Question

Use the following data for potassium chloride to estimate \(\Delta E\) for the reaction: $$\mathrm{K}(s)+\frac{1}{2} \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{KCl}(s) \quad \Delta E=?$$ Lattice energy Ionization energy for \(\mathbf{K}\) Electron affinity of Cl Bond energy of \(\mathrm{Cl}_{2}\) Energy of sublimation for \(\mathrm{K}\) \(-2913 \mathrm{kJ} / \mathrm{mol}\) \(735 \mathrm{kJ} / \mathrm{mol}\) \(1445 \mathrm{kJ} / \mathrm{mol}\) -328 kJ/mol \(154 \mathrm{kJ} / \mathrm{mol}\) 150\. kJ/mol

Short answer

The overall energy change, \(\Delta E\), for the reaction of potassium (K) and chlorine gas (Cl₂) to form potassium chloride (KCl) can be estimated using the Born-Haber cycle. We are given the lattice energy, ionization energy, electron affinity, bond energy, and energy of sublimation. Using these values, we can calculate the energy change for the reaction: \(\Delta E_{reaction} = - \Delta E_{lattice} + \Delta E_{sublimation} + \Delta E_{ionization} + \frac{1}{2} \Delta E_{bond} - \Delta E_{electron-affinity}\) Plugging in the given values, we find: \(\Delta E_{reaction} = 2913 + 154 + 735 + 164 - 1445\) \(\Delta E_{reaction} = 2521 - 1445\) \(\Delta E_{reaction} = 1076 \mathrm{kJ/mol}\) So, the overall energy change for the reaction is 1076 kJ/mol.

Step-by-step solution

1. Identify the Born-Haber cycle components

The Born-Haber cycle includes the ionization energy of K, the electron affinity of Cl, the bond energy of Cl₂, the energy of sublimation for K, and the lattice energy of KCl. Using these values, we can determine the overall energy change for the formation of KCl.

2. Calculate the energy change for each process in the Born-Haber cycle

We are given the following energy values: - Lattice energy of KCl: -2913 kJ/mol - Ionization energy of K: 735 kJ/mol - Electron affinity of Cl: 1445 kJ/mol - Bond energy of Cl₂: 328 kJ/mol (to break the Cl-Cl bond, we need positive value) - Energy of sublimation for K: 154 kJ/mol

3. Determine the overall energy change for the reaction

We can now use these values to calculate the overall energy change, \(\Delta E\), for the reaction: \(\Delta E_{reaction} = - \Delta E_{lattice} + \Delta E_{sublimation} + \Delta E_{ionization} + \frac{1}{2} \Delta E_{bond} - \Delta E_{electron-affinity}\) \(\Delta E_{reaction} = - (-2913) + 154 + 735 + \frac{1}{2}(328) - 1445\) Now, plug in the values and calculate the overall energy change: \(\Delta E_{reaction} = 2913 + 154 + 735 + 164 - 1445\) \(\Delta E_{reaction} = 2521 - 1445\) \(\Delta E_{reaction} = 1076 \mathrm{kJ/mol}\) The overall energy change for the reaction of K and Cl₂ to form KCl is 1076 kJ/mol.

Key concepts

Lattice Energy

Lattice energy is a crucial component when understanding ionic compounds like potassium chloride (KCl). This energy represents the strength of the forces holding the ions together in the ionic solid. In the case of KCl, the lattice energy is quite significant and highly negative, specifically \(-2913\ \text{kJ/mol}\), which means it requires a considerable amount of energy to break apart the crystal structure into individual ions.This energy is usually released when a solid lattice forms from gaseous ions and is pivotal in the Born-Haber cycle for determining the stability of an ionic compound. When calculating energy changes in reactions, such as the formation of KCl, we add the lattice energy because it provides energy to the reaction, making it favorable and exothermic. Understanding this helps explain why some salts are very stable and difficult to decompose.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom, transforming it into a cation. For potassium (K), this process involves the removal of the outermost electron, requiring \(735\ \text{kJ/mol}\) of energy.Ionization energy is critical in the formation of ionic bonds as it directly influences how easily an atom like potassium can lose an electron and form a positive ion. Potassium's relatively low ionization energy compared to other elements in its group makes it especially reactive and likely to form compounds such as KCl readily.In the Born-Haber cycle, the ionization energy is a key step, as it accounts for the conversion of gaseous potassium atoms into potassium ions, which can then interact electrostatically with chloride ions to form the ionic compound.

Electron Affinity

Electron affinity refers to the change in energy when an electron is added to a neutral atom to form an anion. For chlorine (Cl), this energy, \(-328\ \text{kJ/mol}\), expresses how much closer an atom is pulled to an electron added to its outer shell.A negative electron affinity implies that the process is exothermic—energy is released when chlorine gains the electron. This process is essential for forming ionic bonds, as it shows chlorine’s strong tendency to attract electrons and form anions.In the Born-Haber cycle, electron affinity is taken into account to form a complete picture of how energy change occurs during compound formation, emphasizing the thermodynamically favorable conditions for KCl formation due to chlorine's strong electron affinity.

Sublimation Energy

Sublimation energy is the energy required to transform a solid element into gaseous atoms. For potassium, this process uses\(154\ \text{kJ/mol}\), transitioning from its solid metallic state to individual gaseous atoms.This energy is fundamental in the initial steps of forming an ionic bond in the Born-Haber cycle. Before potassium can donate an electron to chlorine, it must be converted into a gaseous state. Once sublimated, potassium can undergo ionization and further steps to form ionic compounds.By including sublimation energy, the Born-Haber cycle provides a comprehensive understanding of the step-by-step energy changes occurring during the reaction, ensuring the formation of stable salts like KCl from their elemental states is accurately modeled.