Lattice Energy and the Born–Haber Cycle

The Thermodynamic Pathway

Imagine you want to travel from London to Paris. You can take a direct flight, or you can fly to Brussels and then take a train. The total change in your position is the same regardless of the route. In thermodynamics, enthalpy behaves similarly; it is a state function. To find the lattice energy, we define two paths starting from the elements in their standard states and ending with the ionic solid. Path A is the direct formation of the solid from the elements, which gives us the standard enthalpy of formation. Path B is a stepwise process: first, we convert the elements into gaseous atoms; second, we ionize the atoms to form cations and anions; and finally, we combine these gaseous ions to form the solid. The final step of Path B is the lattice energy. Because the start and end points are identical for both paths, the sum of the enthalpy changes in Path B must equal the enthalpy change of Path A. This allows us to solve for the unknown lattice energy by rearranging the known values.

Defining the Components

To apply the cycle, we must identify the specific thermodynamic quantities involved in the stepwise path. The first step involves converting solid elements into gaseous atoms. For metals, this is the enthalpy of sublimation, \(\Delta H_{\text{sub}}\). For non-metals that exist as diatomic molecules, such as chlorine, we must first break the covalent bond. This requires half the bond dissociation energy, \(D_{X_2}\), because one mole of the solid contains one mole of anions derived from half a mole of \(X_2\) molecules.

The second step involves the formation of ions. For the cation, we remove electrons from the gaseous atom. The energy required for the first electron is the first ionisation energy, \(\mathrm{IE_1}\). If the ion has a charge of \(+2\), we also need the second ionisation energy, \(\mathrm{IE_2}\). For the anion, we add electrons to the gaseous atom. The energy change for the first electron is the first electron affinity, \(\mathrm{EA_1}\). Note that electron affinity is often defined as the energy released, so it is typically negative in enthalpy calculations, but conventions vary. In the Born–Haber equation, we usually sum the magnitudes and adjust signs based on the direction of energy flow.

The Born–Haber Equation

Combining these steps, we can write the enthalpy of formation, \(\Delta H_f\), as the sum of the sublimation, ionisation, bond dissociation, electron affinity, and lattice energy terms. The lattice energy, \(U\), is defined here as the energy released when gaseous ions form the solid, which is an exothermic process. Therefore, in the summation, it appears as a negative value if we are calculating \(\Delta H_f\), or we can define \(U\) as a positive magnitude and subtract it.

Using the convention where lattice energy \(U\) is the positive magnitude of energy released, the relationship is:

Rearranging to solve for the lattice energy \(U\):

Here, \(\sum \mathrm{IE}\) is the sum of all ionisation energies required to form the cation, and \(\sum \mathrm{EA}\) is the sum of electron affinities (usually just the first one for halides). The term \(\frac{1}{2}D_{X_2}\) represents the energy to break half a mole of \(X_2\) bonds. This equation is a direct application of Hess's law and allows for the precise calculation of lattice energy from measurable data.

Factors Influencing Lattice Energy

The magnitude of the lattice energy is primarily determined by Coulomb's law, which states that the electrostatic potential energy is proportional to the product of the charges and inversely proportional to the distance between them. Therefore, ions with higher charges (e.g., \(\mathrm{Mg^{2+}}\) and \(\mathrm{O^{2-}}\) vs \(\mathrm{Na^+}\) and \(\mathrm{Cl^-}\)) exhibit significantly stronger lattice energies. Additionally, smaller ionic radii lead to shorter internuclear distances, which also increases the lattice energy. This explains why magnesium oxide has a much higher melting point and lattice energy than sodium chloride, despite both being ionic solids. The Born–Haber cycle provides the quantitative tool to verify these trends by comparing calculated values against experimental observations of physical properties.