Question

(a) Does the lattice energy of an ionic solid increase or decrease (i) as the charges of the ions increase, (ii) as the sizes of the ions increase? (b) Arrange the following substances not listed in Table 8.1 according to their expected lattice energies, listing them from lowest lattice energy to the highest: MgS, Ki, GaN, LiBr.

Short answer

(a) (i) The lattice energy of an ionic solid increases as the charges of the ions increase. (ii) The lattice energy decreases as the sizes of the ions increase. (b) The substances can be arranged in ascending order of lattice energy as follows: Ki < LiBr < MgS < GaN.

Step-by-step solution

Understanding lattice energy

Lattice energy is the energy required to break apart one mole of an ionic solid into its gaseous ions. This energy is affected by both the charges of the ions and their sizes. The lattice energy can be calculated using the Born-Haber cycle or Coulomb's Law. In general, stronger lattice energy indicates a more stable ionic compound.

Effect of charges on lattice energy

(i) As the charges of the ions increase, the interaction between the oppositely charged ions becomes stronger. Consequently, the lattice energy increases. This can be explained using Coulomb's law, which states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Therefore, a higher charge will result in a stronger force and increased lattice energy.

Effect of ion size on lattice energy

(ii) As the sizes of the ions increase, the distance between the ions also increases, which results in a decrease in the lattice energy. Again, this can be explained using Coulomb's law, which states that the force between two charged objects is inversely proportional to the square of the distance between them. As the ions get larger, the distance between them increases, leading to weaker ionic interaction and lower lattice energy.

Arranging substances based on lattice energy

(b) The lattice energy of a substance depends on the charges of the ions and their sizes. To arrange the given substances (MgS, Ki, GaN, and LiBr) based on the lattice energy, we must consider their respective charges and ion sizes. MgS: Mg²⁺ and S²⁻ Ki: K⁺ and I⁻ GaN: Ga³⁺ and N³⁻ LiBr: Li⁺ and Br⁻ Comparing the charges, GaN has the highest charges (³⁺, ³⁻), followed by MgS (²⁺, ²⁻), and Ki and LiBr have the lowest charges (¹⁺, ¹⁻). However, we also need to consider the ion sizes, where the larger ion size leads to lower lattice energy. Ion sizes (in approximate order, from smallest to largest): N³⁻ < S²⁻ < Li⁺ < Mg²⁺ < Ga³⁺ < Br⁻ < K⁺ < I⁻ Considering both ion charges and sizes, we can arrange the lattice energies in ascending order as follows: Ki < LiBr < MgS < GaN So, the lattice energies increase in the order: Ki, LiBr, MgS, and GaN.

Key concepts

Ionic Compound Stability

Ionic compounds are held together by the electrostatic attraction between positively and negatively charged ions. This force is fundamental to the stability of the compound. The more energy required to break this attraction and turn the compound into gaseous ions, the more stable the compound is said to be. This is where lattice energy comes into play.

Lattice energy can be thought of as a measure of the bond strength in ionic compounds: the higher the lattice energy, the stronger the bond and the more stable the compound. In addition, the ratio of the charges and the size of the constituent ions greatly influence this stability. As the charge ratio of the ions increases while keeping the ionic sizes consistent, the overall stability of the compound typically increases. On the other hand, if the ions are larger—even with the same charge—a reduction in stability is generally observed due to the lower lattice energy. These concepts apply whether we’re discussing common table salt (NaCl) or more complex ionic structures.

Born-Haber Cycle

The Born-Haber cycle is a thermochemical cycle that helps us to understand and quantify the factors affecting lattice energy in the formation of ionic compounds. It represents a series of hypothetical steps that include sublimation, dissociation, ionization, and electron affinity. The cycle combines these steps to calculate the overall lattice energy of a compound.

The cycle starts with the elements in their standard states and ends with the formation of an ionic solid, allowing us to consider each individual energetic step along the way. For example, we look at the energy necessary to vaporize the metallic element (sublimation), the energy to break diatomic molecules into atoms (dissociation), the ionization energy for the removal of electrons from the metal, and the energy released when the non-metal gains electrons (electron affinity).

Using Hess's Law, which states that the total enthalpy change is the same regardless of the path taken, the Born-Haber cycle allows us to calculate the lattice energy indirectly, even though it can’t be measured directly. Understanding this cycle is crucial for chemists who need to predict the stability of newly synthesized or yet-to-be-discovered ionic compounds.

Coulomb's Law

Coulomb's Law is pivotal to understanding the interactions between charged particles, which in turn helps to interpret the reasons behind variations in lattice energy. Named after the French physicist Charles-Augustin de Coulomb, the law mathematically expresses how the force of attraction or repulsion between two charged particles is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between their centers.

The law is represented by the formula: \[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]where \( F \) denotes the force between the charges, \( q_1 \) and \( q_2 \) are the charges, \( r \) is the distance between the centers of the charges, and \( k \) is Coulomb's constant.

This fundamental physics principle explains the trend observed in lattice energies: when the charges of the ions in a compound increase, the force of attraction rises, resulting in higher lattice energy, and conversely, as ions become larger and distances increase, the force weakens and lattice energy drops. Understanding Coulomb's Law not only aids in explaining observable trends in ionic compounds but also in the broader field of electrostatics and electric fields in various physics and engineering applications.