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 \(E^{\circ} .\)

The Gibbs free energy change, denoted as \( \Delta G^{\box} \), is a pivotal concept in thermodynamics and chemistry, which indicates the spontaneity of a chemical reaction at constant temperature and pressure. In simple terms, it helps predict whether a reaction will occur without additional energy. If \( \Delta G^{\box} < 0 \), the reaction is spontaneous; if \( \Delta G^{\box} > 0 \), the reaction is non-spontaneous; and if \( \Delta G^{\box} = 0 \), the system is at equilibrium.

To calculate \( \Delta G^{\box} \) for a reaction, you can use the relationship between free energy change and the equilibrium constant \( K \) at a particular temperature \( T \) provided by the equation \( \Delta G^{\box} = -RT \ln{K} \). The negative sign indicates that as the equilibrium constant increases, indicating a greater ratio of products to reactants, the free energy decreases, making the reaction more likely to proceed. The gas constant \( R = 8.314 \frac{J}{K\cdot mol} \) links the reaction to absolute temperature in Kelvins.

In the context of the exercise, one can deduce that the small value of \( K = 1.5 \times 10^{-4} \) suggests the reaction favors the reactants at equilibrium, leading to a positive \( \Delta G^{\box} \) value, meaning the reaction is non-spontaneous under standard conditions.

Standard cell potential \( E^{\box} \) represents the voltage or electromotive force of a cell when all components are in their standard states (which usually means concentrations of 1 molar and gases at 1 atmosphere pressure) at a temperature of 298 K (25°C). It's a measure of how much voltage, or potential energy, a cell can produce, which is directly linked to the spontaneity of the redox reaction taking place within the cell.

The standard cell potential can be linked to the Gibbs free energy change via the equation \( \Delta G^{\box} = -nFE^{\box} \), where \( n \) is the number of moles of electrons transferred, and \( F \) is the Faraday constant. By calculating \( \Delta G^{\box} \) first, as done in the exercise, one can then find \( E^{\box} \) to determine if the cell will release or require energy to operate. A positive value of \( E^{\box} \) signifies a spontaneous redox reaction, while a negative value signifies a non-spontaneous reaction.

The importance of \( E^{\box} \) in electrochemistry cannot be overstated, as it underpins the operation of batteries, fuel cells, and electrolysis processes, all of which convert chemical energy into electrical energy or vice-versa.

The Faraday constant \( F \) is a fundamental physical constant used in calculations involving electrochemistry, named after the English scientist Michael Faraday. It represents the total electric charge carried by one mole of electrons, with a magnitude of approximately \( 96485 \frac{C}{mol} \) (coulombs per mole).

This constant is crucial when translating between the chemical scale of moles and the physical scale of electric charge in coulombs. It allows for the quantification of the electrical work—which is directly related to Gibbs free energy change—involving redox reactions, as showcased in our exercise.

In the context of the standard cell potential equation, \( F \) is used to highlight the relationship between energy change and electric charge: \( \Delta G^{\box} = -nFE^{\box} \). It is fundamental in calculating how much electrical energy is produced or required in a reaction involving electron transfer, such as the operation of galvanic or electrolytic cells.

The Nernst equation is another cornerstone of electrochemistry, enabling the determination of cell potential under non-standard conditions, particularly when concentrations are not at standard conditions. It states that the voltage of an electrochemical cell depends logarithmically on the ratio of the concentrations of the products to the reactants.

The equation is given by: \( E = E^{\box} - \frac{RT}{nF} \ln{\frac{[products]}{[reactants]}} \) where \( E \) is the cell potential, \( E^{\box} \) is the standard cell potential, \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvins, \( n \) is the number of moles of electrons transferred, \( F \) is the Faraday constant, and \( [products]/[reactants] \) represents the reaction quotient.

The Nernst equation elegantly combines thermodynamics with equilibrium to describe how electrochemical potentials shift as concentrations change, crucial for understanding batteries, corrosion, sensors, and any application where the chemical composition affects electrical properties. While the Nernst equation isn't directly applied in our exercise with given standard conditions, in practical scenarios, it is omnipresent in the adaptation of standard potentials to real-world conditions.