Lattice energy and Born-Haber cyclesEdexcel A-Level Chemistry: Revision notes

The lattice energy is the energy change when one mole of an ionic solid is formed from its gaseous ions, for example Na⁺(g) + Cl⁻(g) → NaCl(s). It is always exothermic (negative), because strong electrostatic attractions form between the oppositely charged ions. It measures the strength of ionic bonding but cannot be found directly, so it is calculated using a Born–Haber cycle.

Enthalpy change of atomisation: the enthalpy change when one mole of gaseous atoms is formed from the element in its standard state, e.g. Na(s) → Na(g) (+107 kJ mol⁻¹) or ½Cl₂(g) → Cl(g) (+122 kJ mol⁻¹). Always endothermic.

Electron affinity (first): the enthalpy change when one mole of gaseous atoms gains one electron each to form one mole of gaseous 1− ions, e.g. Cl(g) + e⁻ → Cl⁻(g) (−349 kJ mol⁻¹). Second electron affinity: adding an electron to a 1− ion, O⁻(g) + e⁻ → O²⁻(g) (+744 kJ mol⁻¹), which is endothermic because of repulsion between the ion and electron.

A Born–Haber cycle is a Hess's law cycle for an ionic solid. Build it in order, with state symbols:

1. Elements in standard states
2. Atomisation of the metal and the non-metal to gaseous atoms
3. Ionisation of the metal atoms (first, and second for 2+ ions)
4. Electron affinity of the non-metal
5. Lattice energy forming the solid

The enthalpy change of formation gives the direct route from the elements to the solid. Take care with multiples: for MgCl₂ use two Cl atoms and two electron affinities.

$\Delta_fH$ = $\Delta_{at}H$(Na) + $\Delta_{at}H$(Cl) + IE₁(Na) + EA₁(Cl) + lattice energy

−411 = +107 + 122 + 496 − 349 + lattice energy

−411 = +376 + lattice energy, so the lattice energy = −787 kJ mol⁻¹.

MgCl₂: lattice energy = −641 − [148 + 738 + 1451 + 2(122) + 2(−349)] = −641 − 1883 = −2524 kJ mol⁻¹.

MgO: O²⁻ needs the first and second electron affinities, EA₁ = −141 and EA₂ = +744 kJ mol⁻¹. If the lattice energy (−3791) is known, rearrange to find an unknown step: EA₂ = −602 − (148 + 738 + 1451 + 249 − 141 − 3791) = +744 kJ mol⁻¹.