Foundations — Ionic bonding — Born-Haber cycle, lattice energy (Kapustinskii equation)

This page builds every symbol the parent Ionic Bonding note uses, from the ground up. Nothing is assumed. We go in build-order: each idea only uses ideas already defined.

0. The two characters: $+$ and $-$ ions

An atom is neutral: equal numbers of protons (positive) and electrons (negative). If an atom loses an electron it now has more protons than electrons → net positive → a cation (written like ${\text{Na}}^{+}$). If it gains an electron → net negative → an anion (${\text{Cl}}^{-}$).

1. Charge on one electron: $e$

Every electron carries the same fixed amount of charge. We call that amount $e$.

Why the topic needs it: the force law that glues ions together (next section) is written in coulombs, so we convert "$z$ electron-charges" into real charge by multiplying by $e$.

2. Why opposite charges pull: Coulomb's Law

Two charges feel a force along the line joining them: like charges push apart, opposite charges pull together. The energy stored in a pair of charges $q_1,q_2$ a distance $r$ apart is:

$E_{pair}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r}$

Let's earn every symbol in that expression.

See Coulomb's Law for the full force-vs-energy story.

3. From one pair to a whole grid: the Madelung constant $A$

A real crystal isn't two ions — it's a repeating 3D checkerboard. Any one ion is surrounded by nearest neighbours of opposite charge (pull, good), then next-nearest of same charge (push, bad), then more opposite ones further out, and so on forever.

If you add up all those pulls and pushes — an infinite alternating sum — it settles down to a single number that depends only on the shape of the grid, not the ions in it.

4. Why they don't collapse: the Born exponent $n$

If only attraction mattered, ions would fall into each other ($r\to 0$, energy $\to -\infty$). They don't — because when electron clouds start to overlap they shove back hard (a quantum "you can't be here" repulsion). Born modelled this short-range push as growing like $1/r^n$, super-steep.

Why the topic needs it: without it the Born–Landé equation would over-predict the binding energy. The $\left(1-\frac{1}{n}\right)$ is literally "attraction minus the bit repulsion cancels".

5. Counting ions in a formula unit: $\nu$

6. Ion sizes: $r_{+}$, $r_{-}$

Every ion has an effective radius. In Kapustinskii we use $r_{+}+r_{-}$ (in picometres, $1\text{ pm}=10^{-12}$ m) as a stand-in for the ion separation $r_0$: two touching balls, so centre-to-centre = radius + radius.

Why: it's the denominator in Kapustinskii — the size lever that competes with the charge lever $z_{+}{z}_{-}$.

7. Amounts and moles: $N_A$

8. The energy-ledger symbols ($\Delta H$ family)

The Born–Haber cycle is an accounting sheet. Each row is an energy change, all written with the symbol $\Delta H$ ("change in enthalpy" = heat energy in/out at constant pressure).

Here is every ledger symbol the parent uses, in build order:

9. The rule that ties it together: Hess's Law

$\underset{\text{direct route}}{\underbrace{\Delta H_f}}=\underset{\text{long route}}{\underbrace{\Delta H_{atom}+I{E}_1+\frac{1}{2}D+EA+U}}$

That single equation is the Born–Haber cycle. See Hess's Law for the general principle, and once you have $U$, Lattice Energy → Solubility and Fajans' Rules tell you what it means.

Prerequisite map

Equipment checklist

Cover the right side; each item you can explain out loud means you're ready.