Question

The lattice energies of \(\mathrm{KBr}\) and \(\mathrm{CsCl}\) are nearly equal (Table 8.2). What can you conclude from this observation?

Short answer

The lattice energies of KBr and CsCl are nearly equal because their coulombic attractions are similar, which is due to their equal charges (+1 and -1) and similar ionic radii (KBr: \(r_{K^+} = 1.51 \thinspace \text{Å}\) and \(r_{Br^-} = 1.95 \thinspace \text{Å}\); CsCl: \(r_{Cs^+} = 1.67 \thinspace \text{Å}\) and \(r_{Cl^-} = 1.81 \thinspace \text{Å}\)).

Step-by-step solution

Understanding lattice energy

Lattice energy is the energy released when one mole of an ionic compound is formed from its gaseous ions. The factors affecting lattice energy are coulombic attraction, size of the ions, and the arrangement of ions in the crystal lattice. The most important factor is coulombic attraction, which can be calculated using Coulomb's law: \(E = k \frac{q_1 \cdot q_2}{r}\) where E is the energy, k is a constant, q1 and q2 are the charges of the ions, and r is the distance between the ions. Now, let's analyze KBr and CsCl:

Potassium Bromide (KBr)

KBr is formed from potassium ion (K+, charge +1) and bromide ion (Br-, charge -1). The charges of the ions are +1 and -1, respectively. The ionic radii of these ions are: \(r_{K^+} = 1.51 \thinspace \text{Å}\) and \(r_{Br^-} = 1.95 \thinspace \text{Å}\).

Cesium Chloride (CsCl)

CsCl is formed from cesium ion (Cs+, charge +1) and chloride ion (Cl-, charge -1). The charges of the ions are +1 and -1, respectively. The ionic radii of these ions are: \(r_{Cs^+} = 1.67 \thinspace \text{Å}\) and \(r_{Cl^-} = 1.81 \thinspace \text{Å}\).

Comparison of Electronegativities and Ionic Radii

The charges of the ions in both compounds are the same (+1 and -1). Therefore, from Coulomb's law, we can see that the lattice energies will largely depend on the distance r. Ionic radii of the ions: - \(r_{K^+} = 1.51 \thinspace \text{Å}\) - \(r_{Br^-} = 1.95 \thinspace \text{Å}\) - \(r_{Cs^+} = 1.67 \thinspace \text{Å}\) - \(r_{Cl^-} = 1.81 \thinspace \text{Å}\) Comparing the radii, we can conclude that the distances between ions in both compounds are pretty similar.

Conclusion

Based on the observation that the lattice energies of KBr and CsCl are nearly equal, we can conclude that the coulombic attractions between the ions in both compounds are similar due to their equal charges and similar ionic radii.

Key concepts

Coulomb's Law in Chemistry

Coulomb's law is a principle that is fundamental to understanding the forces acting between charged particles, and it plays a pivotal role in chemistry, particularly in the study of ionic compounds. The law states that the force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. The formula is expressed as:

\[ E = k \frac{q_1 \cdot q_2}{r^2} \]
Here, \(E\) represents the electrostatic force, \(k\) is Coulomb's constant, \(q_1\) and \(q_2\) are the charges on the ions, and \(r\) is the distance between the centers of the two ions.

In the context of lattice energy, which is the energy released when ions come together to form a solid lattice, Coulomb's law helps us understand how the strength of the ionic bond is influenced by the charges and the distance between ions. The greater the charges and the closer the distance, the stronger the bond and the higher the lattice energy. However, for ions with equal charges, as noted in KBr and CsCl, the difference in lattice energy is predominantly due to the difference in ionic radii, which affects the distance \(r\) in the Coulomb's law equation.

Ionic Radii

Ionic radii refer to the size of an ion in an ionic compound and are crucial in determining the physical properties of the compound, such as lattice energy. The ionic radius is affected by the number of electrons and the nuclear charge. When an atom loses an electron to form a cation, the radius decreases due to reduced electron-electron repulsion and increased attraction from the nucleus. Conversely, when an atom gains an electron to form an anion, the radius increases as electron-electron repulsion increases.

This change in size affects how closely ions can pack together in a lattice structure, influencing the lattice energy according to Coulomb's law. For instance, in our discussion about KBr and CsCl, studying ionic radii enables us to understand the similarities in lattice energies. With similar charges, the determining factor in the lattice energy formula is the distance between the ion centers, which is affected by ionic radii. The closer the ions, or the smaller their combined radii, the stronger the coulombic attraction and the greater the energy released upon formation of the lattice.

Coulombic Attraction

Coulombic attraction is the force experienced between oppositely charged particles. In ionic compounds, it is the pull that cations feel towards anions and vice versa, which holds the compound together. This electrostatic attraction is the cornerstone for explaining why ions aggregate to form solid structures. Lattice energy, which we measure as the energy given off when a mole of an ionic solid forms from its ions, is a direct result of coulombic attraction.

The strength of this attraction, as informed by Coulomb's law, is determined by both the magnitude of the charges and the distance between the ions. Higher charges and smaller distances result in stronger coulombic attraction. Therefore, when analyzing the nearly equal lattice energies of \(\mathrm{KBr}\) and \(\mathrm{CsCl}\), we attribute this similarity to their corresponding ions having the same charges and their ionic radii being close in size, leading to a comparable amount of coulombic attraction between them in their respective lattices.