Lattice Energy

What lattice energy is

Imagine vaporising one mole of solid sodium chloride into a gas of separated Na⁺ and Cl⁻ ions, all infinitely far apart. That requires energy. The reverse process — letting those gas-phase ions fall together into the cubic crystal lattice — releases the same energy. That energy is the lattice energy U:

Na⁺(g) + Cl⁻(g)  →  NaCl(s)        U = −787 kJ/mol

By the standard chemistry sign convention, U is negative because the formation step is exothermic. Some textbooks (notably older British ones) report the magnitude as positive, defining lattice energy as the endothermic dissociation step. Always check.

The size of U sets a great many physical properties. Crystals with very negative U are hard, refractory, and barely soluble. Salts with smaller-magnitude U dissolve in water, melt readily, and behave more like everyday table salt.

Coulomb's law as a starting point

The simplest model is two point charges separated by distance r₀:

U_pair = −k · z⁺z⁻ · e² / r₀

  k = 1 / (4πε₀) = 8.988 × 10⁹ N·m²/C²
  e = 1.602 × 10⁻¹⁹ C  (elementary charge)
  z⁺, z⁻ = ion charges (use absolute values; sign handled separately)
  r₀ = sum of cation + anion radii

Two simple consequences fall out immediately. First, U scales linearly with the product of charges. Doubling both charges (from ±1 to ±2) quadruples U. Second, U scales inversely with distance. Smaller ions pack tighter and bind harder. Both rules are visible in the periodic table: across a row of an alkali halide table, fluorides (small) beat chlorides which beat bromides which beat iodides.

Worked example — NaCl from first principles

Compute the lattice energy of NaCl from the Born–Landé equation and compare to the measured −787 kJ/mol. Take r₀ = r(Na⁺) + r(Cl⁻) = 1.02 + 1.81 = 2.83 Å = 2.83 × 10⁻¹⁰ m. Madelung constant for the rock-salt structure is M = 1.7476. Born exponent n = 9 (average of 7 for Na⁺ and 9 for Cl⁻).

U = −(N_A · M · z⁺z⁻ · e²) / (4πε₀ · r₀) · (1 − 1/n)

  N_A · e² / (4πε₀) = 1.389 × 10⁻⁴ J·m/mol   (handy combined constant)

U = −(1.389 × 10⁻⁴) · (1.7476) · (1)(1) / (2.83 × 10⁻¹⁰) · (1 − 1/9)
  = −(1.389 × 10⁻⁴ · 1.7476 / 2.83 × 10⁻¹⁰) · (0.889)
  = −(8.578 × 10⁵ J/mol) · (0.889)
  = −7.62 × 10⁵ J/mol
  = −762 kJ/mol

Measured (via Born–Haber cycle): U_NaCl = −787 kJ/mol. The Born–Landé prediction is within 3.2% — remarkable for a model whose only physics is Coulomb's law and a one-parameter repulsive correction. The remaining discrepancy comes from neglected dispersion forces and the assumption of perfectly spherical ions.

For MgO with z⁺ = +2, z⁻ = −2, r₀ = 2.10 Å, the same machinery gives roughly U ≈ −3923 kJ/mol against measured −3795 kJ/mol — about 3% high. The pattern holds across hundreds of ionic crystals.

Lattice-energy specs

Compound	Lattice energy (kJ/mol)	m.p. (°C)	Solubility (g/L, 25 °C)	Use
LiF	−1030	848	2.7	Optics, glassware
NaF	−910	993	40	Toothpaste additive
NaCl	−787	801	360	Table salt, deicer
NaBr	−747	747	905	Drilling fluid, sedative
NaI	−704	661	1840	Scintillator crystals
MgO	−3795	2852	0.0086	Refractory bricks, antacid
CaO	−3414	2613	1.19 (reactive)	Cement, steel making
Al₂O₃	−15,916	2072	~0	Sapphire, abrasives

The Al₂O₃ value reflects the multiplicity: five ions per formula unit, with charges +3 and −2. Lattice energies of trivalent oxides explain why aluminium and chromium oxides form the protective passivation layers that make stainless steel and aluminium cookware corrosion-resistant.

Madelung sum visualisation

NaCl rock-salt structure (cubic)

           ●───○───●           ● = Na⁺
           │   │   │           ○ = Cl⁻
           ○───●───○           Pair distance r₀
           │   │   │
           ●───○───●

  Madelung sum at central Na⁺:
    +6 × (−1/r₀)        nearest neighbours (Cl⁻ at r₀)
    +12 × (+1/(r₀√2))   next-nearest (Na⁺ at r₀√2)
    +8 × (−1/(r₀√3))    third shell (Cl⁻ at r₀√3)
    + ...
  Sum (after geometric convergence):  M = 1.7476

Lattice energy estimation methods

	Born–Landé	Born–Mayer	Kapustinskii	Born–Haber (experimental)
Form	Coulomb · (1 − 1/n)	Coulomb · (1 − ρ/r₀)	Coulomb · 1.21·10⁵·z⁺z⁻·ν / r₀	Sum of measured ΔH steps
Need crystal structure?	Yes (M from geometry)	Yes	No (ν = ions per formula)	No
Repulsion model	r⁻ⁿ (Born exponent)	e^(−r/ρ) (Buckingham)	Empirical correction	—
Typical accuracy	±3–5%	±2–3%	±5–10%	±0.5%
Strength	Transparent physics	Better at short r₀	Works for unknown structures	Direct, structure-free
Weakness	Ignores dispersion, polarisability	Same	Empirical	Needs all sub-step ΔHs

Variants and refinements

• Kapustinskii equation. When you don't know the crystal structure, use U ≈ 1.21 × 10⁵ · ν · z⁺z⁻ / (r⁺ + r⁻) kJ/mol·Å, where ν is the number of ions per formula unit. Works to ~10% for any new ionic compound.
• Born–Mayer refinement. Replaces the 1/n repulsive term with an exponential, fitting compressibility data better. The two parameters (e^(−r/ρ)) usually give a percent better than Born–Landé.
• Polarisability corrections. Real ions are not perfectly spherical; high-charge cations distort anions. Adding a London-dispersion term (the Mayer correction) tightens predictions for "soft" ions like Cs⁺ or I⁻.
• Modern DFT calculations. Periodic density-functional theory computes lattice energies from electronic structure, achieving ±1% accuracy for simple binary compounds and serving as a benchmark for the analytic equations.
• Madelung-constant catalogue. Each lattice type has its own M: NaCl 1.7476, CsCl 1.7627, ZnS (zincblende) 1.6381, fluorite 2.5194, rutile 2.408. Different M values explain why CaF₂ is so much more stable than CaCl₂.

Common pitfalls

• Sign-convention confusion. Some textbooks report U as positive (energy needed to break the lattice), others as negative (energy released when it forms). Always confirm before comparing values across sources.
• Ignoring the charge product. Students often quote "smaller ions = bigger lattice energy" but forget that doubling charges has roughly four times the effect of halving the radius. CaO beats LiF for both reasons.
• Forgetting that ionic radii are model-dependent. Shannon, Pauling, and Goldschmidt radii differ by ~10%. The same Born–Landé calculation can change by 5% depending on which set you use — quote the source.
• Comparing ionic and covalent solids on the same scale. Lattice energy is well-defined only for predominantly ionic compounds. AgI and HgS have substantial covalent character; their measured "lattice energies" diverge wildly from Born–Landé predictions because the model assumption fails.
• Forgetting that lattice energy is a thermodynamic state function, not an experimental observable. You can't put NaCl in a calorimeter and read off −787 kJ/mol directly. The Born–Haber cycle backs it out from independently measured quantities — and any error in those quantities propagates.
• Mistaking Madelung sums for simple geometric series. The lattice sum is conditionally convergent — different summation orders give different "answers" until you use Ewald-style techniques. The 1.7476 figure is the regularised result.