Lattice energy measures the energy required to completely separate one mole of a solid ionic compound into its constituent gaseous ions. This quantity represents the strength of the electrostatic attractions holding the ions together in the crystal structure. A higher lattice energy indicates a stronger ionic bond and a more stable compound, often correlating with a higher melting point. Predicting and ranking the magnitude of this energy is fundamental to understanding ionic compound properties. This ranking is governed by two physical properties of the ions: their magnitude of charge and their physical size.
The Mathematical Basis of Lattice Energy
The stability of an ionic lattice is governed by electrostatic forces, described by Coulomb’s Law. This law explains the attractive force between oppositely charged particles, specifically positive cations and negative anions. The magnitude of the lattice energy (\(LE\)) is directly proportional to the product of the ionic charges (\(Q_1\) and \(Q_2\)). Conversely, \(LE\) is inversely proportional to the distance (\(r\)) separating the centers of the ions.
This relationship is simplified as \(LE \propto \frac{Q_1 Q_2}{r}\), where \(r\) is the sum of the cation and anion radii. This formula shows that stronger attractions result from increasing the charge on the ions or decreasing the distance between them. This mathematical proportionality provides a framework for ranking lattice energies. The comparison process is hierarchical, with one factor having a significantly greater influence than the other.
Ranking by Ionic Charge Magnitude
The product of the ionic charges (\(Q_1 \times Q_2\)) is the most significant factor determining lattice energy magnitude. Since the force of attraction is directly proportional to the charge product, increasing the charge results in a large increase in lattice energy. A compound with \(+2\) and \(-2\) ions will exhibit a greater lattice energy than one with \(+1\) and \(-1\) ions, even if ionic sizes are similar.
For example, magnesium oxide (\(\text{MgO}\)) involves \(\text{Mg}^{2+}\) and \(\text{O}^{2-}\) ions, resulting in a charge product of \(2 \times 2 = 4\). Sodium chloride (\(\text{NaCl}\)) has \(\text{Na}^{+}\) and \(\text{Cl}^{-}\) ions, yielding a charge product of \(1 \times 1 = 1\). The lattice energy of \(\text{MgO}\) is approximately four times greater than that of \(\text{NaCl}\), a direct consequence of the higher ionic charges. Charge dominance is so pronounced that it must always be the first consideration. The compound with the higher charge product will consistently have the greater lattice energy, regardless of differences in ionic size.
Ranking by Ionic Radius
Ionic radius becomes the deciding factor only when the ionic charge products of the compounds being compared are identical. When the charge product is the same, the ranking depends on the inverse relationship between lattice energy and the distance (\(r\)) between the ion centers. Smaller ions allow the positive and negative nuclei to approach more closely, leading to a stronger electrostatic attraction and a higher lattice energy. The variable \(r\) is the sum of the radii of the cation and the anion.
The compound with the smaller overall sum of ionic radii will possess the higher lattice energy. This trend is observed when examining compounds within the same periodic table group. For example, comparing lithium chloride (\(\text{LiCl}\)) with potassium chloride (\(\text{KCl}\)), the charges are identical (\(+1\) and \(-1\)). Since \(\text{Li}^{+}\) is smaller than \(\text{K}^{+}\), \(\text{LiCl}\) has a shorter interionic distance, resulting in a higher lattice energy. Moving down a group increases ionic radius, which decreases the lattice energy for compounds sharing the same charge product.
Strategy for Comparing Multiple Compounds
A structured approach is required to accurately rank the lattice energies of multiple compounds by synthesizing the rules of charge and radius.
Step 1: Compare Charge Products
The first step involves determining the product of the absolute charges (\(|Q_1 \times Q_2|\)) for each compound. The compound with the largest charge product will automatically possess the highest lattice energy. This charge comparison serves as the primary determinant for ranking.
Step 2: Compare Ionic Radii (If Charges are Equal)
If two or more compounds share the same charge product, the second step is to compare their ionic radii. The compound with the smallest sum of ionic radii will have the greater lattice energy.
For example, to rank \(\text{MgS}\), \(\text{NaF}\), and \(\text{KCl}\): \(\text{MgS}\) has a charge product of 4, while \(\text{NaF}\) and \(\text{KCl}\) both have a product of 1. \(\text{MgS}\) is ranked highest. The tie between \(\text{NaF}\) and \(\text{KCl}\) is broken by comparing their radii, where the smaller ions in \(\text{NaF}\) give it the advantage, leading to a final ranking of \(\text{MgS} > \text{NaF} > \text{KCl}\).