Lattice energy and Born-Haber cycle — how to calculate step by step

Question

Using the Born-Haber cycle, calculate the lattice energy of NaCl given:

Sublimation energy of Na = +108 kJ/mol
Ionisation energy of Na = +496 kJ/mol
Bond dissociation energy of Cl₂ = +242 kJ/mol
Electron affinity of Cl = -349 kJ/mol
Enthalpy of formation of NaCl = -411 kJ/mol

(JEE Main / JEE Advanced pattern)

Solution — Step by Step

The Born-Haber cycle breaks the formation of an ionic compound into individual thermodynamic steps. By Hess’s law, the sum of all steps must equal the enthalpy of formation.

flowchart TD
    A["Na(s) + ½Cl₂(g)"] -->|"ΔfH = -411"| B["NaCl(s)"]
    A -->|"Step 1: Sublimation\n+108"| C["Na(g) + ½Cl₂(g)"]
    C -->|"Step 2: Ionisation\n+496"| D["Na⁺(g) + ½Cl₂(g) + e⁻"]
    D -->|"Step 3: Bond dissociation\n+121 (half of 242)"| E["Na⁺(g) + Cl(g) + e⁻"]
    E -->|"Step 4: Electron affinity\n-349"| F["Na⁺(g) + Cl⁻(g)"]
    F -->|"Step 5: Lattice energy\nU = ?"| B

\Delta_f H = \Delta_{\text{sub}}H + \text{IE} + \frac{1}{2}\text{BDE} + \text{EA} + U

where $U$ is the lattice energy (the value we want to find).

-411 = 108 + 496 + \frac{242}{2} + (-349) + U

-411 = 108 + 496 + 121 - 349 + U

-411 = 376 + U

U = -411 - 376 = \mathbf{-787 \text{ kJ/mol}}

The negative sign indicates energy is released when gaseous ions come together to form the crystal lattice. This makes sense — forming bonds always releases energy.

Lattice energy is always negative (exothermic) when defined as the energy released during lattice formation. Some textbooks define it as the energy required to break the lattice apart, in which case it would be +787 kJ/mol. Always check which convention the question uses.

Why This Works

The Born-Haber cycle is Hess’s law applied to ionic compound formation. We cannot measure lattice energy directly — there is no experiment that forms a crystal from isolated gaseous ions. But we can measure every other step independently (sublimation, ionisation, electron affinity, bond dissociation, formation enthalpy). Since enthalpy is a state function, the missing step (lattice energy) is fixed by the constraint that all steps must add up to the formation enthalpy.

This is a powerful example of how thermodynamics lets us calculate quantities that cannot be measured directly.

Alternative Method — Kapustinskii Equation

For a quick estimate without the full cycle:

U = K \cdot \frac{\nu \cdot z^+ \cdot z^-}{r^+ + r^-}

where $K = 1202$ kJ pm/mol, $\nu$ = number of ions per formula unit, $z$ = ion charges, $r$ = ionic radii. This gives approximate values useful for comparing lattice energies of different compounds.

For JEE Advanced, the Born-Haber cycle is a reliable 4-mark question. The key is to draw the cycle diagram first and then write the equation. If you jump to the equation directly, you will likely mess up signs. Remember: sublimation, ionisation, and bond dissociation are always positive (energy input). Electron affinity is usually negative for halogens. Lattice energy is negative (energy output).

Common Mistake

The most common error: using the full bond dissociation energy of Cl₂ (242 kJ/mol) instead of half of it (121 kJ/mol). The formula unit NaCl requires only one Cl atom, not two. Since Cl₂ gives 2 Cl atoms, you must divide the BDE by 2. This single mistake changes the final answer by 121 kJ/mol and costs you full marks.

Question

Solution — Step by Step

Why This Works

Alternative Method — Kapustinskii Equation

Common Mistake

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