Question

Consider the following galvanic cell at \(25^{\circ} \mathrm{C} :\) $$\text { Pt }\left|\mathrm{Cr}^{2+}(0.30 M), \mathrm{Cr}^{3+}(2.0 M)\right|\left|\mathrm{Co}^{2+}(0.20 M)\right| \mathrm{Co}$$ The overall reaction and equilibrium constant value are $$2 \mathrm{Cr}^{2+}(a q)+\mathrm{Co}^{2+}(a q) \rightleftharpoons_{2 \mathrm{Cr}^{3+}}(a q)+\mathrm{Co}(s) \quad K=2.79 \times 10^{7}$$ Calculate the cell potential, \(\mathscr{E},\) for this galvanic cell and \(\Delta G\) for the cell reaction at these conditions.

Short answer

The cell potential for the given galvanic cell is 1.70 V, and the change in Gibbs free energy for the cell reaction is -3.28 x 10⁵ J/mol.

Step-by-step solution

Identify the Half-Reactions

First, let's identify the half-reactions involved. The anode half-reaction has a Cr²⁺/Cr³⁺ couple, and the cathode half-reaction has a Co/Co²⁺ couple. Anode: \(Cr^{2+} \to Cr^{3+} + e^-\) Cathode: \(Co^{2+} + 2e^- \to Co\)

Write the balanced reaction and overall cell reaction

To obtain the overall reaction, we need to balance the electrons in both half-reactions. \(2\ [Cr^{2+} \to Cr^{3+} + e^-]\) \(1\ [Co^{2+} + 2e^- \to Co]\) Now, we can write the overall cell reaction: \(2\ Cr^{2+}(aq) + Co^{2+}(aq) \to 2\ Cr^{3+}(aq) + Co(s)\)

Calculate the cell potential using the Nernst equation

Now, let's use the Nernst equation: \(\mathscr{E} = \mathscr{E}^\circ - \frac{RT}{zF} \ln{Q}\) where \(\mathscr{E}^\circ\) is the standard cell potential, R is the gas constant, T is the absolute temperature, z is the number of electrons transferred, F is the Faraday constant, and Q is the reaction quotient. Given the equilibrium constant, we can calculate the standard cell potential using: \(\Delta G^\circ = -RT \ln{K}\) \(\mathscr{E}^\circ = \frac{-\Delta G^\circ}{zF}\) First, we need to find \(\Delta G^\circ\): From the information given, K = 2.79 x 10⁷, T = 298 K, n = 2, F = 96,485 C/mol \(\Delta G^\circ = -RT \ln{K}\) \(\Delta G^\circ = -(8.314\ \mathrm{J/K \cdot mol}) (298\ \mathrm{K}) \ln{(2.79 \times 10^{7})}\) \(\Delta G^\circ = -3.66 \times 10^5\ \mathrm{J/mol}\) Now, let's find \(\mathscr{E}^\circ\): \(\mathscr{E}^\circ = \frac{-\Delta G^\circ}{zF}\) \(\mathscr{E}^\circ = \frac{-(-3.66 \times 10^5\ \mathrm{J/mol})}{(2)(96,485\ \mathrm{C/mol})}\) \(\mathscr{E}^\circ = 1.90\ \mathrm{V}\) Next, we will calculate the reaction quotient, Q: \(Q = \frac{[\mathrm{Cr}^{3+}]^2}{ [\mathrm{Cr}^{2+}]^2 [\mathrm{Co}^{2+}]} = \frac{(2.0\ \mathrm{M})^2}{(0.30\ \mathrm{M})^2(0.20\ \mathrm{M})}\) Now, we can find the cell potential \(\mathscr{E}\): \(\mathscr{E} = 1.90\ \mathrm{V} - \frac{(8.314\ \mathrm{J/K \cdot mol}) (298\ \mathrm{K})}{(2)(96,485\ \mathrm{C/mol})} \ln{Q}\) \(\mathscr{E} = 1.90\ \mathrm{V} - 0.198\ \mathrm{V}\) \(\mathscr{E} = 1.70\ \mathrm{V}\)

Calculate the change in Gibbs free energy

Finally, we can calculate the change in Gibbs free energy, \(\Delta G\), using the cell potential \(\mathscr{E}\): \(\Delta G = -zF\mathscr{E}\) \(\Delta G = -(2)(96,485\ \mathrm{C/mol})(1.70\ \mathrm{V})\) \(\Delta G = -3.28 \times 10^5\ \mathrm J/mol\) Therefore, the cell potential for this galvanic cell is 1.70 V, and the change in Gibbs free energy for the cell reaction is -3.28 x 10⁵ J/mol.

Key concepts

Galvanic Cell

A galvanic cell, also known as a voltaic cell, is a type of electrochemical cell that converts chemical energy into electrical energy through a spontaneous redox reaction. In a galvanic cell, two different metals or metal ions undergo oxidation and reduction reactions at separate electrodes.

These reactions occur in separate half-cells, which are connected by a salt bridge that allows for the flow of ions. The anode is where oxidation occurs, losing electrons, while the cathode is where reduction takes place, gaining electrons.

This cell generates an electric current as electrons flow from the anode to the cathode through an external circuit. The generated voltage or cell potential is a result of the difference in electrochemical potential between the two electrodes.

Nernst Equation

The Nernst equation is a fundamental tool in electrochemistry that relates the cell potential to the concentration of reactants and products in a galvanic cell. This equation allows for the calculation of the electromotive force (emf) under non-standard conditions.

The equation is expressed as:
\[\mathscr{E} = \mathscr{E}^\circ - \frac{RT}{zF} \ln{Q}\]
where:

• \(\mathscr{E}\) is the cell potential under non-standard conditions.
• \(\mathscr{E}^\circ\) is the standard cell potential.
• \(R\) is the universal gas constant (8.314 J/K·mol).
• \(T\) is the temperature in Kelvin.
• \(F\) is Faraday's constant (96,485 C/mol).
• \(z\) is the number of moles of electrons transferred in the reaction.
• \(Q\) is the reaction quotient.

This equation demonstrates how the potential of a reaction changes with varying concentrations of reactants and products. It is especially useful for calculating the potential of a cell under different conditions from the standard state.

Gibbs Free Energy

Gibbs Free Energy (\(\Delta G\)) is a thermodynamic quantity that represents the maximum reversible work a system can perform at constant temperature and pressure. It helps determine the spontaneity of a reaction.

For galvanic cells, Gibbs Free Energy is directly related to the cell potential by the equation:
\[\Delta G = -zF\mathscr{E}\]
where:

• \(\Delta G\) is the change in Gibbs Free Energy.
• \(z\) is the number of moles of electrons exchanged.
• \(F\) is Faraday's constant.
• \(\mathscr{E}\) is the cell potential.

A negative \(\Delta G\) indicates that the reaction is spontaneous, meaning it can occur without any additional input of energy. For the given galvanic cell, the negative value confirms the spontaneous nature of the reaction.

Cell Potential

Cell potential, also known as electromotive force (emf, \(\mathscr{E}\)), is the measure of the voltage produced by a galvanic cell. It arises from the differences in potential energy between the electrodes due to their tendency to gain or lose electrons.

The cell potential under standard conditions (denoted as \(\mathscr{E}^\circ\)) can be measured when all reactants and products are in their standard states, typically 1 M concentration for solutions and 1 atm pressure for gases.

In practice, the actual cell potential can differ from the standard potential as it depends on the concentrations of the ions involved, temperature, and pressure. The Nernst equation is used to adjust the standard potential to the actual, non-standard conditions.

This potential is important because it indicates the direction in which a chemical reaction can proceed spontaneously and provides a way to utilize chemical reactions to generate electric current, as seen in devices like batteries.