Born-Haber cycle revisited — calculating lattice energy: Foundations

This is the "toolbox" page for the Born–Haber topic. The parent note throws a dozen symbols at you — $U$ , $Δ H_{f}$ , $Δ H_{s u b}$ , $I E$ , $D$ , $E A$ — and assumes you already read them. Here we build every one from zero, in the order they stack on top of each other.

0. What is a "state" and a "state (s), (g)" label?

Before any energy, learn the little letters in brackets.

Why the topic needs it. Lattice energy is defined as going from $(g)$ ions to an $(s)$ crystal. If you ignore these labels you will mix up "atoms floating apart" with "atoms locked in a crystal" — and those differ by a huge amount of energy.

1. The plus/minus superscript — an ion and its charge

Figure — Born-Haber cycle revisited — calculating lattice energy

Look at the figure: the neutral atom on the left has balanced charge; the red electron leaves; the atom on the right is now short one negative, so it reads $+$ . Opposite signs attract — that attraction is the whole reason a crystal holds together.

Why the topic needs it. Every step in the cycle either makes an ion (ionisation, electron affinity) or assembles ions (lattice energy). You must read $Mg^{2 +}$ as "lost 2 electrons" to know you need two ionisation steps.

2. The mole — "one recipe's worth"

Why the topic needs it. All energies are quoted in $kJ mol^{- 1}$ = kilojoules per mole. This keeps every step on the same footing so we can add them.

3. Enthalpy $H$ and enthalpy change $Δ H$

Now the central quantity. See Hess's Law and state functions for the deeper version.

The red arrow in the figure points down: that step (like snapping ions into a crystal) releases energy, so its $Δ H$ carries a minus sign. Every step in the cycle is one such arrow, up or down.

Why the topic needs it. The entire Born–Haber cycle is a bookkeeping of $Δ H$ values, and getting the sign right (up vs down) is where nearly all mistakes happen.

4. State function — why the route doesn't matter

Why the topic needs it. This is the permission slip for the whole method: because $Δ H$ is path-independent, the direct road (formation) and the long road (via gaseous ions) must have the same total — and that equality is the equation we solve.

5. Enthalpy of formation $Δ H_{f}$

See Enthalpy of formation for its full definition and sign conventions.

Why the topic needs it. This is the direct road — one single measurable step from elements to salt. It is the "known total" we compare the long road against.

6. Sublimation / atomisation $Δ H_{s u b}$

Why the topic needs it. The long road must first free the metal atoms from their solid before they can be ionised. It is the first "up" step.

7. Ionisation energy $I E$

See Ionisation energy trends for why $I E_{2} > I E_{1}$ .

Why the topic needs it. This is the step that actually makes the cation. For $Mg^{2 +}$ you need two removals, so $I E_{1} + I E_{2}$ — a fact the parent's MgCl₂ example hinges on.

8. Bond dissociation enthalpy $D$ (and the $\frac{1}{2} D$ trick)

The red bond in the figure snaps into two free atoms. Notice one $Cl_{2}$ gives two $Cl$ atoms.

See Bond dissociation enthalpy.

Why the topic needs it. This is how the non-metal atoms are freed before they can grab electrons. Getting the $\frac{1}{2}$ vs full factor wrong is the classic MgCl₂ trap.

9. Electron affinity $E A$

See Electron affinity.

Why the topic needs it. This step makes the anion. For two chlorides you need $2 \times E A$ .

10. Lattice energy $U$ — the unknown we want

Why the topic needs it. This is the step we cannot measure (free gaseous ions can't be bottled), so the whole cycle exists to compute it.

11. Coulomb's law $U \propto \frac{q _{+} q _{-}}{r}$ — why $U$ has the size it does

See Ionic bonding and Coulomb's law; the fast estimate of $U$ from this idea is the Kapustinskii equation.

Why the topic needs it. It explains why MgCl₂ ( $q_{+} = 2$ , small ion) has a much larger $U$ than NaCl — a "why" the parent note quotes without deriving.

Putting it together — the master equation preview

Once every symbol above is defined, the parent's headline equation is just "long road = direct road" rearranged: $U = Δ H_{f} - Δ H_{s u b} - I E - \frac{1}{2} D - E A$ Every term on the right is one of the defined steps. You now know what each letter means, its picture, and its sign — so the algebra is safe.

Prerequisite map

Equipment checklist

Read

Na^{+} (g)

out loud — what does every part mean?

A sodium atom that has lost one electron (so charge

+ 1

), floating as a gas.

What does

Δ H

measure and what does a negative value mean?

Change in enthalpy (heat content); negative = exothermic, energy released, "downhill" on the ladder.

Why can we swap the long road for the direct road?

Enthalpy is a state function (Hess's Law) — total change depends only on start and end, not path.

When do you use

\frac{1}{2} D

instead of full

D

When the formula unit needs only one atom from a diatomic molecule (e.g. NaCl uses one Cl, so half a

Cl_{2}

What sign does electron affinity of Cl carry in the cycle?

Negative (

- 349

kJ/mol) — it releases energy.

For

Mg^{2 +}

how many ionisation steps?

Two —

I E_{1} + I E_{2}

, because two electrons are removed.

Which single step in the cycle cannot be measured directly?

The lattice energy

U

(gaseous ions cannot be isolated to condense in a calorimeter).

What makes

U

larger (more negative)?

Bigger ion charges

q_{+} q_{-}

and smaller separation

r

, via

U \propto q_{+} q_{-} / r

Connections

2.5.10 Born-Haber cycle revisited — calculating lattice energy (Hinglish)
Hess's Law and state functions
Enthalpy of formation
Ionisation energy trends
Electron affinity
Bond dissociation enthalpy
Ionic bonding and Coulomb's law
Kapustinskii equation
Solubility and enthalpy of hydration