Question

Based on the following standard reduction potentials: $$\begin{aligned}\mathrm{Fe}^{2+}(a q)+2 e^{-} & \longrightarrow \mathrm{Fe}(s) & & E_{1}^{\circ}=-0.44 \mathrm{~V} \\ \mathrm{Fe}^{3+}(a q)+e^{-} \longrightarrow \mathrm{Fe}^{2+}(a q) & E_{2}^{\circ} &=0.77 \mathrm{~V} \end{aligned}$$ calculate the standard reduction potential for the halfreaction:$$\mathrm{Fe}^{3+}(a q)+3 e^{-} \longrightarrow \mathrm{Fe}(s) \quad E_{3}^{\circ}=?$$

Short answer

The standard reduction potential for \(\mathrm{Fe}^{3+}(aq) + 3e^{-} \rightarrow \mathrm{Fe}(s)\) is 0.33 V.

Step-by-step solution

Identify the Half-Reactions

We are given two half-reactions: 1) \( \mathrm{Fe}^{2+}(aq) + 2e^{-} \longrightarrow \mathrm{Fe}(s) \) with \( E_{1}^{\circ} = -0.44 \mathrm{~V} \) and 2) \( \mathrm{Fe}^{3+}(aq) + e^{-} \longrightarrow \mathrm{Fe}^{2+}(aq) \) with \( E_{2}^{\circ} = 0.77 \mathrm{~V} \). We need to find the potential for the reaction \( \mathrm{Fe}^{3+}(aq) + 3e^{-} \longrightarrow \mathrm{Fe}(s) \).

Understand the Combined Reaction

To derive the needed half-reaction, combine the two given reactions such that electrons cancel out and lead to the final desired equation \( \mathrm{Fe}^{3+}(aq) + 3e^{-} \rightarrow \mathrm{Fe}(s) \). Do this by adding the given reactions in a way that the electrons are balanced.

Balance Electrons

Multiply the first equation by 1 (no change needed) and the second equation by 2: \( 2\mathrm{Fe}^{3+}(aq) + 2e^{-} \longrightarrow 2\mathrm{Fe}^{2+}(aq) \). Add these adjusted reactions together: \[ \mathrm{Fe}^{3+}(aq) + 3e^{-} \longrightarrow \mathrm{Fe}(s) \].

Calculate the Overall Potential

Add the potentials of the two adjusted reactions to find the overall standard reduction potential. Using the formula: \( E^{\circ}_{\text{{total}}} = E^{\circ}_{2} + E^{\circ}_{1} \). Substitute the given values: \[ E^{\circ}_{3} = 0.77 \mathrm{~V} + (-0.44 \mathrm{~V}) = 0.33 \mathrm{~V} \].

Conclusion

The standard reduction potential for the half-reaction \( \mathrm{Fe}^{3+}(aq) + 3e^{-} \longrightarrow \mathrm{Fe}(s) \) is \( 0.33 \mathrm{~V} \).

Key concepts

Understanding Half-Reaction Calculations

Half-reaction calculations are an essential part of determining the electrochemical behavior of substances. A half-reaction shows the reduction or oxidation process occurring at an electrode. It involves either the gain of electrons (reduction) or the loss of electrons (oxidation). In the given exercise, we are focusing on reduction potentials.

The key steps in half-reaction calculations include:

• Identifying the relevant half-reactions from given data.
• Understanding how these reactions combine to achieve the desired half-reaction.
• Balancing the number of electrons to reflect the accurate electrochemical process.

Reductions, like in the reaction for converting oindent\( \text{Fe}^{3+} \) to \( \text{Fe} \) involve gaining three electrons.

By effectively combining the given reactions, you can find the unknown potential, bridging concepts of electron transfer with thermodynamic energy changes.

Exploring Fundamentals of Electrochemistry

Electrochemistry bridges the principles of chemistry and electricity, playing a key role in countless applications. The primary focus is on reactions that involve electron transfer, critical to understanding processes like battery operations and corrosion.

The driving force behind these electron transfers is the standard reduction potential, indicating how readily a species gains electrons. Reactions with a higher reduction potential occur more spontaneously.

In electrochemistry,

• Electrodes and electrolytes come together to form an electrochemical cell.
• Electrons move through an external circuit from an anode to a cathode.
• Reduction occurs at the cathode where species gain electrons.

The exercise provided utilizes these principles to calculate a standard reduction potential for a complex half-reaction, emphasizing how electrochemical balances are achieved in reactions.

Mastering Electron Balancing in Redox Reactions

Electron balancing is crucial in composing accurate chemical equations for reactions, especially redox reactions involving reduction and oxidation. For the desired reaction conversion \( \text{Fe}^{3+} + 3e^{-} \rightarrow \text{Fe} \), balancing ensures that the number of electrons lost equals those gained.

To achieve this:

• Identify how many electrons are involved in each given reaction.
• Multiply the reactions by necessary coefficients to match electron numbers.
• Add balanced reactions to reflect the overall process.

In the example, combining individual reactions with one losing electrons and the other gaining, the electrons naturally balance out at three.

Effectively balancing electrons not only simplifies calculation of the overall potential but also ensures consistency with the conservation of charge principle, making the derived potentials physically accurate.