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

Short Answer

Expert verified
The standard Gibbs free energy of formation (螖Gf掳) for Fe虏鈦(aq) and Fe鲁鈦(aq) can be calculated using the equation 螖G掳 = -nF鈩奥 and the standard reduction potentials. The balanced half-reactions are: \(Fe^{2+}(aq) + 2e^- \rightarrow Fe(s)\) with \(E掳_{Fe^{2+}/Fe} = -0.44 V\) \(Fe^{3+}(aq) + 3e^- \rightarrow Fe(s)\) with \(E掳_{Fe^{3+}/Fe} = -0.036 V\) Using these values, we find that the 螖Gf掳 for Fe虏鈦(aq) is approximately 84965.6 J/mol and the 螖Gf掳 for Fe鲁鈦(aq) is approximately 10460.42 J/mol.

Step by step solution

01

Write the balanced half-reactions for Fe虏鈦 and Fe鲁鈦

The balanced half-reactions for Fe虏鈦 and Fe鲁鈦 are: \(Fe^{2+}(aq) + 2e^- \rightarrow Fe(s)\) \(Fe^{3+}(aq) + 3e^- \rightarrow Fe(s)\) Now we have the half-reactions, and we know the number of electrons involved (n) in each reaction.
02

Find the standard reduction potentials for the half-reactions

You can find the standard reduction potentials for these half-reactions in a standard reduction potential table: For Fe虏鈦: \(E掳_{Fe^{2+}/Fe} = -0.44 V\) For Fe鲁鈦: \(E掳_{Fe^{3+}/Fe} = -0.036 V\)
03

Calculate the 螖Gf掳 for Fe虏鈦 and Fe鲁鈦

Use the equation 螖G掳 = -nF鈩奥 to find the 螖Gf掳 for each species. Faraday's constant (F) is equal to 96485 C/mol. For Fe虏鈦: 螖G掳 = -nF鈩奥 螖G掳 = -(2 mol)(96485 C/mol)(-0.44 V) 螖G掳 = 84965.6 J/mol For Fe鲁鈦: 螖G掳 = -nF鈩奥 螖G掳 = -(3 mol)(96485 C/mol)(-0.036 V) 螖G掳 = 10460.42 J/mol Thus, the 螖Gf掳 for Fe虏鈦(aq) is approximately 84965.6 J/mol and the 螖Gf掳 for Fe鲁鈦(aq) is approximately 10460.42 J/mol.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Reduction Potentials
When studying electrochemistry, understanding standard reduction potentials is crucial. These potentials measure the tendency of a chemical species to gain electrons and thus be reduced. They are typically measured in volts (V). Standard reduction potentials are determined under standard conditions, which include 25掳C, 1 atmosphere for gases, and 1 mol/L concentrations for solutions.
Consider a reduction half-reaction like the one for iron, where ions are reduced to solid iron:
  • Fe虏鈦 + 2e鈦 鈫 Fe(s)
Each half-reaction is assigned a standard reduction potential, derived from experimental measurements. In our example, the standard reduction potential for Fe虏鈦 to Fe is E掳 = -0.44 V.
These values allow us to compare the electron affinity of different ions. More positive values indicate a greater tendency to gain electrons. On the other hand, a more negative potential means the ion is less inclined to be reduced. This information enables us to predict reaction spontaneity when paired with the concept of Gibbs Free Energy.
Half-Reactions
Half-reactions break down the steps of redox (reduction-oxidation) reactions, crucial for understanding how electrons are transferred between substances. Unlike complete reactions, half-reactions separately depict the oxidation or reduction process.
In a half-reaction, you show either the loss or gain of electrons. This clarity helps in balancing reactions and calculating potentials or energies. Let鈥檚 examine the oxidation half-reaction for iron:
  • Fe(s) 鈫 Fe虏鈦 + 2e鈦
For this process, two electrons are lost as iron oxidizes from a solid form to ions. In the reduction half-reaction:
  • Fe虏鈦 + 2e鈦 鈫 Fe(s)
Here, the reverse occurs as electrons are gained. It's important to note that the number of electrons gained in the reduction half-reaction must match those lost in oxidation to maintain charge balance. By using half-reactions, chemists can derive equations describing complex chemical processes in simpler terms.
Faraday's Constant
Faraday's Constant is a key concept in electrochemistry, representing the electric charge per mole of electrons. It is one of the factors used to calculate Gibbs Free Energy changes in redox reactions.
The constant is approximately equal to 96485 C/mol. It connects the charge flow in electrochemical reactions to the amount of substance undergoing change, facilitating the calculation of energy changes in mole-based terms. In essence, it quantifies the total electric charge carried by one mole of electrons.
Applying Faraday's Constant in Gibbs Free Energy calculations, we use the formula:
  • 螖G掳 = -nF鈩奥
Here, 螖G掳 represents the change in Gibbs Free Energy, n is the number of moles of electrons transferred, and 鈩奥 is the electromotive force or cell potential.
This formula helps to quantify how much free energy is available or required for a reaction under standard conditions, thus predicting the reaction's spontaneity in electrochemical systems.

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Most popular questions from this chapter

Combine the equations $$ \Delta G^{\circ}=-n F \mathscr{C}^{\circ} \quad \text { and } \quad \Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ} $$ to derive an expression for \(\mathscr{E}^{\circ}\) as a function of temperature. Describe how one can graphically determine \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) from measurements of \(\mathscr{E}^{\circ}\) at different temperatures, assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature. What property would you look for in designing a reference half-cell that would produce a potential relatively stable with respect to temperature?

Balance the following equations by the half-reaction method. a. \(\mathrm{Fe}(s)+\mathrm{HCl}(a q) \longrightarrow \mathrm{HFeCl}_{4}(a q)+\mathrm{H}_{2}(g)\) b. \(\mathrm{IO}_{3}^{-}(a q)+\mathrm{I}^{-}(a q) \stackrel{\text { Acid }}{\longrightarrow} \mathrm{I}_{3}^{-}(a q)\) c. \(\mathrm{Cr}(\mathrm{NCS})_{6}^{4-}(a q)+\mathrm{Ce}^{4+}(a q) \stackrel{\text { Acid }}{\longrightarrow}\) \(\mathrm{Cr}^{3+}(a q)+\mathrm{Ce}^{3+}(a q)+\mathrm{NO}_{3}^{-}(a q)+\mathrm{CO}_{2}(g)+\mathrm{SO}_{4}{ }^{2-}(a q)\) d. \(\operatorname{CrI}_{3}(s)+\mathrm{Cl}_{2}(g) \stackrel{\text { Base }}{\longrightarrow}\) \(\operatorname{CrO}_{4}{ }^{2-}(a q)+\mathrm{IO}_{4}^{-}(a q)+\mathrm{Cl}^{-}(a q)\) e. \(\mathrm{Fe}(\mathrm{CN})_{6}^{4-}(a q)+\mathrm{Ce}^{4+}(a q) \stackrel{\text { Base }}{\longrightarrow}\) \(\mathrm{Ce}(\mathrm{OH})_{3}(s)+\mathrm{Fe}(\mathrm{OH})_{3}(s)+\mathrm{CO}_{3}^{2-}(a q)+\mathrm{NO}_{3}^{-}(a q)\)

An experimental fuel cell has been designed that uses carbon monoxide as fuel. The overall reaction is $$ 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g) $$ The two half-cell reactions are $$ \begin{array}{l} \mathrm{CO}+\mathrm{O}^{2-} \longrightarrow \mathrm{CO}_{2}+2 \mathrm{e}^{-} \\\ \mathrm{O}_{2}+4 \mathrm{e}^{-} \longrightarrow 2 \mathrm{O}^{2-} \end{array} $$ The two half-reactions are carried out in separate compartments connected with a solid mixture of \(\mathrm{CeO}_{2}\) and \(\mathrm{Gd}_{2} \mathrm{O}_{3} . \mathrm{Ox}-\) ide ions can move through this solid at high temperatures (about \(\left.800^{\circ} \mathrm{C}\right) . \Delta G\) for the overall reaction at \(800^{\circ} \mathrm{C}\) under certain concentration conditions is \(-380 \mathrm{~kJ}\). Calculate the cell potential for this fuel cell at the same temperature and concentration conditions.

It takes \(15 \mathrm{kWh}\) (kilowatt-hours) of electrical energy to produce \(1.0 \mathrm{~kg}\) aluminum metal from aluminum oxide by the Hall-Heroult process. Compare this to the amount of energy necessary to melt \(1.0 \mathrm{~kg}\) aluminum metal. Why is it economically feasible to recycle aluminum cans? [The enthalpy of fusion for aluminum metal is \(10.7 \mathrm{~kJ} / \mathrm{mol}(1 \mathrm{watt}=1 \mathrm{~J} / \mathrm{s}) .]\)

In 1973 the wreckage of the Civil War ironclad USS Monitor was discovered near Cape Hatteras, North Carolina. [The Monitor and the CSS Virginia (formerly the USS Merrimack) fought the first battle between iron-armored ships.] In 1987 investigations were begun to see if the ship could be salvaged. It was reported in Time (June 22,1987 ) that scientists were considering adding sacrificial anodes of zinc to the rapidly corroding metal hull of the Monitor. Describe how attaching zinc to the hull would protect the Monitor from further corrosion.

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