Born–Landé Equation for Lattice Energy
$$ U_0 = -\frac{N_A \, M \, |Z_+ Z_-| \, e^2}{4\pi\varepsilon_0 \, r_0}\left(1 - \frac{1}{n}\right) $$
The lattice energy $U_0$ is the energy released when one mole of an ionic crystal is formed from gaseous ions at infinite separation. It is the single most important quantity in ionic-bonding thermodynamics — it determines melting points, solubilities, and the very stability of an ionic compound.
Symbol Definitions
| Symbol | Meaning | Unit (SI) |
|---|
| $U_0$ | Lattice energy (formation, negative) | J/mol or kJ/mol |
| $N_A$ | Avogadro's number = 6.022×10²³ | mol⁻¹ |
| $M$ | Madelung constant (geometry-dependent) | dimensionless |
| $Z_+, Z_-$ | Integer charges of cation, anion | dimensionless |
| $e$ | Elementary charge = 1.602×10⁻¹⁹ | C |
| $\varepsilon_0$ | Vacuum permittivity = 8.854×10⁻¹² | C²·N⁻¹·m⁻² |
| $r_0$ | Equilibrium internuclear distance ($r_+ + r_-$) | m (or pm) |
| $n$ | Born exponent (Pauli repulsion) | dimensionless (5–12) |
Step-by-Step Derivation
STEP 1 — Coulomb potential of one ion pair
The electrostatic potential energy between a cation ($+Z_+e$) and anion ($-Z_-e$) at distance $r$ is:
$$ V_{\text{pair}}(r) = -\frac{Z_+ Z_- e^2}{4\pi\varepsilon_0 \, r} $$
The minus sign comes from opposite charges attracting.
STEP 2 — Sum over the entire lattice (Madelung)
In a crystal, each ion is surrounded by nearest-neighbour counter-ions, second-nearest co-ions (repulsive), third-nearest counter-ions, and so on. The infinite alternating series converges to the Madelung constant $M$:
$$ M = \sum_{i \neq j} \pm \frac{r_0}{r_{ij}} \approx 1.7476 \text{ for NaCl} $$
So the Coulombic energy per ion-pair becomes $V_{\text{Coul}} = -\frac{M\,Z_+Z_-e^2}{4\pi\varepsilon_0\,r_0}$.
STEP 3 — Add Pauli repulsion (Born term)
Pure $1/r$ attraction would collapse the lattice to a point. Closed-shell electron clouds repel at short range; Born modelled this empirically as $V_{\text{rep}} = B/r^n$ where $n$ is the Born exponent (≈ 5 for He, 7 for Ne, 9 for Ar/Cu⁺, 12 for Xe).
$$ V_{\text{total}}(r) = -\frac{M\,Z_+Z_-e^2}{4\pi\varepsilon_0\,r} + \frac{B}{r^n} $$
STEP 4 — Find equilibrium ($dV/dr = 0$)
$$ \frac{dV}{dr}\bigg|_{r_0} = \frac{M\,Z_+Z_-e^2}{4\pi\varepsilon_0\,r_0^2} - \frac{nB}{r_0^{n+1}} = 0 $$
Solving for $B$:
$$ B = \frac{M\,Z_+Z_-e^2 \, r_0^{n-1}}{4\pi\varepsilon_0 \, n} $$
STEP 5 — Substitute back & multiply by $N_A$
Putting $B$ into $V_{\text{total}}(r_0)$ and converting from per-pair to per-mole gives the famous Born-Landé equation:
$$ \boxed{\,U_0 = -\frac{N_A \, M \, |Z_+ Z_-| \, e^2}{4\pi\varepsilon_0 \, r_0}\left(1 - \frac{1}{n}\right)\,} $$
The factor $(1 - 1/n)$ shows that Pauli repulsion
reduces the binding energy by ~10–15%.
STEP 6 — Born-Haber cycle (alternative experimental route)
Hess's law lets us measure $U_0$ indirectly: $\Delta H_f^\circ = \Delta H_{\text{sub}} + \tfrac{1}{2}D + IE - EA + U_0$. Comparing experimental $U_0$ with Born-Landé tests the ionic model — large deviations (e.g. AgCl, ZnS) indicate
covalent character.
Mapping to Simulation
- Z₊, Z₋ sliders → directly set $|Z_+Z_-|$ — note the dramatic scaling (MgO ≈ 4× NaCl)
- r₊, r₋ sliders → set $r_0 = r_+ + r_-$; observe the inverse-distance scaling
- Born exponent n → controls the $(1-1/n)$ correction; varies with electron configuration
- Preset selector → loads experimental ionic radii (Shannon) and structural Madelung constants
Worked Example — NaCl
For NaCl: $r_0 = 102 + 181 = 283$ pm = $2.83 \times 10^{-10}$ m, $M = 1.7476$, $|Z_+Z_-| = 1$, $n = 9$.
$$ U_0 = -\frac{(6.022\times 10^{23})(1.7476)(1)(1.602\times 10^{-19})^2}{4\pi(8.854\times 10^{-12})(2.83\times 10^{-10})}\left(1-\tfrac{1}{9}\right) $$
$$ U_0 \approx -755 \text{ kJ/mol} \quad (\text{exp: } -787 \text{ kJ/mol}) $$
The small ~4% gap is the covalent contribution not captured by a purely ionic model.
📚 Reference: Housecroft & Sharpe — Inorganic Chemistry, 5th Ed., Chapter 6.16: "The Born–Landé Equation"; Atkins & de Paula — Physical Chemistry, 11th Ed., Section 18B.1: "Lattice enthalpy"; Silberberg — Chemistry: The Molecular Nature of Matter and Change, Ch. 9.2: "The Born-Haber Cycle".