Lattice Energy (Born-Haber Cycle)

The Born-Haber cycle is a thermodynamic trick to determine the lattice enthalpy of an ionic crystal, which cannot be measured directly. It relies on Hess's law: because enthalpy is a state function, the total energy depends only on the initial and final states, not on the path. You compare two paths from the elements to the solid salt: the direct path via the standard enthalpy of formation and a detour via gaseous atoms and ions. The detour consists of measurable partial steps: sublimation of the metal, ionization, dissociation of the non-metal, electron affinity and finally the sought lattice formation. Setting both paths equal lets you calculate the lattice enthalpy as the only unknown.

For NaCl the detour runs through five partial steps. First the sublimation of solid sodium to gaseous Na atoms, $\Delta{}H_{\mathrm{Sub}} = +\frac{108 kJ}{mol}$. Second the ionization of the Na atoms to Na⁺ ions, ionization energy $I = +\frac{496 kJ}{mol}$. Third the dissociation of the chlorine molecule; since only one Cl atom is needed per formula unit, half the bond energy enters, $\frac{1}{2}D = +\frac{122 kJ}{mol}$. Fourth the uptake of an electron by the Cl atom, electron affinity $EA = -\frac{349 kJ}{mol}$, exothermic. Fifth the union of the gaseous ions into the solid crystal lattice, the sought lattice enthalpy. The sum of all five steps must equal the directly measured standard enthalpy of formation ΔH°$f = -\frac{411 kJ}{mol}$. From this the lattice enthalpy follows.

The lattice enthalpy is defined as the energy released when isolated gaseous ions assemble into one mole of solid ionic crystal, or conversely the energy needed to separate them. There is no experiment that carries out just this one step: you cannot simply produce one mole of Na⁺ and one mole of Cl⁻ gaseous ions and let them condense in a controlled way to measure the heat. Therefore one takes the detour via the Born-Haber cycle, in which all the other partial steps are individually measurable. The lattice enthalpy then follows by calculation as the difference. This approach is a classic example of how Hess's law makes inaccessible quantities calculable indirectly.

The lattice energy is essentially determined by the Coulomb attraction between the ions and therefore depends on two factors: the charge of the ions and their distance, that is their radii. The higher the charges, the stronger the attraction; that is why magnesium oxide MgO with doubly charged ions has a very large lattice energy in magnitude of about $-\frac{3800 kJ}{mol}$, while sodium chloride with singly charged ions is only around $-\frac{788 kJ}{mol}$. The smaller the ions, the shorter the distance and the stronger the attraction; that is why the lattice energy of NaCl is larger in magnitude than that of KCl, because the potassium ion is larger. These relationships explain trends in melting points, hardness and solubility of salts.

In the Born-Haber cycle all partial steps must be added with the correct sign, otherwise the result is wrong. The electron affinity of chlorine is exothermic, the Cl atom releases energy when it takes up an electron, so $EA = -\frac{349 kJ}{mol}$. If you accidentally insert a positive value, the calculated lattice enthalpy shifts by almost $\frac{700 kJ}{mol}$, a gross error. When rearranging the cycle equation for the lattice enthalpy, the sign of the subtracted terms also flips, so that −349 becomes +349 there. This is exactly where most calculation errors happen. It helps to draw the cycle completely as a diagram and label each arrow with magnitude and sign before forming the sum.