Problem 23 - Chemistry: Principles and Reactions

Question

Calculate \(E^{\circ}\) for the following cells: (a) \(\mathrm{Mn}\left|\mathrm{Mn}^{2+} \| \mathrm{H}^{+}\right| \mathrm{H}_{2} \mid \mathrm{Pt}\) (b) \(\mathrm{Au}\left|\mathrm{AuCl}_{4}^{-} \| \mathrm{Co}^{3+}, \mathrm{Co}^{2+}\right| \mathrm{Pt}\) (c) \(\mathrm{Pt}\left|\mathrm{S}^{2-}\right| \mathrm{S} \| \mathrm{NO}_{3}^{-}|\mathrm{NO}| \mathrm{Pt} \quad\) (basic medium)

Step-by-Step Solution

Answer

In summary, the standard cell potentials for the given electrochemical cells are: Cell (a): \(E^{\circ}_\text{cell(a)} = 1.18\,\mathrm{V}\) Cell (b): \(E^{\circ}_\text{cell(b)} = 1.21\,\mathrm{V}\) Cell (c): \(E^{\circ}_\text{cell(c)} = 1.456\,\mathrm{V}\)

Step 1: Identify the half-cell reactions

The given cell describes the following half-cell reactions: 1. \(\mathrm{Mn} \rightarrow \mathrm{Mn}^{2+} + 2e^-\) (Oxidation half-cell) 2. \(\mathrm{H}^{+} + e^- \rightarrow \frac{1}{2} \mathrm{H}_{2}\) (Reduction half-cell)

Step 2: Find the standard reduction potentials

The standard reduction potentials can be found in a standard reduction potential table: 1. \(\mathrm{Mn}^{2+} + 2e^- \rightarrow \mathrm{Mn}\): \(E^{\circ} = -1.18\,\mathrm{V}\) 2. \(\mathrm{H}^{+} + e^- \rightarrow \frac{1}{2} \mathrm{H}_{2}\): \(E^{\circ} = 0\,\mathrm{V}\)

Step 3: Calculate the standard cell potential

Now we can calculate the standard cell potential by subtracting the standard reduction potentials of the oxidation reaction from the reduction reaction: \(E^{\circ}_\text{cell(a)} = E^{\circ}_\text{reduction} - E^{\circ}_\text{oxidation} = 0\,\mathrm{V} - (-1.18\,\mathrm{V}) = \boxed{1.18\,\mathrm{V}}\) _Cell (b):_ Au|AuCl₄⁻||Co³⁺, Co²⁺|Pt

Step 1: Identify the half-cell reactions

The given cell describes the following half-cell reactions: 1. \(\mathrm{AuCl}_{4}^{-} + e^- \rightarrow \mathrm{Au} + 4\mathrm{Cl}^-\) (Reduction half-cell) 2. \(\mathrm{Co}^{3+} + e^- \rightarrow \mathrm{Co}^{2+}\) (Reduction half-cell)

Step 2: Find the standard reduction potentials

The standard reduction potentials can be found in a standard reduction potential table: 1. \(\mathrm{AuCl}_{4}^{-} + e^- \rightarrow \mathrm{Au} + 4\mathrm{Cl}^-\): \(E^{\circ} = 0.93\,\mathrm{V}\) 2. \(\mathrm{Co}^{3+} + e^- \rightarrow \mathrm{Co}^{2+}\): \(E^{\circ} = -0.28\,\mathrm{V}\)

Step 3: Calculate the standard cell potential

Now we can calculate the standard cell potential by subtracting the standard reduction potentials of the oxidation reaction from the reduction reaction: \(E^{\circ}_\text{cell(b)} = E^{\circ}_\text{reduction} - E^{\circ}_\text{oxidation} = 0.93\,\mathrm{V} - (-0.28\,\mathrm{V}) = \boxed{1.21\,\mathrm{V}}\) _Cell (c):_ Pt|S²⁻|S||NO₃⁻|NO|Pt (basic medium)

Step 1: Identify the half-cell reactions

The given cell describes the following half-cell reactions: 1. \(\mathrm{S}^{2-} \rightarrow \mathrm{S} + 2e^-\) (Oxidation half-cell) 2. \(\mathrm{NO}_{3}^{-} + 3e^- + 3\mathrm{OH}^- \rightarrow \mathrm{NO} + 2\mathrm{H}_{2}\,\mathrm{O}\) (Reduction half-cell, balanced in basic medium)

Step 2: Find the standard reduction potentials

The standard reduction potentials can be found in a standard reduction potential table: 1. \(\mathrm{S}^{2-} \rightarrow \mathrm{S} + 2e^-\): \(E^{\circ} = -0.48\,\mathrm{V}\) 2. \(\mathrm{NO}_{3}^{-} + 3e^- + 3\mathrm{OH}^- \rightarrow \mathrm{NO} + 2\mathrm{H}_{2}\,\mathrm{O}\): \(E^{\circ} = 0.976\,\mathrm{V}\)

Step 3: Calculate the standard cell potential

Now we can calculate the standard cell potential by subtracting the standard reduction potentials of the oxidation reaction from the reduction reaction: \(E^{\circ}_\text{cell(c)} = E^{\circ}_\text{reduction} - E^{\circ}_\text{oxidation} = 0.976\,\mathrm{V} - (-0.48\,\mathrm{V}) = \boxed{1.456\,\mathrm{V}}\)

Key Concepts

Standard Cell Potential

In electrochemistry, the standard cell potential, denoted as \(E^{\circ}_{\text{cell}}\), is a measure of the voltage potential difference between two half-cells in an electrochemical cell under standard conditions. These conditions include concentrations of 1 M for each ion participating in the reaction, a pressure of 1 atm for any gases, and a temperature of 25°C.

The standard cell potential is calculated by taking the difference between the standard reduction potentials of the cathode and anode half-cell reactions. This can be mathematically expressed as:

• \(E^{\circ}_{\text{cell}} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}}\)

The sign of \(E^{\circ}_{\text{cell}}\) indicates the spontaneity of the reaction. A positive \(E^{\circ}_{\text{cell}}\) value means the reaction can occur spontaneously, making it favorable. A negative value would imply a non-spontaneous reaction under standard conditions.

Half-Cell Reactions

Half-cell reactions are the individual oxidation or reduction reactions that occur in the separate sections (half-cells) of an electrochemical cell. These reactions show how electrons are transferred between the species.

Each half-cell contains two components:

• An oxidizing agent, which gains electrons, thereby undergoing reduction.
• A reducing agent, which loses electrons, thereby undergoing oxidation.

The movement of electrons from the reducing agent in one half-cell to the oxidizing agent in the other half-cell generates an electric current. This flow of electrons is harnessed in batteries and other electrical devices. Remember, each half-reaction should be balanced to ensure that the number of electrons lost in the oxidation process equals the number gained in the reduction process.

Reduction Potentials

Reduction potentials, expressed in volts, are measures of the tendency of a chemical species to acquire electrons and thus be reduced. They are essential in determining voltages for cells and predicting the direction of redox reactions.

Each reduction potential corresponds to a specific half-reaction, which is typically referenced from a standard hydrogen electrode (SHE), assigned a potential of 0 V. Positive reduction potentials indicate a greater tendency for a species to gain electrons, while negative values suggest a lower tendency.

• For example, a high positive value indicates a strong oxidizing agent.
• Conversely, a low or negative value indicates a strong reducing agent.

Reduction potential tables are invaluable in any electrochemical analysis as they help identify which substances will be reduced and which will be oxidized under standard conditions.

Nernst Equation

The Nernst Equation allows the calculation of cell potential under non-standard conditions. It takes into account the concentrations of the reacting ions, temperature, and the number of electrons transferred in the redox reaction.

The Nernst Equation is written as:

• \(E = E^{\circ}_{\text{cell}} - \frac{RT}{nF} \ln Q\)

Where:

• \(E\) is the cell potential at non-standard conditions,
• \(E^{\circ}_{\text{cell}}\) is the standard cell potential,
• \(R\) is the universal gas constant (8.314 J/mol·K),
• \(T\) is the temperature in Kelvin,
• \(n\) is the number of moles of electrons transferred,
• \(F\) is Faraday's constant (96485 C/mol),
• \(Q\) is the reaction quotient.

This equation is pivotal for understanding and calculating the voltage output of an electrochemical cell in real-world conditions.