From the lattice energy of \(\mathrm{KCl}\) in Table 9.1 and the ionization energy of \(\mathrm{K}\) and electron affinity of \(\mathrm{Cl}\) in Tables 8.2 and \(8.3,\) calculate the \(\Delta H^{\circ}\) for the reaction $$ \mathrm{K}(g)+\mathrm{Cl}(g) \longrightarrow \mathrm{KCl}(s) $$

Understanding Hess's law is fundamental when we need to calculate the total enthalpy change of a reaction. This law states that the total enthalpy change during the complete course of a chemical reaction is the same, whether the reaction is made in one step or several steps.

This principle is based on the conservation of energy, implying that the energy change is path-independent as long as the initial and final conditions are the same. For our KCl example, Hess's law allows us to break down the reaction into multiple steps and add up the individual enthalpy changes, such as the ionization energy of potassium, the electron affinity of chlorine, and the formation of the KCl lattice. Even if these processes occur separately in stages, by Hess's law, their summed changes in enthalpy will lead to the same total energy change as the direct transformation from reactants to products.

Ionization energy refers to the energy required to remove an electron from an atom or ion in its gaseous state. For alkali metals, like potassium (K), this process involves taking away an electron from their outermost shell to form a cation.

The ionization energy is always positive, as it represents the input of energy. In chemical equations, it's important to keep track of energy direction: adding energy to a system is indicated by a positive sign, while the release of energy is marked by a negative sign. When calculating the enthalpy change for the formation of KCl, the ionization energy of K is considered a positive value, reflecting the endothermic nature of this process.

On the flip side of ionization energy, we have electron affinity. This is the energy change when an electron is added to a neutral atom, usually in the gaseous state, to form an anion. For halogens, such as chlorine (Cl), this process is exothermic, meaning it releases energy to the surrounding system.

In thermodynamic terms, electron affinity can have either a positive or negative value, but for chemical elements that readily gain an electron (like chlorine), the electron affinity is negative. In the context of our KCl example, when an electron is added to chlorine to form Cl⁻, energy is liberated, so we record electron affinity as a negative value in the enthalpy change equation.

Enthalpy change, often denoted as \( \Delta H \), is the heat content change occurring at a constant pressure when a reaction takes place. It's a crucial concept in thermodynamics and chemistry for understanding how much energy is absorbed or released during a chemical transformation.

In an equation, a negative \( \Delta H \) implies that a reaction is exothermic (releasing heat), while a positive \( \Delta H \) indicates that the reaction is endothermic (absorbing heat). When calculating enthalpy changes, we consider the direction of heat flow—towards the system for endothermic reactions, and away from the system for exothermic reactions. Thus, for the formation of KCl, the overall \( \Delta H \) will be the sum of the ionization energy, electron affinity, and lattice energy, aligning with the principles set by Hess's law.