Born-Haber Cycles: Lattice Energy and Ionic Bond Formation

What Is a Born-Haber Cycle?

A Born-Haber cycle is an energy cycle that uses Hess's Law to calculate the lattice energy of an ionic compound — a quantity that cannot be measured directly.

Lattice energy ($\Delta_{lat}H$) is the enthalpy change when gaseous ions come together to form one mole of an ionic solid. It's a measure of how strong the ionic bonds are in the lattice.

Learning Goals: By the end of this guide, you should be able to:

1. Define all five enthalpy changes in a Born-Haber cycle.
2. Construct a Born-Haber cycle diagram for any simple ionic compound.
3. Use Hess's Law to calculate lattice energy.
4. Explain how ionic charge and radius affect lattice energy.

The Five Enthalpy Terms

Constructing the Cycle: NaCl Example

The Born-Haber cycle for NaCl connects these steps:

Route 1 (direct): $Na(s) + \frac{1}{2}Cl_2(g) \rightarrow NaCl(s)$ with $\Delta_f H° = -411\ kJ/mol$

Route 2 (indirect):

1. Atomise sodium: $Na(s) \rightarrow Na(g)$ — $\Delta_{at} H° = +107\ kJ/mol$
2. Atomise chlorine: $\frac{1}{2}Cl_2(g) \rightarrow Cl(g)$ — $\Delta_{at} H° = +122\ kJ/mol$
3. Ionise sodium: $Na(g) \rightarrow Na^+(g) + e^-$ — $IE_1 = +496\ kJ/mol$
4. Add electron to chlorine: $Cl(g) + e^- \rightarrow Cl^-(g)$ — $EA = -349\ kJ/mol$
5. Form lattice: $Na^+(g) + Cl^-(g) \rightarrow NaCl(s)$ — $\Delta_{lat} H° = ?$

Applying Hess's Law

By Hess's Law, Route 1 = Route 2:

The lattice energy of NaCl is -787 kJ/mol (exothermic — energy is released when the lattice forms).

Factors Affecting Lattice Energy

Lattice energy is more exothermic (stronger ionic bonding) when:

This follows Coulomb's Law: $F \propto \frac{q^+ \times q^-}{r^2}$

Worked Examples

Example 1: Calculate Lattice Energy of MgCl₂

Given: $\Delta_f H° = -641$, $\Delta_{at}H°(Mg) = +148$, $\Delta_{at}H°(Cl) = +122$, $IE_1(Mg) = +738$, $IE_2(Mg) = +1451$, $EA(Cl) = -349$

Note: MgCl₂ needs two atomisation of Cl and two electron affinities, plus both $IE_1$ and $IE_2$ for Mg²⁺.

Example 2: Why Is MgO's Lattice Energy Larger Than NaCl's?

$Mg^{2+}$ has a higher charge (+2 vs +1) and a smaller radius than $Na^+$. $O^{2-}$ has a higher charge (-2 vs -1) than $Cl^-$. By Coulomb's Law, the electrostatic attraction is much stronger, so the lattice energy is much more exothermic.

Common Mistakes

1. Forgetting to multiply for stoichiometry — In $MgCl_2$, you need 2× atomisation of Cl, 2× electron affinity, AND both first and second ionisation energies.

2. Getting the sign of electron affinity wrong — First electron affinity is usually negative (exothermic). Second electron affinity is positive (endothermic) because you're adding an electron to an already negative ion.

3. Confusing lattice formation with lattice dissociation — Formation is exothermic (ions → solid, negative). Dissociation is endothermic (solid → ions, positive). Check which definition your syllabus uses.

4. Not using Hess's Law correctly — Always set up the cycle: $\Delta_f H° =$ sum of all other steps. Rearrange to find the unknown.

Exam Tips (A-Level / AP / IB)

• Draw the cycle before calculating. Label every arrow with the enthalpy term and its value.
• Always check your sign conventions — different exam boards define lattice energy differently (formation vs. dissociation).
• For multi-step ionisation (like $Mg^{2+}$), include each IE step separately.
• State Hess's Law explicitly in your answer: "By Hess's Law, the enthalpy change is independent of the route taken."

Frequently Asked Questions

Why can't lattice energy be measured directly?

You would need to create gaseous ions and bring them together to form a crystal — this isn't experimentally feasible. Born-Haber cycles use measurable quantities (formation, atomisation, ionisation, electron affinity) to calculate it indirectly.

What is the difference between lattice formation and lattice dissociation enthalpy?

Lattice formation: gaseous ions → solid (exothermic, negative). Lattice dissociation: solid → gaseous ions (endothermic, positive). They have the same magnitude but opposite signs.

How does lattice energy relate to thermal stability?

Higher lattice energy → more energy needed to break the lattice → higher melting point and greater thermal stability. This is why MgO (mp 2852°C) melts at a much higher temperature than NaCl (mp 801°C).

Related Topics

• Gibbs Free Energy — Thermodynamic feasibility extends beyond just enthalpy.
• Chemical Bonds — Understand ionic bond formation before studying energy cycles.
• Periodic Trends — How ionisation energy and atomic radius trends feed into Born-Haber cycles.