In physical chemistry, a Born-Haber cycle is an energy cycle representation of Hess's Law. It allows us to calculate thermodynamic values that cannot be measured directly in a laboratory, most notably the **Lattice Enthalpy** of an ionic solid. Understanding how to set up these cycles and avoid arithmetic sign errors is essential for top-tier exam performance.
- Purpose: An application of Hess's Law used to find experimental lattice enthalpy.
- Upward Arrows: Endothermic processes (Atomisation, Ionisation Energy, Second Electron Affinity).
- Downward Arrows: Exothermic processes (Formation, First Electron Affinity, Lattice Enthalpy of Formation).
- Boards: A core component of OCR A (Module 5) and AQA (Topic 3.1.8).
Enthalpy Definitions & Equation Table
Before constructing a cycle, you must memorize the key enthalpy changes. Use this table as a quick reference sheet:
| Enthalpy Change | Symbol | Standard Definition Equation (e.g., Sodium) | Typical Sign |
|---|---|---|---|
| Standard Enthalpy of Formation | ΔHfθ | Na(s) + ½Cl2(g) → NaCl(s) | Exothermic (-) |
| Standard Enthalpy of Atomisation | ΔHatθ | Na(s) → Na(g) | Endothermic (+) |
| First Ionisation Energy | IE1 | Na(g) → Na+(g) + e- | Endothermic (+) |
| First Electron Affinity | EA1 | Cl(g) + e- → Cl-(g) | Exothermic (-) |
| Second Electron Affinity | EA2 | O-(g) + e- → O2-(g) | Endothermic (+) |
| Lattice Enthalpy of Formation | ΔHLEθ | Na+(g) + Cl-(g) → NaCl(s) | Exothermic (-) |
Step-by-Step Construction Layout
Establish the Baseline (Elements in standard states)
Write the constituent elements in their standard states at a middle level on your diagram (e.g. Mg(s) + ½O2(g)).
Draw the Formation Arrow (Downwards)
Draw a downward arrow from the baseline elements to the solid ionic lattice at the bottom level. Label this ΔHfθ.
Atomise and Ionise (Upwards)
Create upward steps representing the energy required to turn the metal and non-metal elements into gaseous atoms (ΔHatθ), and then into gaseous ions (IE1, IE2).
Add Electron Affinities and Lattice Enthalpy
Add the non-metal's first electron affinity (downwards). If a 2- ion is formed (like O2-), draw the second electron affinity upwards. Finally, complete the loop with a massive downward arrow from the gaseous ions directly to the solid lattice, representing ΔHLEθ.
Worked Example: Calculating Lattice Enthalpy for MgO
Calculate the lattice enthalpy of formation (ΔHLEθ) for Magnesium Oxide (MgO) using the following data:
- ΔHfθ(MgO) = -602 kJ mol-1
- ΔHatθ(Mg) = +148 kJ mol-1
- IE1(Mg) = +738 kJ mol-1
- IE2(Mg) = +1450 kJ mol-1
- ΔHatθ(O) = +249 kJ mol-1
- EA1(O) = -141 kJ mol-1
- EA2(O) = +798 kJ mol-1
Applying Hess's Law
Hess's Law states that the total enthalpy change for a reaction is independent of the route taken. Looking at the cycle, we can equate the indirect route (atomisation, ionisation, affinities, and lattice enthalpy) to the direct route (formation):
Substitute the values, keeping close track of the positive and negative signs:
Sum the known terms on the right-hand side:
Rearrange to solve for ΔHLEθ:
Common Pitfall: Diatomic atomisation values
Exam papers often supply the bond dissociation enthalpy of Oxygen (O=O bond) instead of the enthalpy of atomisation. Remember: ΔHat of oxygen is half the bond dissociation enthalpy (ΔHatθ = ½ E(O=O)). Do not double-count or forget to divide by 2!
Pro-Tip: Lattice Enthalpy sign check
Lattice enthalpy of formation must always be a large negative value (exothermic), because bringing opposite gaseous ions together to form a solid ionic lattice releases substantial electrostatic energy. If your calculated value is positive, check for simple arithmetic sign flips in your algebraic rearrangement!
Frequently Asked Questions
Adding a negatively charged electron to a neutral oxygen atom releases energy because of the nuclear attraction. However, adding a second electron to a negatively charged O- ion requires energy to overcome the strong electrostatic repulsion between the negative ion and the incoming negative electron.
Lattice enthalpy is determined by charge density: ionic charge and ionic radius. Higher ionic charges (e.g. Mg2+ vs Na+) and smaller ionic radii lead to stronger electrostatic attraction, resulting in a more exothermic (larger negative) lattice enthalpy.
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