Born-Haber cycles are used in physical chemistry to relate the lattice energy of an ionic compound to its enthalpy of formation and to the energies of atomization, ionization and electron gain of its elements.

The lattice energy of an ionic compound is the potential energy holding its ions together. It is expressed as a negative value - the more negative, the stronger the ionic bonding. It is equal to the amount of energy given out when the ions come together to form the compound. It cannot be measured directly, but a Born-Haber cycle relates it to all the other energies involved in the formation of an ionic compound, which can be measured. This is how the lattice energy of a crystal is derived experimentally.

The atoms that are to form the compound must first be separated from each other to become gaseous - this is atomization. Oxygen, for example, must be converted from O2 molecules to free oxygen atoms, O. This requires that a certain amount of energy is provided - the atomization energy of the elements.

The atomized elements must now be converted to ions. In the case of the metal, e.g. sodium in sodium chloride, this involves removing electrons to create a positive cation. The energy required to remove the outermost electron from a mole of atoms is the first ionization energy of the element. If the cation is to have multiple charge, such as Al3+, there will be further ionization energies.

The non-metal, e.g. chlorine in sodium chloride, must gain electrons to become a negative anion. The energy taken in or released during each successive gain of electrons by a mole of atoms is the electron affinity of that element. The first electron affinity of non-metals is usually negative, meaning that it gives out energy. However the subsequent values are always positive: endothermic.

Finally, the gaseous ions come together to form the crystal. This releases a lot of energy as it is much more energetically favourable for gaseous counterions to convert to an ionic solid than to remain in the highly unstable gas phase.

The energies of atomization, ionization and electron affinity can all be measured experimentally, as can the enthalpy of formation - the energy change when the compound is produced directly from its elements. These energies are related through a Born-Haber cycle:

 GASEOUS IONS - - - - - - - 
      |                   |
      |2                  |
      |                   |
      |                   |3
      |1                  |
      |                   |
 ELEMENTS AT              |
 AMBIENT STATE            |
      |                   |
      - - - - - - - IONIC COMPOUND
1 = atomization energies
2 = ionization energies and electron affinities
3 = lattice energy
4 = enthalpy of formation

To determine the lattice energy of an ionic compound, the energies of atomization, ionization and electron affinity are subtracted from its enthalpy of formation. This, like a Hess cycle, is simply an application of basic thermodynamics - energy is conserved. The enthalpy of formation is the complete energy change accompanying the formation of the compound, so if you measure individually all the steps but the lattice energy, the difference between this sum and the enthalpy of formation is equal to the lattice energy.

This can be summarised as lattice energy = enthalpy of formation - [atomization energy of metal + atomization energy of non-metal + ionization energies of metal + electron affinities of non-metal]

With lots of thanks to discofever.

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