For each pair of standard free energy change and electron stoichiometry values below, calculate a corresponding standard cell potential. (a) \(12 \mathrm{kJ} / \mathrm{mol}, \mathrm{n}=3\) (b) \(-45 \mathrm{kJ} / \mathrm{mol}, \mathrm{n}=1\)

Gibbs free energy change, denoted as \( \Delta G^\circ \), is a thermodynamic quantity that measures the maximum amount of reversible work that may be performed by a system at constant temperature and pressure. In simple terms, it helps predict whether a reaction will spontaneously occur under standard conditions. A negative value of \( \Delta G^\circ \) indicates that a reaction is exergonic and will proceed spontaneously, while a positive value suggests that the reaction is endergonic and will not occur without input of energy.

Electrochemistry ties into this concept through electrochemical cells, where the \( \Delta G^\circ \) of a redox reaction can determine the cell potential. By calculating the change in Gibbs free energy, one can determine the feasibility and direction of electron flow in the cell. This relationship is quantitatively expressed by the equation \( \Delta G^\circ = -nFE^\circ \) where \( n \) represents the number of moles of electrons transferred, and \( F \) is the Faraday constant. The ability to connect these two properties is fundamental in predicting the voltage output of electrochemical cells.

The Faraday constant, represented by \( F \), is a fundamental constant in electrochemistry and represents the charge of one mole of electrons, approximately 96,485 Coulombs/mol. This constant is pivotal when converting between electrical energy and chemical potential because it links the macroscopic quantities of charge and substance.

The Faraday constant comes into play when determining the electrical work reversible of a chemical reaction (\( \Delta G^\circ \)) through the relationship \( \Delta G^\circ = -nFE^\circ \). For every mole of electrons exchanged in an electrochemical reaction, a fixed quantity of charge, equivalent to \( F \), is transported. Hence, knowing the value of the Faraday constant allows for precise calculations of the cell potential. It is an essential aspect of solving electrochemical problems and understanding how the flow of electrons relates to chemical change.

Electrochemistry is the branch of chemistry that deals with the chemical changes produced by electricity and the production of electricity by chemical reactions. It is a study of redox reactions that involve electron transfer between reactants and the conversion between chemical energy and electrical energy.

Key concepts within electrochemistry include the standard cell potential (\( E^\circ \)), which is an indicator of a cell's ability to drive an electrical current through an external circuit. It is influenced by the intrinsic properties of the substances involved in the redox reactions. The standard cell potential can be calculated from the Gibbs free energy change, reinforcing the close relationship between thermodynamics and electrochemical processes. Electrochemistry finds applications in a variety of fields including battery technology, corrosion prevention, and electroplating.

Electron stoichiometry refers to the balance of electron transfer in chemical reactions, particularly in redox reactions. Stoichiometry, in general, deals with the quantitative relationships between reactants and products in chemical reactions.

In electrochemistry, understanding electron stoichiometry is crucial because it defines the exact number of electrons involved in the oxidation-reduction reactions occurring within an electrochemical cell. This number, denoted by \( n \), is required when we use the formula \( \Delta G^\circ = -nFE^\circ \) to calculate the standard cell potential or to determine Gibbs free energy change from a known cell potential. The ability to accurately count the number of electrons transferred allows chemists and engineers to build precise models of electrochemical cells and predict their behavior in practical applications.