This lattice energy calculator estimates the lattice energy of an ionic crystal with the Kapustinskii equation — U ≈ 1202.5 × ν × z⁺z⁻ ÷ r₀ × (1 − 0.345/r₀) kJ/mol — needing only the ion charges, their radii sum, and the number of ions per formula unit.

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Coulomb’s Law, Crystallized

Lattice energy is the energy released when gas-phase ions assemble into a crystal — the quantitative core of “why ionic solids are stable,” and its drivers are transparent: charge product in the numerator, ion distance in the denominator. The Kapustinskii equation packages the full Born–Landé treatment into a structure-free estimator, and its two calibration points bracket its honesty: NaCl computes to 746 kJ/mol (experimental 786, a 5% underestimate) while MgO computes to 3,799 (experimental 3,795 — dead on). The trends the formula makes inevitable are the ones every exam tests: doubling both charges (NaCl → MgO style) roughly quadruples U; smaller ions (LiF, r₀ ≈ 2.09 Å) beat larger ones (CsI) at equal charge; and the ν term rewards formulas with more ions per unit. These trends then explain melting points (MgO melts at 2,852 °C, NaCl at 801 °C), hardness, and why lattice energy is the term that makes Born–Haber cycles close.

Context for the numbers: the “experimental” values are themselves derived, not measured directly — the Born–Haber cycle infers lattice energy by Hess’s law around measurable steps (sublimation, ionization, dissociation, electron affinity, formation enthalpy). Kapustinskii’s shortcuts agree with those cycles within a few percent for genuinely ionic compounds, and drift where covalency creeps in (AgCl, with its polarizable ions, is the textbook misfit). For radii, use Shannon ionic radii matched to coordination number (Na⁺ 1.02 Å and Cl⁻ 1.81 Å at coordination 6) and enter the SUM — the equation’s r₀ is the internuclear distance, not either radius alone.

Worked Examples (Verified)

  • NaCl (defaults): ν 2, z 1×1, r₀ 2.83 Å → 746 kJ/mol (Born–Haber: 786).
  • MgO: ν 2, z 2×2, r₀ 2.12 Å → 3,799 kJ/mol (experimental ≈ 3,795).
  • LiF: ν 2, z 1×1, r₀ 2.09 Å → ≈960 kJ/mol — the small-ion premium (experimental ≈ 1,046).

Lattice Energy Discipline

  • r₀ is the SUM of cation + anion radii (nearest-neighbor distance) in Ångströms — entering one radius halves the denominator’s honesty.
  • ν counts ions per formula unit (NaCl 2, CaCl₂ 3, Al₂O₃ 5) — it’s Kapustinskii’s stand-in for the Madelung constant.
  • Sign conventions differ by text: formation-released (positive here) vs dissociation-required. Magnitude is the shared truth.
  • Covalent character (small highly-charged cation + large polarizable anion — Fajans’ rules) makes real crystals deviate from the ionic model, usually upward in stability.
  • Radii depend on coordination number: use Shannon radii for the structure’s actual CN when accuracy matters.

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Frequently Asked Questions

How is lattice energy calculated with Kapustinskii?

U ≈ 1202.5 × ν × z⁺z⁻ ÷ r₀ × (1 − 0.345 ÷ r₀) kJ/mol, where ν is ions per formula unit, z⁺z⁻ the charge product, and r₀ the sum of ionic radii in Ångströms. NaCl: 1202.5 × 2 × 1 ÷ 2.83 × (1 − 0.345/2.83) ≈ 746 kJ/mol, within ~5% of the Born–Haber value 786 — remarkable for an equation needing no crystal structure. Enter charges, ν and radii sum above for your compound.

What is lattice energy?

The energy released when one mole of an ionic compound’s gas-phase ions come together to form the solid crystal (equivalently, the energy needed to separate the crystal back into gas ions, with sign flipped by convention). It’s the stability ledger of ionic bonding: large lattice energies mean high melting points, low solubility tendencies (against hydration), and formations that pay for expensive ionization steps — as in MgO forming despite oxygen’s second electron affinity being endothermic.

Why is MgO’s lattice energy so much bigger than NaCl’s?

The charge product: Mg²⁺ and O²⁻ give z⁺z⁻ = 4 against NaCl’s 1, at a SHORTER distance (r₀ 2.12 vs 2.83 Å). The two factors compound — 4× from charge, ~1.4× from distance — landing MgO near 3,800 kJ/mol vs NaCl’s ~790. Consequences cascade: melting point 2,852 °C vs 801 °C, refractory applications for MgO, and Born–Haber arithmetic where the lattice term single-handedly justifies the compound’s existence.

Kapustinskii vs Born-Landé — what’s the difference?

Born–Landé is the parent theory: U ∝ M·z⁺z⁻/r₀ × (1 − 1/n), requiring the crystal’s Madelung constant M (structure-specific: 1.748 for rock salt, 2.519 for fluorite…) and Born exponent n. Kapustinskii noticed M/ν is nearly constant across structures and folded everything into the 1202.5 constant and ν — trading a few percent accuracy for needing no structural data at all. Use Born–Landé when the structure and M are known; Kapustinskii for estimates and unfamiliar compounds.

How is lattice energy measured experimentally?

It isn’t, directly — it’s extracted from the Born–Haber cycle: the formation enthalpy of the crystal (measured) minus the sum of sublimation, ionization energies, bond dissociation and electron affinities (all measurable) leaves lattice energy as the closing term. The agreement between those cycle values and model equations like Kapustinskii’s is one of the ionic model’s great validations — and the discrepancies (Ag halides) map exactly where covalency enters.

Does bigger lattice energy mean less soluble?

It’s half the tug-of-war: dissolution pays the lattice cost and collects hydration energy in return. Big lattice energies resist dissolving (MgO, CaCO₃), but big hydration energies (small, highly charged ions again) pull the other way — solubility is the small difference of two large numbers, which is why it’s hard to predict from lattice energy alone. The trends still guide: among similar salts, the one with the much larger U is usually the precipitate.

What radii should I enter?

Shannon ionic radii for the ions’ actual coordination numbers, summed. Working values (CN 6): Li⁺ 0.76, Na⁺ 1.02, K⁺ 1.38, Mg²⁺ 0.72, Ca²⁺ 1.00, F⁻ 1.33, Cl⁻ 1.81, O²⁻ 1.40, S²⁻ 1.84 Å. So CaO: r₀ = 1.00 + 1.40 = 2.40 Å, ν = 2, charges 2×2 → the calculator returns ≈ 3,400 kJ/mol (experimental ≈ 3,414 — Kapustinskii at its best).

Is this lattice energy calculator free?

Yes — free, instant, no sign-up. Enter ions per formula unit, both charges and the radii sum for a Kapustinskii lattice energy estimate in kJ/mol.