Question

Explain the following trends in lattice energy: (a) \(\mathrm{CaF}_{2}>\mathrm{BaF}_{2} ;\) (b) \(\mathrm{NaCl}>\mathrm{RbBr}>\mathrm{CsBr} ;\) (c) \(\mathrm{BaO}>\mathrm{KF}\)

Short answer

The lattice energies follow the trends: (a) \(\mathrm{CaF}_{2}>\mathrm{BaF}_{2}\) because the ionic radius of Ca is smaller than that of Ba; (b) \(\mathrm{NaCl}>\mathrm{RbBr}>\mathrm{CsBr}\) due to the increasing ionic radii down the group (Na < Rb < Cs) and across the period (Cl < Br); and (c) \(\mathrm{BaO}>\mathrm{KF}\) as the ionic radius of O is much smaller than that of F, causing the sum of their ionic radii with Ba and K to be smaller for \(\mathrm{BaO}\). Smaller sums of ionic radii result in larger lattice energies.

Step-by-step solution

Understand Lattice Energy Definition

Lattice energy is defined as the energy required to separate one mole of an ionic solid into its gaseous ions. It depends on the size of the ions and the charge on the ions. A higher lattice energy indicates stronger attractive forces between the ions in the crystal lattice, and conversely, a lower lattice energy indicates weaker attractive forces.

Lattice Energy Formula

The lattice energy can be calculated using the formula: \[LE \propto \frac{(q_{1} \times q_{2})}{(r_{1} + r_{2})}\] Where: - \(LE\) is the lattice energy - \(q_{1}\) and \(q_{2}\) are the charges on the cations and anions, respectively - \(r_{1}\) and \(r_{2}\) are the ionic radii of the cations and anions, respectively

Use the Formula for the Comparison (a)

(a) \(\mathrm{CaF}_{2}>\mathrm{BaF}_{2}\) \(\mathrm{CaF}_{2}\) and \(\mathrm{BaF}_{2}\) both have a cation with a charge of +2 and anion (F) with a charge of -1. Therefore, the product of their charges will be the same for both of them. However, the ionic radius of Ca is smaller than that of Ba, which means the sum of their ionic radii with F will be smaller for \(\mathrm{CaF}_{2}\) than for \(\mathrm{BaF}_{2}\). The smaller the sum of ionic radii, the larger the lattice energy. Therefore, \(\mathrm{CaF}_{2}>\mathrm{BaF}_{2}\) in terms of lattice energy.

Use the Formula for the Comparison (b)

(b) \(\mathrm{NaCl}>\mathrm{RbBr}>\mathrm{CsBr}\) All three compounds have a cation with a charge of +1 and an anion with a charge of -1. Therefore, the product of their charges will be the same for all of them. However, the ionic radius increases as we move down the group (Na < Rb < Cs) and increases as we move across the period (Cl < Br). Hence, the sum of the ionic radii will be smallest for NaCl, followed by RbBr and largest for CsBr. A smaller sum of ionic radii means a larger lattice energy. Therefore, \(\mathrm{NaCl}>\mathrm{RbBr}>\mathrm{CsBr}\) in terms of lattice energy.

Use the Formula for the Comparison (c)

(c) \(\mathrm{BaO}>\mathrm{KF}\) In both \(\mathrm{BaO}\) and \(\mathrm{KF}\), the cation has a charge of +2, and the anion has a charge of -2. Therefore, the product of their charges will be the same for both of them. However, the ionic radius of Ba is larger than that of K, but the ionic radius of O is much smaller than that of F. As a result, the sum of their ionic radii will be smaller for \(\mathrm{BaO}\) than for \(\mathrm{KF}\). A smaller sum of ionic radii means a larger lattice energy. Therefore, \(\mathrm{BaO}>\mathrm{KF}\) in terms of lattice energy.

Key concepts

Ionic Compounds

Ionic compounds are chemical substances composed of positively charged ions, called cations, and negatively charged ions, called anions. The electrostatic attraction between these oppositely charged ions forms an ionic bond, holding the compound together in a solid state.

These compounds have high melting and boiling points, are typically hard and brittle, and conduct electricity when melted or dissolved in water due to the movement of free ions. Understanding the lattice energy of ionic compounds is crucial, as it provides insight into the strength of the bonds within the crystal lattice structure.

Ionic Radii

The ionic radius is a measure of the size of an ion in a crystal lattice. It's important to understand that the radius of an ion can change depending on its environment and its charge. Cations, being positively charged, lose electrons and thus, typically have smaller radii compared to their neutral atoms. Conversely, anions, with added electrons and negative charge, have larger radii.

The lattice energy of an ionic compound is inversely proportional to the sum of the ionic radii of the constituent ions; the closer the ions are to each other, the stronger the electrostatic forces between them, leading to higher lattice energies.

Crystal Lattice

A crystal lattice is the symmetrical three-dimensional arrangement of atoms inside a crystal. For ionic compounds, this lattice is made up of a regular pattern of cations and anions. The specific pattern of the lattice depends on the sizes and charges of the ions involved, and affects the overall stability and properties of the compound.

The concept of lattice energy originates from the energy released when the crystal lattice is formed from its gaseous ions. A tightly packed lattice, with smaller ionic radii and higher charges, will have a stronger interionic attraction and thus a higher lattice energy.

Electrostatic Forces

Electrostatic forces are the attractions or repulsions that occur between charged particles. In ionic compounds, these are the forces that hold the cations and anions together in the crystal lattice. The strength of the electrostatic forces is determined by Coulomb's law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

In lattice energy terms, this distance is approximated by the sum of the ionic radii, and the charges are those of the ions themselves. Thus, knowing the charge and size of the ions allows us to predict the relative strength of the electrostatic forces within different ionic compounds, which directly correlates with their lattice energies.