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The '''Born–Haber cycle''' is a particular application of [[Hess's law]] to deduce the changes in [[enthalpy]] for reactions involving gaseous ions, particularly the determination of [[Lattice energy|lattice enthalpies]].<ref name="Atkins">{{Atkins4th|pages=49–52}}.</ref> It is named after the German scientists [[Max Born]] (1882–1970) and [[Fritz Haber]] (1868–1934).

As with other applications of Hess's law, the Born–Haber cycle relies on the fact that enthalpy is a [[state function]], that is that the change in enthalpy between two states of a system is independent of the path taken to change the state. In a Born–Haber cycle, the path is cyclical, with the initial and final states being identical, so Δ''H''&nbsp;=&nbsp;0. This allows a single unknown enthalpy change along the path to be calculated by reference to the others. Born–Haber cycles can also be constructed for other state functions, such as [[internal energy]], [[entropy]] and [[Gibbs energy]].

==Lattice enthalpy==
{{main|Lattice energy}}
The lattice enthalpy Δ<sub>latt</sub>''H'' of an [[ionic compound]] MX is the enthalpy change for the reaction<ref group="note">The lattice enthalpy is conventionally defined as the production of gaseous ions so that it is positive for all solids.</ref>
:MX (cr) &rarr; M<sup>+</sup> (g) + X<sup>−</sup> (g)
It cannot be measured directly, as the ions cannot be sufficiently separated in the gas state: in practice, all ionic compounds vaporize to covalent molecules. However, the enthalpies of formation of each of the ions separately are known: the [[enthalpy change of atomization]] for the element concerned plus the [[ionization energy]] for cations or the [[electron affinity]] for anions.<ref group="note">As measured, ionization energies and electron affinities are [[internal energy]] changes and not [[enthalpy]] changes. However, the difference is small enough to be ignored in these applications.</ref>

{|
|-
| colspan=3 align=center | Na<sup>+</sup> (g) + Cl (g) + e<sup>−</sup>
|-
| Δ<sub>f</sub>''H''[Na<sup>+</sup> (g)] || || Δ<sub>f</sub>''H''[Cl<sup>−</sup> (g)]
|-
| Na (g) + Cl (g) || || Na<sup>+</sup> (g) + Cl<sup>−</sup> (g)
|-
| Δ<sub>f</sub>''H''[Cl (g)] || rowspan=5 | || rowspan=5 | −Δ<sub>latt</sub>''H''[NaCl (cr)]
|-
| Na (g) + ½Cl<sub>2</sub> (g)
|-
| Δ<sub>f</sub>''H''[Na (g)]
|-
| Na (cr) + ½Cl<sub>2</sub> (g)
|-
| −Δ<sub>f</sub>''H''[NaCl (cr)]
|-
| colspan=3 align=center | NaCl (cr)
|-
|}

==Notes and references==
===Notes===
{{reflist|group=note}}

===References===
{{reflist}}

==External links==
{{wikipedia|Born–Haber cycle}}

[[Category:Thermodynamics]]

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