Problem 53 - Chemistry The Central Science

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 \(E_{\mathrm{red}}^{\circ}\).

Step-by-Step Solution

Answer

The standard Gibbs free energy change (ΔG°) for the two-electron redox reaction at 298 K with an equilibrium constant of \(1.5 \times 10^{-4}\) is approximately -41.48 kJ mol⁻¹. The corresponding standard reduction potential (E°(red)) is approximately 0.215 V.

Step 1: Calculate the standard Gibbs free energy change (ΔG°)

To calculate the standard Gibbs free energy change, we can use the relationship between ΔG° and K: ΔG° = -RT ln(K) Where: R is the gas constant (8.314 J mol⁻¹ K⁻¹) T is the temperature in Kelvin (298 K) K is the equilibrium constant (1.5 × 10⁻⁴) Plugging these values into our equation, we get: ΔG° = -(8.314 J mol⁻¹ K⁻¹) * (298 K) * ln(1.5 × 10⁻⁴) ΔG° = -41.4779 kJ mol⁻¹ Therefore, the standard Gibbs free energy change (ΔG°) is approximately -41.48 kJ mol⁻¹.

Step 2: Calculate the standard reduction potential (E°(red))

To calculate the standard reduction potential, we can use the relationship between ΔG° and E°(red): ΔG° = -nFE°(red) Where: ΔG° is the standard Gibbs free energy change (-41.478 kJ mol⁻¹) n is the number of electrons involved in the reaction (2) F is the Faraday constant (96,485 C mol⁻¹) E°(red) is the standard reduction potential, which we need to solve for. Rearranging the equation to solve for E°(red), we get: E°(red) = -ΔG° / (nF) Plugging our known values into this equation, and remembering to convert ΔG° to J mol⁻¹ by multiplying by 1,000, we get: E°(red) = -(-41,478 J mol⁻¹) / (2 * 96,485 C mol⁻¹) E°(red) = 0.2152 V Therefore, the standard reduction potential (E°(red)) is approximately 0.215 V.

Key Concepts

Understanding Standard Gibbs Free Energy Change

When studying chemical systems, the standard Gibbs free energy change (G^) is a pivotal concept that quantifies the maximum amount of work a system can perform at constant temperature and pressure when it reaches equilibrium. Essentially, it tells us which reactions are spontaneous under standard conditions. Spontaneous reactions have a negative G^, indicating that they can proceed without any input of energy. In contrast, reactions with a positive G^ require external energy to occur.

G^ is calculated by the formula G^ = -RT ln(K), where R is the universal gas constant, T is the temperature in Kelvin, and K is the equilibrium constant. The equilibrium constant itself encodes the ratio of product concentrations to reactant concentrations at equilibrium, for reactions involving gases or solutions.

Moreover, this calculation becomes extremely useful when we examine redox reactions in the context of electrochemistry. By knowing G^, we can predict not only the spontaneity but also the direction and extent to which a redox reaction will proceed. This brings us seamlessly into another key area of interest—the relationship between G^ and standard reduction potentials, which gives us deeper insight into the electrochemical behavior of the reaction.

The Role of Standard Reduction Potential in Redox Reactions

The standard reduction potential (E^(red)) is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. It's essentially a measure of the driving force behind a redox reaction. Each half-reaction in an electrochemical cell has an associated standard reduction potential, which is calculated under standard conditions: 25°C (298 K), 1 M concentration for solutions, and 1 atm pressure for gases.

For a given half-reaction, a higher E^(red) means a greater likelihood that the species will gain electrons, acting as an oxidizing agent. Conversely, a lower E^(red) indicates a better ability to lose electrons, behaving as a reducing agent. Importantly, we use the Nernst Equation to calculate E^(red) when conditions are not standard.

E^(red) is related to the Gibbs free energy change by the equation G^ = -nFE^(red), where n is the number of moles of electrons transferred, and F is the Faraday constant. This important relationship allows us to predict the electrical potential of redox reactions, which is a crucial aspect of battery technology and corrosion science, among other fields. Thus, E^(red) not only helps in understanding the spontaneity and directionality of redox reactions but also enables the practical applications of such reactions in various technologies.

Faraday's Constant and Its Significance in Electrochemistry

The Faraday constant (F), significant in the realms of physical chemistry and electrochemistry, represents the total electric charge carried by one mole of electrons, approximately 96,485 coulombs per mole. Michael Faraday, for whom the constant is named, was a pioneering scientist whose work in electricity and magnetism laid the groundwork for modern electrochemistry.

When relating electrical energy to chemical change, the Faraday constant is used to convert between moles of electrons and the amount of electric charge they represent. This makes F a key player in equations like G^ = -nFE^(red), bridging the gap between the chemical and electrical quantities in electrochemical cells.

Applications of Faraday’s Constant

Faraday’s constant finds its application in various calculations, such as determining the amount of substance produced or consumed in an electrochemical reaction (electrolysis) based on the quantity of electricity passed through the substance. It also helps in calculating the energy efficiency of batteries and fuel cells. Being a fundamental value, F is indispensable for students and researchers who deal with processes where electric current causes chemical changes, underlining its significance in both theoretical studies and industrial applications.