Question

If the equilibrium constant for a two-electron redox reaction at \(298 \mathrm{~K}\) is \(1.5 \times 10^{-4},\) calculate the corresponding \(\Delta G^{\circ}\) and \(E_{\text {red }}^{\circ}\).

Short answer

In this two-electron redox reaction with an equilibrium constant (K) of \(1.5 \times 10^{-4}\) at 298 K, we can calculate the standard Gibbs free energy change (ΔG°) using the formula ΔG° = -RT ln(K) and find that ΔG° ≈ 43625.3 J/mol. Using the Nernst equation (ΔG° = -nFE°), we can then calculate the standard reduction potential (E°) and find that E° ≈ 0.226 V.

Step-by-step solution

Calculate ΔG° using the equilibrium constant (K)

Use the formula ΔG° = -RT ln(K) to calculate the standard Gibbs free energy change: ΔG° = - (8.314 J/(mol K)) * (298 K) * ln(1.5 * 10^{-4}) Calculate the value: ΔG° ≈ 43625.3 J/mol

Calculate E° using the Nernst equation

Use the formula ΔG° = -nFE° to calculate the standard reduction potential E°. First, rearrange the equation to solve for E°: E° = - (ΔG°) / (nF) Now plug in the values we know: E° = - (43625.3 J/mol) / (2 * 96485 C/mol) Calculate the value: E° ≈ 0.226 V So, the standard Gibbs free energy change (ΔG°) is approximately 43625.3 J/mol and the standard reduction potential (E°) is approximately 0.226 V.

Key concepts

Equilibrium Constant

The equilibrium constant (K) is a dimensionless value that quantifies the ratio of concentrations of products to reactants at equilibrium for a chemical reaction. It is a critical concept in electrochemistry and thermodynamics as it helps to predict the extent to which a reaction will proceed under certain conditions. The magnitude of K can indicate whether the products or reactants are favored in a reaction. A large K (much greater than 1) signifies that the reaction strongly favors the formation of products, while a small K (much less than 1) indicates that the reactants are favored.

For example, in the given exercise, an equilibrium constant of \(1.5 \times 10^{-4}\) suggests that at equilibrium, the concentration of reactants is much higher than that of the products. This information is vital when calculating other related thermodynamic properties, like the Gibbs free energy.

Gibbs Free Energy

Gibbs free energy (\(\Delta G\)), named after Josiah Willard Gibbs, is an essential thermodynamic quantity that represents the maximum amount of work that can be performed by a system at constant temperature and pressure. It is a key determinant of spontaneity for a chemical reaction. A negative \(\Delta G\) value indicates that a process is spontaneous, while a positive value suggests non-spontaneity.

The relationship between the equilibrium constant and Gibbs free energy (\(\Delta G\)) is expressed as \(\Delta G^\circ = -RT \ln(K)\). In the given exercise, by substituting the given equilibrium constant into this equation, we can calculate the standard Gibbs free energy change for the two-electron redox reaction, helping us to understand the energy dynamics of the process.

Standard Reduction Potential

The standard reduction potential (\(E^\circ\)) is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. Its value is measured in volts (V) and is determined under standard conditions, which include a temperature of 298 K, a pressure of 1 bar, and 1 M concentrations for all solutes. Reduction potentials are fundamental to electrochemistry as they help to predict the direction of electron flow in a redox reaction.

Each half-reaction in electrochemistry has its own standard reduction potential. When comparing two half-reactions, the one with the higher standard reduction potential will more readily gain electrons and therefore act as an oxidizing agent. In the context of the exercise, the calculation of the standard reduction potential for the redox reaction gives us insight into the electron transfer process and the electrochemical properties of the reaction.

Nernst Equation

The Nernst equation enables the calculation of the cell potential under non-standard conditions by incorporating the effect of temperature and concentration on the cell's electromotive force (EMF). The equation is given as \(E = E^\circ - (RT/nF) \ln(Q)\), where E is the cell potential, \(E^\circ\) is the standard reduction potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons transferred in the reaction, F is the Faraday's constant, and Q is the reaction quotient.

The Nernst equation is a pivotal tool for understanding how changes in conditions affect the potential of an electrochemical cell. In our exercise, by isolating \(E^\circ\) from the relationship \(\Delta G^\circ = -nFE^\circ\), the standard cell potential is computed, which links the thermodynamic and electrochemical aspects of the reaction together.

Redox Reaction

Redox reactions, short for reduction-oxidation reactions, are a type of chemical process involving the transfer of electrons between two species. Reduction describes the gain of electrons, while oxidation refers to the loss. These reactions are essential to numerous processes, including metabolic pathways in biology, corrosion, and battery operation.

In a redox reaction, there is always an oxidizing agent that accepts electrons and becomes reduced, and a reducing agent that donates electrons and becomes oxidized. The flow of electrons from the reducing agent to the oxidizing agent generates an electric current, which is harnessed in electrochemical cells. Understanding redox reactions is crucial for analyzing the exercise in question, as it provides the base knowledge required to comprehend the transfer of electrons that leads to a change in Gibbs free energy and the establishment of an electrochemical potential.