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.

Example: formation of lithium fluoride

The enthalpy of formation of lithium fluoride from atomic lithium and fluorine components is modeled in five steps in the diagram:
# Atomisation enthalpy of lithium
# Ionisation enthalpy of lithium
# Atomisation enthalpy of fluorine
# Electron affinity of fluorine
# Lattice enthalpy

The same calculation applies for any metal other than lithium or any non-metal other than fluorine.

The sum of the energies for each step of the process must equal the enthalpy of formation of the metal and non-metal, $Delta\; ext\{H\}\_\{\; ext\{f$.

$Delta\; ext\{H\}\_\{\; ext\{f\; =\; ext\{V\}\; +\; frac\{1\}\{2\}\; ext\{B\}\; +\; ext\{IE\}\_\{\; ext\{M\; -\; ext\{EA\}\_\; ext\{X\}\; -\; ext\{U\}\_\; ext\{L\}$

*V is the enthalpy of vaporization for metal atoms (lithium)
*B is the bond energy (of F₂). The coefficient 1/2 is used because the formation reaction is Li + 1/2 F₂ → LiF.
*$ext\{IE\}\_\{\; ext\{M$ is the ionization energy of the metal atom: $ext\{M\}\; +\; ext\{IE\}\_\; ext\{M\}\; o\; ext\{M\}^+\; +\; ext\{e\}^-$
*$ext\{EA\}\_\; ext\{X\}$ is the electron affinity of non-metal atom X (fluorine)
*$ext\{U\}\_\; ext\{L\}$ is the lattice energy

The net enthalpy of formation and the first four of the five energies can be determined experimentally, but the lattice energy cannot be measured directly. Instead, the lattice energy is calculated by subtracting the other four energies in the Born-Haber cycle from the net enthalpy of formation. [Moore, Stanitski, and Jurs. "Chemistry: The Molecular Science." 3rd edition. 2008. ISBN 0-495-10521-X. pages 320-321.]

References

ee also

*ionic crystal
*ionic compound
*ionic liquids