Born-Haber cycle
/ born HAH-ber /
Lattice enthalpy cannot be measured directly — you cannot watch gaseous sodium and chloride ions snap together in a calorimeter. The Born-Haber cycle is the clever accounting trick that gets the number anyway. It is an application of Hess's law: because energy is a state function, the total energy change going from elements to compound is the same whichever route you take, so an immeasurable step can be found by adding up all the measurable ones around a closed loop.
Picture two routes from the elements in their standard states (solid sodium and chlorine gas) to solid NaCl. The direct route is just the enthalpy of formation, easily measured. The long way around builds the compound step by step through gaseous ions: turn solid sodium into gaseous atoms (atomization or sublimation enthalpy), ionize those atoms (ionization energy), split chlorine molecules into atoms (bond dissociation), add electrons to those atoms (electron affinity), and finally let the gaseous Na+ and Cl- ions assemble into the solid (the lattice enthalpy we want). Set the two routes equal — the formation enthalpy must equal the sum of all the long-way steps — and solve for the one unknown. Every other quantity has been measured independently, so lattice enthalpy drops out.
The Born-Haber cycle matters in two directions. First, it delivers the experimental lattice enthalpy used to test the ionic model against the Born-Lande calculation. Second, run in reverse it lets chemists estimate quantities that are themselves hard to measure, such as an electron affinity, or predict whether a hypothetical compound (say, NaCl2 or NeF) could ever be stable by checking if its overall formation enthalpy comes out favourable. It is one of the most elegant demonstrations in chemistry that the conservation of energy is a powerful tool, not just a principle.
Why is there no NaCl2? Running a Born-Haber cycle for the imaginary compound, the second ionization energy of sodium (stripping a core electron) is so enormous that even the larger lattice enthalpy of a 2+ salt cannot pay for it, so the formation enthalpy comes out strongly positive. The cycle thus explains why sodium is resolutely +1.
Run in reverse, a Born-Haber cycle shows why NaCl2 cannot form — the second ionization energy is unaffordable.
The cycle relies on Hess's law and on every other step being known. Be careful with signs (ionization energy positive, electron affinity usually negative for the first electron) and with whether you want the formation or dissociation lattice enthalpy.