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# Born-Haber cycle

The Born-Haber Cycle is an approach to analyzing reaction energies. It was named after and developed by the two German scientists Max Born and Fritz Haber.

The Born-Haber cycle involves the formation of an ionic compound from the reaction of a metal (often a Group I or Group II element) with a non-metal. Born-Haber cycles are used primarily as a means of calculating lattice enthalpies, which cannot otherwise be measured directly.

The lattice enthalpy is the enthalpy change involved in formation of the ionic compound from gaseous ions. Some chemists define it as the energy to break the ionic compound into gaseous ions. The former definition is invariably exothermic and the latter is endothermic.

A Born-Haber cycle calculates the lattice enthalpy by comparing the standard enthalpy change of formation of the ionic compound (from the elements) to the enthalpy required to make gaseous ions from the elements. This is an application of Hess's Law.

It is this latter calculation that is complex. To make gaseous ions from elements it is necessary to atomise the elements (turn each into gaseous atoms) and then to ionise the atoms. If the element is normally a molecule then we have to consider its bond dissociation enthalpy (see also bond energy). The energy involved in removing electrons to make a cation is called the ionization energy. The enthalpy of adding electrons to an atom to make it an anion is called the electron affinity.

Diagram Explanation

1. Atomisation enthalpy of metal (in this case lithium)
2. Ionisation enthalpy of metal
3. Atomisation enthalpy of non-metal (in this case fluorine)
4. Electron affinity of non-metal
5. Lattice enthalpy

The sum of the energies for each step of the process must equal the enthalpy of formation of the metal and non-metal, ΔHf. $\Delta\text{H}_{\text{f}} = \text{V} + \frac{1}{2}\text{B} + \text{IE}_{\text{M}} - \text{EA}_\text{X} - \text{U}_\text{L}$

V is the enthalpy of vaporization for metal atoms

B is the bond energy

IEM is the ionization energy of metal atom $\text{M} + \text{IE}_\text{M} \to \text{M}^+ + \text{e}^-$

EAX is the electron affinity of non-metal atom X

UL is the lattice energy