Question
Write the Born-Haber cycle for the formation of NaCl from its elements and use it to calculate the lattice energy of NaCl, given the following data:
- Heat of formation of NaCl: −411 kJ/mol
- Sublimation energy of Na: +108 kJ/mol
- Bond dissociation energy of Cl₂: +244 kJ/mol
- First ionisation energy of Na: +496 kJ/mol
- Electron affinity of Cl: −349 kJ/mol
Solution — Step by Step
The formation reaction we want is:
The Born-Haber cycle breaks this one step into a series of steps that we can measure experimentally — then uses Hess’s Law to find the one we can’t measure directly (lattice energy).
Going from elements in standard states to the ionic solid, we need five intermediate steps:
- Sublimation of Na(s) → Na(g): kJ/mol
- Dissociation of ½Cl₂(g) → Cl(g): kJ/mol
- Ionisation of Na(g) → Na⁺(g): kJ/mol
- Electron gain by Cl(g) → Cl⁻(g): kJ/mol
- Lattice formation Na⁺(g) + Cl⁻(g) → NaCl(s): (this is what we want)
Note we take half the bond dissociation energy because the formula has ½Cl₂, not Cl₂.
The key principle: the overall enthalpy change is path-independent. So:
Substituting:
Lattice energy of NaCl = −788 kJ/mol
The negative sign tells us energy is released when the ionic lattice forms — which makes sense because ions of opposite charge attract each other strongly.
Why This Works
The Born-Haber cycle is just Hess’s Law in disguise. We can’t measure lattice energy directly in a lab — you can’t just pull apart a crystal and measure the energy. But we can measure formation enthalpies, ionisation energies, and electron affinities.
Since enthalpy is a state function, the total energy change is the same regardless of the path taken. We construct a cycle where every step except the lattice energy is known, then solve for the unknown. Think of it as a thermodynamic equation with one variable.
The large negative value (−788 kJ/mol) explains why NaCl is such a stable compound. The electrostatic attraction between Na⁺ and Cl⁻ ions in the crystal releases a massive amount of energy — that’s the lattice energy working for you.
Alternative Method (Energy Diagram Approach)
Instead of the algebraic approach, you can draw a proper energy level diagram:
- Place Na(s) + ½Cl₂(g) at the reference level (zero)
- Go up for endothermic steps (sublimation, dissociation, ionisation)
- Go down for exothermic steps (electron affinity, lattice formation, overall ΔHf)
The cycle must close — the energy going up equals the energy going down. This visual method is less error-prone for signs because you literally draw arrows up and down.
In JEE Advanced, they sometimes give lattice energy and ask you to find electron affinity or ionisation energy instead. The algebra is identical — just rearrange the same equation for whichever unknown they’ve hidden.
Common Mistake
Taking the full bond dissociation energy of Cl₂ instead of half.
The formation equation uses ½Cl₂, so you only need to break half a mole of Cl–Cl bonds. Using +244 kJ/mol instead of +122 kJ/mol gives U = −910 kJ/mol — completely wrong. Always check your stoichiometry in the formation equation before plugging in bond energies. This exact slip has cost marks in JEE Main multiple times.
Where is the lattice energy (always negative for ionic compounds).