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

Question

How do we calculate lattice energy using the Born-Haber cycle? Walk through the complete cycle for NaCl as an example.

Solution — Step by Step

Lattice energy ( $U$ ) is the energy released when gaseous cations and anions come together to form one mole of an ionic solid:

\text{Na}^+(g) + \text{Cl}^-(g) \rightarrow \text{NaCl}(s) \qquad \Delta H = -U

It cannot be measured directly — so we use Hess’s law through the Born-Haber cycle.

The overall formation reaction:

\text{Na}(s) + \frac{1}{2}\text{Cl}_2(g) \rightarrow \text{NaCl}(s) \qquad \Delta H_f = -411 \text{ kJ/mol}

We will break this into steps whose energies we know, and solve for the unknown lattice energy.

Step	Process	Symbol	Value (kJ/mol)
1	Sublimation of Na(s) to Na(g)	$\Delta H_{sub}$	+108
2	Ionisation of Na(g) to Na $^+$ (g)	IE $_1$	+496
3	Dissociation of $\frac{1}{2}$ Cl $_2$ (g) to Cl(g)	$\frac{1}{2}\Delta H_{diss}$	+121
4	Electron gain by Cl(g) to Cl $^-$ (g)	$\Delta H_{eg}$	-349
5	Formation of lattice NaCl(s)	$-U$	?

By Hess’s law, the sum of all steps equals $\Delta H_f$ :

\Delta H_f = \Delta H_{sub} + \text{IE}_1 + \frac{1}{2}\Delta H_{diss} + \Delta H_{eg} + (-U)

Substituting values:

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

-411 = 376 - U

U = 376 + 411 = \boxed{787 \text{ kJ/mol}}

flowchart TD
    A["Na(s) + 1/2 Cl2(g)"] -->|"Delta Hf = -411"| B["NaCl(s)"]
    A -->|"Step 1: Sublimation +108"| C["Na(g) + 1/2 Cl2(g)"]
    C -->|"Step 2: IE +496"| D["Na+(g) + 1/2 Cl2(g) + e-"]
    D -->|"Step 3: Bond dissociation +121"| E["Na+(g) + Cl(g) + e-"]
    E -->|"Step 4: Electron affinity -349"| F["Na+(g) + Cl-(g)"]
    F -->|"Step 5: Lattice energy -U"| B

Why This Works

The Born-Haber cycle is just Hess’s law applied to ionic compound formation. Since enthalpy is a state function, the total energy change from elements to ionic solid must be the same regardless of the path taken. We know the overall $\Delta H_f$ and all intermediate steps except lattice energy — so we solve for it algebraically.

The cycle works because every step corresponds to a measurable thermodynamic quantity: sublimation enthalpy, ionisation energy, bond dissociation energy, and electron gain enthalpy are all experimentally determined.

Alternative Method

For a quick estimation (not exact calculation), use the Kapustinskii equation:

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

where $\nu$ = number of ions per formula unit, $z$ = charges, $r$ = ionic radii, and $K$ = 1202 kJ pm/mol. This gives approximate lattice energies without needing the full Born-Haber data.

Common Mistake

Students frequently confuse the sign of electron gain enthalpy with electron affinity. Electron gain enthalpy for Cl is -349 kJ/mol (exothermic — energy is released). Electron affinity is often reported as a positive number (349 kJ/mol) in older textbooks. Mixing up signs in the Born-Haber cycle gives a lattice energy that is off by ~700 kJ/mol. Always check which convention your data uses before plugging in. JEE Advanced 2021 had a Born-Haber calculation where this sign error was the primary trap.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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