Calculating Lattice Energies

The lattice energy of nearly any ionic solid can be calculated rather accurately using a modified form of Equation 1.1:

U, which is always a positive number, represents the amount of energy required to dissociate 1 mol of an ionic solid into the gaseous ions. As before, Q₁ and Q₂ are the charges on the ions and r₀ is the internuclear distance. We see from Equation 1.1 that lattice energy is directly related to the product of the ion charges and inversely related to the internuclear distance. The value of the constant k′ depends on the specific arrangement of ions in the solid lattice and their valence electron configurations, topics that will be discussed in more detail in the second semester. Representative values for calculated lattice energies, which range from about 600 to 10,000 kJ/mol, are listed in Table 1.1. Energies of this magnitude can be decisive in determining the chemistry of the elements.

Because the lattice energy depends on the product of the charges of the ions, a salt having a metal cation with a +2 charge (M²⁺) and a nonmetal anion with a −2 charge (X²⁻) will have a lattice energy four times greater than one with M⁺ and X⁻, assuming the ions are of comparable size (and have similar internuclear distances). For example, the calculated value of U for NaF is 910 kJ/mol, whereas U for MgO (containing Mg²⁺ and O²⁻ ions) is 3795 kJ/mol.

Because lattice energy is inversely related to the internuclear distance, it is also inversely proportional to the size of the ions. This effect is illustrated in Figure 1.1, which shows that lattice energy decreases for the series LiX, NaX, and KX as the radius of X⁻ increases. Because r₀ in Equation 1.1 is the sum of the ionic radii of the cation and the anion (r₀ = r⁺ + r⁻), r₀ increases as the cation becomes larger in the series, so the magnitude of U decreases. A similar effect is seen when the anion becomes larger in a series of compounds with the same cation.