Calculate the standard enthalpy change for the reaction $$ 2 \mathrm{Al}(s)+\mathrm{Fe}_{2} \mathrm{O}_{3}(s) \longrightarrow 2 \mathrm{Fe}(s)+\mathrm{Al}_{2} \mathrm{O}_{3}(s) $$ given that $$ \begin{aligned} 2 \mathrm{Al}(s)+\frac{3}{2} \mathrm{O}_{2}(g) \longrightarrow & \mathrm{Al}_{2} \mathrm{O}_{3}(s) \\ \Delta H_{\mathrm{rxn}}^{\circ} &=-1669.8 \mathrm{~kJ} / \mathrm{mol} \\ 2 \mathrm{Fe}(s)+\frac{3}{2} \mathrm{O}_{2}(g) \longrightarrow & \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \\ \Delta H_{\mathrm{rxn}}^{\circ} &=-822.2 \mathrm{~kJ} / \mathrm{mol} \end{aligned} $$

Understanding Hess's Law in thermochemistry is crucial when it comes to determining the enthalpy change of a chemical reaction that is not directly measurable. The enthalpy of a reaction depends solely upon the initial and final states of the system and not on the path taken. This principle is famously encapsulated in Hess's Law, which states that the total enthalpy change for a chemical reaction is the same, regardless of the number of steps taken to perform the reaction.

Hess's Law makes it possible to calculate the enthalpy changes for complex chemical reactions by breaking them down into a series of simpler steps with known enthalpy changes. To apply this law, equations representing individual steps of a reaction are manipulated — reversed or multiplied to match the number of moles with those in the final reaction — and their respective enthalpy changes are adjusted accordingly. The overall enthalpy change is then obtained by summing the enthalpy changes for each step.

By using Hess's Law, we can apply known thermochemical equations to find the enthalpy change of the main reaction, much like completing a puzzle with pieces representing simpler reactions. In essence, Hess's Law serves as a bookkeeping tool, allowing for the balance of energy changes just as one would balance atoms in a chemical equation.

Thermochemistry is the branch of chemistry that deals with the study of the heat energy involved in chemical reactions and physical transformations. The central concept in thermochemistry is the enthalpy change, often symbolized as \(\Delta H\). It's a measure of the heat change at constant pressure and is expressed in units of kilojoules per mole (kJ/mol).

Every chemical reaction either absorbs or releases energy in the form of heat, leading to two primary classifications of reactions:

• Endothermic reactions: These absorb heat, resulting in a positive \(\Delta H\).
• Exothermic reactions: These release heat, resulting in a negative \(\Delta H\).

Understanding the energetics of chemical reactions is fundamental to various applications in science and engineering, such as designing batteries, understanding metabolism in living organisms, and predicting the impacts of reactions on the environment.

The enthalpy of reaction, or reaction enthalpy, is the heat change that occurs during a reaction at constant pressure and is represented by the symbol \(\Delta H_{\text{rxn}}^\circ\). It is an essential quantity in the study of chemical reactions, as it provides insight into the energetics and spontaneity of the process. When the reaction releases heat, the enthalpy of reaction is negative, indicating it is exothermic. Conversely, if heat is absorbed, the enthalpy of reaction is positive, suggesting an endothermic nature.

The standard enthalpy change \(\Delta H^\circ\) refers to the enthalpy change when all reactants and products are in their standard states, usually at 1 bar of pressure. Standard enthalpy changes provide a reference point to compare the relative energetics of different reactions and are fundamental when applying Hess's Law, as shown in our original exercise.

To calculate the standard enthalpy change, it's important to ensure that the stoichiometry of the reactions matches the overall process being analyzed. The steps provided in our textbook solution illustrate the method to obtain the enthalpy change for the desired reaction, often using tabulated values of standard enthalpy changes for individual reactions or formation enthalpies of compounds.