Lattice energy and Born-Haber cycles

Lattice energy (ΔHlatt) is the enthalpy change when one mole of an ionic solid is formed from its gaseous ions under standard conditions:

$\text{Na}^+(g) + \text{Cl}^-(g) \rightarrow \text{NaCl}(s) \qquad \Delta H_{latt}$

It is always exothermic (negative) because strong electrostatic attractions form as the oppositely charged gaseous ions come together into the ordered lattice. The more energy released, the more stable the lattice.

You can't put gaseous ions in a beaker and measure this directly, so lattice energy is found indirectly using a Born–Haber cycle (an application of Hess's law).

A Born–Haber cycle relates the enthalpy of formation of an ionic solid to all the steps that turn the elements into gaseous ions and then into the lattice:

• Enthalpy of atomisation (ΔHat): element → gaseous atoms (endothermic). For a diatomic, ΔHat(Cl) uses ½ the bond energy (½Cl₂ → Cl).
• Ionisation energy (IE): gaseous atom → gaseous cation (endothermic).
• Electron affinity (EA): gaseous atom + e⁻ → gaseous anion (1st EA exothermic; 2nd EA endothermic).
• Lattice energy (ΔHlatt): gaseous ions → solid (exothermic).

Worked (NaCl). ΔHf = −411; ΔHat(Na) = +107; ΔHat(Cl) = +122; IE₁(Na) = +496; EA₁(Cl) = −349 (kJ mol⁻¹): $\Delta H_{latt} = \Delta H_f - (\Delta H_{at}(\text{Na}) + IE_1 + \Delta H_{at}(\text{Cl}) + EA_1)$ $= -411 - (107 + 496 + 122 - 349) = -411 - 376 = -787\,\text{kJ mol}^{-1}$

For a 2:1 compound like MgCl₂ you need both ionisation energies (to make Mg²⁺) and you double the chlorine atomisation and electron affinity (two Cl⁻).

Lattice energy depends on the electrostatic attraction between ions:

$\Delta H_{latt} \propto \frac{q^+ \times q^-}{r^+ + r^-}$

So lattice energy is more exothermic when:

• the ionic charges are higher (e.g. MgO with 2+/2− is far more exothermic than NaCl with 1+/1−), and
• the ions are smaller (e.g. LiF more exothermic than KI).

Theoretical vs experimental. A theoretical lattice energy is calculated assuming perfectly spherical, purely ionic ions. The experimental (Born–Haber) value is the real one. When the experimental value is more exothermic than theoretical, the bonding has covalent character: the cation polarises (distorts) the anion's electron cloud, adding extra bonding. The bigger the gap, the more covalent character (e.g. AgI shows a larger difference than AgCl).