# Born–Haber Cycle Calculator - Lattice Energy, Diagram & Kapustinskii

> Born–Haber cycle for NaCl, MgO, CaF₂, Al₂O₃ and more: built-in data, lattice energy or any unknown via Hess’s law, a scaled diagram and Kapustinskii comparison.

- URL: https://qilingtools.com/en/math/born-haber
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Chemistry](https://qilingtools.com/en/tags/chemistry)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Lattice energy is the energy needed to separate one mole of an ionic solid into gaseous ions. It cannot be measured directly, so it is found with a Born–Haber cycle and Hess’s law: the standard enthalpy of formation ΔfH° equals the sum of atomising the metal, ionising it, atomising the non-metal, adding electrons and finally forming the lattice from gaseous ions. For NaCl, ΔfH° = −411 kJ/mol, sublimation of Na 107, first ionisation energy 496, half the Cl–Cl bond enthalpy 121.7 and the electron affinity of Cl −349, so the lattice formation enthalpy is −411 − (107 + 496 + 121.7 − 349) ≈ −787 kJ/mol and the lattice energy is about 787 kJ/mol. Pick one of 28 compounds (NaCl, KCl, LiF, CsCl, AgCl, MgO, CaO, MgCl₂, CaF₂, Na₂O, Al₂O₃ …) and the calculator lists every step with its equation, multiplier and CRC/NIST value, computes the lattice energy and draws a to-scale energy ladder. Every value is editable, and you can solve for any one unknown instead — for example an electron affinity from a known lattice energy. A second panel estimates the lattice energy from a purely ionic model with the Born–Landé equation (Madelung constant, Born exponent) and the Kapustinskii equation (Shannon radii) and compares it with the thermochemical value: about 5% apart for NaCl but over 20% for AgCl, a sign of covalent character. Switch between positive lattice energy and negative lattice formation enthalpy conventions.

## How to use

1. Choose an ionic compound and the sign convention (lattice energy U > 0 or lattice enthalpy ΔlatH < 0)
2. Check or edit the data for each step, and choose which quantity to solve for
3. Read the Hess’s-law working, the scaled Born–Haber diagram and the Born–Landé / Kapustinskii comparison

## Features

- 28 compounds: alkali halides, silver halides, alkaline-earth oxides, MX₂, M₂O and Al₂O₃
- Built-in 298 K enthalpies of formation and atomisation, successive ionisation energies, bond enthalpies and electron affinities (including oxygen’s second EA)
- Solve for lattice energy or back-calculate an electron affinity, ionisation energy or ΔfH°
- Each step shown with its equation and multiplier; every value editable
- To-scale Born–Haber energy diagram with endothermic steps up and exothermic steps down
- Born–Landé and Kapustinskii estimates with Shannon ionic radii and Madelung constants
- Theory vs experiment comparison flags covalent character (AgCl, AgI)
- Toggle between positive lattice energy and negative lattice formation enthalpy

## FAQ

### Is lattice energy positive or negative?

It depends on the definition. Lattice dissociation energy (used in most Chinese and many US texts) is the energy absorbed to break 1 mol of solid into gaseous ions, so it is positive (NaCl ≈ +787 kJ/mol). Lattice formation enthalpy (UK A-level) is the heat released when gaseous ions form the solid, so it is negative (−787 kJ/mol). The magnitudes are the same; the tool lets you switch.

### Why is the lattice energy of MgO so much larger than NaCl’s?

Lattice energy scales roughly with the product of the ion charges divided by the inter-ionic distance. Mg²⁺O²⁻ has a charge product of 4, four times that of Na⁺Cl⁻, and the ions are smaller, so MgO’s lattice energy is about 3800–3900 kJ/mol — nearly five times NaCl’s — which is why MgO melts at 2852 °C. Note that oxygen’s second electron affinity is endothermic (about +844 kJ/mol).

### What is the difference between the Kapustinskii and Born–Landé equations?

Born–Landé, U = NₐA|z₊z₋|e²/(4πε₀r₀)·(1 − 1/n), needs the Madelung constant A of the crystal structure and the Born exponent n. Kapustinskii, U = 1.2025×10⁵·ν|z₊z₋|/r₀·(1 − 34.5/r₀), needs only the number of ions per formula unit ν and the ionic radii, so it works when the structure is unknown. Both assume purely ionic bonding and usually agree with Born–Haber values within 5–10% for typical ionic solids.

### Why do theoretical and Born–Haber values differ for AgCl?

Ag⁺ is an 18-electron ion with strong polarising power and Cl⁻ is fairly polarisable, so their electron clouds overlap and the bonding has substantial covalent character, which adds extra stability. The ionic model ignores this, so the Born–Landé estimate for AgCl (about 730 kJ/mol) falls well below the Born–Haber value (about 916 kJ/mol). The same effect explains why AgCl, AgBr and AgI become less soluble and more coloured down the group.

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