamics and electrochemistry by linking the Gibbs free energy change (?G°) of a reaction to the number of electrons transferred (n), Faraday’s constant (F), and the electrode potential (E°). This equation is central to calculating the overall spontaneity of redox reactions, as free energies of individual half-reactions can be added, which in turn allows for the determination of an overall cell potential when the corresponding electron counts are taken into account.

Additivity of Free Energies in Combining Reactions

When combining half-reactions to form an overall reaction, the Gibbs free energy changes (?G°) are additive, according to Hess's Law. Since the electrode potential is related to free energy by the nFE° factor, the overall electrode potential is not directly additive unless the electron transfer numbers are properly considered. This concept is key when analyzing systems where multiple half-reactions need to be summed to give the total free energy change, allowing one to derive relationships among the potentials of the constituent half-reactions.

Distinction between Half-Reactions and Overall Cell Reactions

Half-reactions represent the oxidation or reduction processes occurring at individual electrodes, whereas overall cell reactions combine these half-reactions to describe the net redox process in an electrochemical cell. In overall cell reactions, the potentials contribute in a weighted manner based on the number of electrons transferred; hence, their additive properties (E° values) may be understood differently when comparing individual half-reactions with net cell reactions. This distinction is critical when building a complete picture of electron transfer processes in electrochemistry.

Please give Ace some feedback