Question

Consider the galvanic cell based on the following halfreactions: $$ \begin{array}{ll} \mathrm{Zn}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Zn} & \mathscr{E}^{\circ}=-0.76 \mathrm{~V} \\ \mathrm{Fe}^{2+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe} & \mathscr{E}^{\circ}=-0.44 \mathrm{~V} \end{array} $$ a. Determine the overall cell reaction and calculate \(\mathscr{E}_{\text {cell }}^{\circ}\) b. Calculate \(\Delta G^{\circ}\) and \(K\) for the cell reaction at \(25^{\circ} \mathrm{C}\). c. Calculate \(\mathscr{E}_{\text {cell }}\) at \(25^{\circ} \mathrm{C}\) when \(\left[\mathrm{Zn}^{2+}\right]=0.10 M\) and \(\left[\mathrm{Fe}^{2+}\right]=1.0 \times 10^{-5} M\)

Short answer

a. The overall cell reaction is Zn + Fe^2+ -> Zn^2+ + Fe and the standard cell potential (\(E_{cell}^°\)) is 0.32 V. b. The Gibbs free energy change (\(\Delta G^°\)) for the cell reaction at 25°C is -61750 J/mol and the equilibrium constant (K) is \(3.86 \times 10^{13}\). c. The cell potential (\(E_{cell}\)) under the given concentrations at 25°C is 0.318 V.

Step-by-step solution

Determine the overall cell reaction

To find the overall cell reaction, we will make the following considerations: (a) the Zn half-reaction has a more negative standard cell potential, so it's the anode and the Fe half-reaction is the cathode; and (b) both half-reactions have the same number of transferred electrons, so they can be directly combined: Zn -> Zn^2+ + 2e- (anode) Fe^2+ + 2e- -> Fe (cathode) Adding both half-reactions, we get the overall cell reaction: $$ \mathrm{Zn}+\mathrm{Fe}^{2+} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Fe} $$#

Calculate the standard cell potential

The standard cell potential can be calculated by finding the difference between the standard reduction potentials of the cathode and the anode: $$ \mathscr{E}_{\text {cell }}^{\circ}=\mathscr{E}_{\text {cathode }}^{\circ}-\mathscr{E}_{\text {anode }}^{\circ} $$ Substituting the given standard reduction potentials: $$ \mathscr{E}_{\text {cell }}^{\circ}=(-0.44 \,V)-(-0.76 \,V)=0.32 \,\mathrm{V} $$

Calculate the Gibbs free energy change

Using the relationship between the standard cell potential and Gibbs free energy change, we can find ΔG°: $$ \Delta G^{\circ}=-n F \mathscr{E}_{\text {cell }}^{\circ} $$ Substituting the values for n (electrons transferred), F (Faraday's constant = 96485 C/mol), and the calculated standard cell potential: $$ \Delta G^{\circ} = -2 \times 96485 \times 0.32 \,\mathrm{J/mol} = -61750 \,\mathrm{J/mol} $$

Calculate the equilibrium constant K

Using the relationship between ΔG° and the equilibrium constant K, we can find K: $$ \Delta G^{\circ}=-RT \ln K $$ Solving for K and substituting the values for ΔG°, R (gas constant = 8.314 J/mol K), and T (temperature = 25°C = 298 K): $$ K = e^{\frac{-\Delta G^{\circ}}{RT}} = e^{\frac{61750}{8.314 \times 298}} = 3.86 \times 10^{13} $$

Calculate the cell potential under given concentrations

Using the Nernst Equation, we can find the cell potential under given concentrations: $$ \mathscr{E}_{\text {cell }}=\mathscr{E}_{\text {cell }}^{\circ}-\frac{RT}{nF}\ln Q $$ Where Q is the reaction quotient and has the form: $$ Q = \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Fe}^{2+}]} $$ Substituting the given concentrations: $$ Q = \frac{0.10}{1.0 \times 10^{-5}} = 1.0 \times 10^{4} $$ Now, substituting the values for the standard cell potential, R, T, n, F, and Q into the Nernst Equation: $$ \mathscr{E}_{\text {cell }}=0.32 - \frac{8.314 \times 298}{2 \times 96485} \ln{(1.0 \times 10^{4})} = 0.318 \,\mathrm{V} $$ To summarize the results: a. The overall cell reaction: Zn + Fe^2+ -> Zn^2+ + Fe b. Standard cell potential: 0.32 V c. Gibbs free energy change: -61750 J/mol d. Equilibrium constant: 3.86 x 10^13 e. Cell potential under given concentrations: 0.318 V

Key concepts

Standard Cell Potential

In the realm of electrochemistry, the term \textbf{standard cell potential}, denoted as \( \mathscr{E}_{\text{cell}}^{\circ} \), is a critical concept that defines the potential difference between two half-cells when they are connected. This measurement is taken under standard conditions, which typically means solutes at 1 M concentration, gases at 1 atm pressure, and a temperature of 25°C (298 K).

The standard cell potential hints at how likely a reaction is to occur; a positive value suggests a spontaneous reaction, while a negative one indicates non-spontaneity. In our galvanic cell example, the half-reactions involving Zn and Fe have their own potentials, and the difference between the cathode and anode potentials gives the overall cell potential, which is positive in this case, indicating a favorable reaction.

Gibbs Free Energy

Gibbs free energy, \( \Delta G^{\circ} \), is an extensive property used to predict the direction of chemical processes and indicate spontaneity at constant temperature and pressure. A negative value of \( \Delta G^{\circ} \) indicates that a process will proceed spontaneously. The link with electrochemistry is provided through the equation \( \Delta G^{\circ} = -n F \mathscr{E}_{\text{cell}}^{\circ} \), where \( n \) is the number of moles of electrons exchanged, and \( F \) is the Faraday's constant.

The calculation for our example yields a negative Gibbs free energy, which corroborates the standard cell potential indication that the reaction should spontaneously occur under standard conditions.

Equilibrium Constant

The equilibrium constant, represented by \( K \), reflects the ratio of the concentration of products to reactants at equilibrium for a particular reaction. It is intimately connected to the Gibbs free energy through the relationship \( \Delta G^{\circ} = -RT \ln K \), with \( R \) being the universal gas constant and \( T \) the temperature in Kelvin.

In our example, we detailed the process of deriving \( K \) from \( \Delta G^{\circ} \), ending up with an extremely large value. This sizable equilibrium constant further supports that the reaction strongly favors the production of \( \text{Zn}^{2+} \) and \( \text{Fe} \), corresponding to a predominance of products at equilibrium.

Nernst Equation

The Nernst Equation is a fundamental equation in electrochemistry that allows us to calculate the cell potential \( (\mathscr{E}_{\text{cell}}) \) under non-standard conditions. It factors in the concentration of reactants and products using the reaction quotient \( Q \). The equation is usually expressed as \( \mathscr{E}_{\text{cell}}=\mathscr{E}_{\text{cell}}^{\circ}-\frac{RT}{nF}\ln Q \), adapting the standard cell potential based on actual conditions.

For our galvanic cell, when we substituted the given concentrations into the Nernst Equation, we found a minimal change in cell potential from standard conditions to the given conditions. This example illustrates how concentrations can influence cell potential, albeit to a small degree in this case due to the logarithmic nature of the Nernst Equation.

Electrochemical Cells

Electrochemical cells, such as the galvanic cell in our exercise, are devices that convert chemical energy into electrical energy through redox reactions. These cells consist of two half-cells, each representing a half-reaction, that are connected via a salt bridge or porous partition to maintain electrical neutrality. They are broadly used in batteries, corrosion prevention, and metallurgical processes, among other applications.

In our example, the galvanic cell uses a zinc half-cell and an iron half-cell, with the flow of electrons generating a potential that we've calculated using the concepts of standard cell potential and the Nernst Equation. The study of such electrochemical reactions is not only fascinating but also serves as a foundation for innovation in energy storage and conversion technologies.