Use the standard free energy of formation data in Appendix G to determine the free energy change for each of the following reactions, which are run under standard state conditions and \(25^{\circ} \mathrm{C}\). Identify each as either spontaneous or nonspontaneous at these conditions. (a) \(\mathrm{MnO}_{2}(s) \longrightarrow \mathrm{Mn}(s)+\mathrm{O}_{2}(g)\) (b) \(\mathrm{H}_{2}(g)+\mathrm{Br}_{2}(l) \longrightarrow 2 \mathrm{HBr}(g)\) (c) \(\mathrm{Cu}(s)+\mathrm{S}(g) \longrightarrow \mathrm{CuS}(s)\) (d) \(2 \mathrm{LiOH}(s)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{Li}_{2} \mathrm{CO}_{3}(s)+\mathrm{H}_{2} \mathrm{O}(g)\) (e) \(\mathrm{CH}_{4}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{C}(s, \text { graphite })+2 \mathrm{H}_{2} \mathrm{O}(g)\) (f) \(\mathrm{CS}_{2}(g)+3 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{CCl}_{4}(g)+\mathrm{S}_{2} \mathrm{Cl}_{2}(g)\)

Understanding Gibbs free energy change is crucial when analyzing chemical reactions. It is represented by the symbol \( \Delta G \) and is a thermodynamic property indicating the amount of energy available to do work during a chemical process at constant temperature and pressure. The formula \( \Delta G = \Delta G_f^{\text{(products)}} - \Delta G_f^{\text{(reactants)}} \) incorporates the standard free energies of formation \( \Delta G_f \) for all reactants and products.

When \( \Delta G \) is negative, it signals that the reaction can occur spontaneously under standard conditions, meaning no external energy input is needed for the reaction to proceed. Conversely, if \( \Delta G \) is positive, the reaction is termed nonspontaneous; it will not proceed on its own without an input of energy. A \( \Delta G \) of zero indicates the system is at equilibrium and no net reaction will occur.

To make these calculations more meaningful for students, consider giving real-world examples of spontaneous and nonspontaneous reactions. For instance, the rusting of iron is a spontaneous process, whereas the electrolysis of water is nonspontaneous and requires external electricity.

Spontaneous reactions are processes that occur without the need for continuous external energy. These reactions have a natural tendency to proceed in a direction that leads to a decrease in free energy, hence a negative \( \Delta G \). It's important to note that 'spontaneous' does not necessarily mean 'immediate'; some reactions may be spontaneous but occur over long periods, like the conversion of diamond to graphite.

Factors like temperature and pressure can affect spontaneity, so a reaction that is spontaneous under one set of conditions might not be under others. For students, a common source of confusion is the misconception that 'spontaneous' equates to 'fast'. Clarifying that spontaneity relates to the thermodynamic favorability rather than the rate of reaction will enhance understanding.

Chemical thermodynamics is the branch of science that studies the energy changes accompanying chemical and physical processes. It lays the groundwork for understanding concepts such as enthalpy (\( \Delta H \)), entropy (\( \Delta S \)), and Gibbs free energy (\( \Delta G \)). The interplay between these quantities decides the spontaneity and equilibrium of processes.

For instance, the second law of thermodynamics posits that the total entropy of an isolated system can only increase over time. In the realm of chemistry, this means that for a reaction to be spontaneous, the Gibbs free energy should decrease (\( \Delta G < 0 \)), signifying an overall increase in entropy. Teachers can solidify this concept by discussing energy diagrams and how systems naturally progress towards lower energy states.

Stoichiometry is a quantitative relationship between the amounts of substances involved in a chemical reaction. It is based on the balanced chemical equation and the conservation of mass and atoms. For calculations involving Gibbs free energy change, the stoichiometry of the reaction must be considered as it directly influences \( \Delta G \) values since each substance contributes to the total free energy in a mole-specific manner.

When teaching stoichiometry, it's key to emphasize the importance of a balanced equation and mole ratios. For example, if a reaction has a coefficient of 2 in front of a product in the balanced equation, its standard free energy of formation must be multiplied by 2 when calculating the overall \( \Delta G \) for the reaction. Applying stoichiometry in solving \( \Delta G \) problems will help students realize the intricate connections between chemical formulas, quantities in moles, and energy changes.