The Born-Haber Cycle

The Born-Haber cycle, named after Max Born and Fritz Haber, is a specific and especially important application of Hess's law that breaks down the formation of an ionic solid from its constituent elements into a sequence of separate, individually measurable enthalpy steps. Its principal use is to calculate lattice enthalpy — the enthalpy released when a mole of ionic solid forms from its gaseous ions (or, equivalently, the energy needed to separate a mole of solid completely into gaseous ions) — a quantity that cannot be measured directly in the laboratory, since gaseous ions cannot simply be combined and observed forming a crystal under controlled calorimetric conditions.

For a general ionic solid $\text{MX}(s)$ formed from a metal $\text{M}$ and a non-metal $\text{X}_2$, the cycle connects the elements in their standard states to the solid compound through five conceptual steps, forming a closed thermochemical loop:

1. Sublimation of the solid metal to gaseous atoms: $\text{M}(s) \rightarrow \text{M}(g)$, enthalpy $\Delta_{sub}H$ (endothermic).
2. Ionization of the gaseous metal atoms: $\text{M}(g) \rightarrow \text{M}^+(g) + e^-$, enthalpy = ionization enthalpy (endothermic).
3. Dissociation of the gaseous non-metal molecule into atoms: $\tfrac{1}{2}\text{X}_2(g) \rightarrow \text{X}(g)$, enthalpy = half the bond dissociation enthalpy (endothermic).
4. Electron gain by the gaseous non-metal atoms: $\text{X}(g) + e^- \rightarrow \text{X}^-(g)$, enthalpy = electron gain enthalpy (usually exothermic for a first electron gain, though often endothermic overall for a second, as with oxide ion formation).
5. Lattice formation, combining the gaseous ions into the solid crystal: $\text{M}^+(g) + \text{X}^-(g) \rightarrow \text{MX}(s)$, enthalpy = $-(\text{lattice enthalpy})$ (strongly exothermic).

Because enthalpy is a state function, the sum of these five step enthalpies must exactly equal the directly measurable standard enthalpy of formation of the compound, $\Delta_f H^\circ$, giving the working equation of the cycle:

$\Delta_f H^\circ = \Delta_{sub}H + (\text{ionization enthalpy}) + \tfrac{1}{2}(\text{bond dissociation enthalpy}) + (\text{electron gain enthalpy}) + (-\text{lattice enthalpy})$ …