Born-Haber Cycles and Lattice Enthalpy

Born-Haber Cycles and Lattice Enthalpy

A Born-Haber cycle is an application of Hess's law to calculate the lattice enthalpy of an ionic compound — a value that cannot be measured directly by experiment.

Lattice Enthalpy

Lattice enthalpy of formation (ΔHL) is the enthalpy change when one mole of an ionic compound is formed from its gaseous ions under standard conditions.

Na⁺(g) + Cl⁻(g) → NaCl(s) ΔHL = −787 kJ mol⁻¹

This value is always exothermic (negative) because forming a lattice from separated ions releases energy as electrostatic attractions are established.

Some textbooks define lattice enthalpy as the enthalpy of dissociation (breaking the lattice apart into gaseous ions), which is the same magnitude but endothermic (positive). Always check which definition is being used.

Enthalpy Terms in a Born-Haber Cycle

To construct a Born-Haber cycle, you need the following enthalpy changes:

Term	Symbol	Definition
Enthalpy of formation	ΔHf	Enthalpy change when 1 mol of compound forms from its elements in their standard states
Enthalpy of atomisation	ΔHat	Enthalpy change when 1 mol of gaseous atoms forms from the element in its standard state
First ionisation energy	IE₁	Energy to remove 1 mol e⁻ from 1 mol gaseous atoms
Second ionisation energy	IE₂	Energy to remove 1 mol e⁻ from 1 mol gaseous 1+ ions
First electron affinity	EA₁	Enthalpy change when 1 mol of gaseous atoms each gain 1 electron
Second electron affinity	EA₂	Enthalpy change when 1 mol of gaseous 1⁻ ions each gain 1 electron

Key points:

• Atomisation enthalpies are always endothermic (bond breaking)
• Ionisation energies are always endothermic (removing electrons)
• First electron affinities are usually exothermic (adding an electron to a neutral atom releases energy)
• Second electron affinities are always endothermic (adding an electron to an already-negative ion requires energy to overcome repulsion)

Constructing a Born-Haber Cycle for NaCl

The cycle connects the formation of NaCl(s) from its elements via two routes:

Route 1 (direct): Na(s) + ½Cl₂(g) → NaCl(s) ΔHf

Route 2 (indirect — the Born-Haber steps):

1. Atomise sodium: Na(s) → Na(g) ΔHat = +107 kJ mol⁻¹

2. Atomise chlorine: ½Cl₂(g) → Cl(g) ΔHat = +122 kJ mol⁻¹

3. Ionise sodium: Na(g) → Na⁺(g) + e⁻ IE₁ = +496 kJ mol⁻¹

4. Add electron to chlorine: Cl(g) + e⁻ → Cl⁻(g) EA₁ = −349 kJ mol⁻¹

5. Form lattice: Na⁺(g) + Cl⁻(g) → NaCl(s) ΔHL = ?

By Hess's law:

ΔHf = ΔHat(Na) + ΔHat(Cl) + IE₁(Na) + EA₁(Cl) + ΔHL

Rearranging:

ΔHL = ΔHf − ΔHat(Na) − ΔHat(Cl) − IE₁(Na) − EA₁(Cl)

ΔHL = (−411) − (+107) − (+122) − (+496) − (−349)

ΔHL = −411 − 107 − 122 − 496 + 349 = −787 kJ mol⁻¹

Born-Haber Cycle for MgO (2+ and 2⁻ Ions)

For compounds with multiply-charged ions, you must include successive ionisation energies and successive electron affinities:

1. Atomise Mg(s) → Mg(g) ΔHat

2. Atomise ½O₂(g) → O(g) ΔHat

3. First ionisation: Mg(g) → Mg⁺(g) + e⁻ IE₁

4. Second ionisation: Mg⁺(g) → Mg²⁺(g) + e⁻ IE₂

5. First electron affinity: O(g) + e⁻ → O⁻(g) EA₁ (exothermic)

6. Second electron affinity: O⁻(g) + e⁻ → O²⁻(g) EA₂ (endothermic, ~+798 kJ mol⁻¹)

7. Form lattice: Mg²⁺(g) + O²⁻(g) → MgO(s) ΔHL

The lattice enthalpy of MgO is much more exothermic (−3850 kJ mol⁻¹) than NaCl because of the higher ionic charges (2+ and 2−) and smaller ionic radii.

Factors Affecting Lattice Enthalpy

Lattice enthalpy becomes more exothermic when:

• Ionic charge increases — greater electrostatic attraction (lattice enthalpy is roughly proportional to the product of charges, q⁺ × q⁻)
• Ionic radius decreases — ions are closer together, so the attraction is stronger

This is why MgO (−3850 kJ mol⁻¹) has a much more exothermic lattice enthalpy than NaCl (−787 kJ mol⁻¹): Mg²⁺ and O²⁻ have higher charges than Na⁺ and Cl⁻, and Mg²⁺ is smaller than Na⁺.

Theoretical vs Experimental Lattice Enthalpies

• Born-Haber (experimental) values come from measurable enthalpy data via Hess's law
• Theoretical values are calculated assuming a purely ionic model (point charges with no polarisation)

If the two values agree closely, the compound is well described by the ionic model (e.g. NaCl: experimental −787, theoretical −770 kJ mol⁻¹).

If the experimental value is significantly more exothermic than the theoretical value, this indicates covalent character — the anion has been polarised by the cation, distorting its electron cloud. This is explained by Fajans' rules: polarisation increases with higher cation charge, smaller cation radius, and larger anion radius.

Enthalpy of Solution and Hydration

Born-Haber cycles can be extended to calculate enthalpy of solution:

ΔHsol = −ΔHL + ΔHhyd(cation) + ΔHhyd(anion)

Where enthalpy of hydration is the enthalpy change when 1 mol of gaseous ions is dissolved in sufficient water to form an infinitely dilute solution. Hydration enthalpies are always exothermic and become more exothermic with increasing charge density.

Exam Tips

• Draw the cycle as an energy level diagram with elements at the top, gaseous ions in the middle, and the ionic compound at the bottom
• Be meticulous with signs — a common error is to forget that EA₁ is exothermic (negative)
• For 2+ or 2− ions, you need TWO ionisation energies or TWO electron affinities
• If asked to compare lattice enthalpies, always refer to charge and radius