Lattice energy is the energy released when gaseous cations and anions come together to form one mole of a crystalline ionic solid – or, with the opposite sign, the energy needed to pull the crystal apart into free ions. It is the main reason ionic compounds have high melting points, and it decides trends in solubility, hydration and the stability of oxides and halides.
Lattice energies cannot be measured directly. Experimentally they come from a Born–Haber cycle, which adds up sublimation, ionisation, bond-dissociation and electron-affinity energies with the enthalpy of formation. Theoretically they come from electrostatics: the Coulomb attraction summed over the whole crystal (the Madelung constant does that sum for each crystal structure) minus a short-range repulsion when electron clouds overlap. Born and Landé modelled the repulsion as a power law with the Born exponent n; Born and Mayer used an exponential with a range parameter ρ. Kapustinskii noticed that the Madelung constant divided by the number of ions is almost the same for every structure, which lets you estimate U without knowing the crystal structure at all.
The equations show the two big trends: lattice energy grows with the product of the ion charges (MgO is about four times NaCl) and falls as the ions get larger (LiF > NaCl > KBr > CsI).
How to use the Lattice Energy Calculator
- Method
- Cation
- Cation charge
- Cation radius
- Cation Born exponent — He-like 5, Ne 7, Ar 9, Kr 10, Xe 12
- Anion
- Anion charge (magnitude)
- Anion radius
- Anion Born exponent
- Crystal structure (Madelung constant)
- Madelung constant
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Pick the cation and anion (or enter a custom charge, radius and Born exponent), choose the crystal structure for the Madelung constant, and choose a method. The calculator adds the ionic radii to get r₀, finds the formula and number of ions per formula unit from the charges, and evaluates the selected equation; “All methods” lists every estimate side by side.
Worked example
Sodium chloride with Shannon radii: r₀ = 102 + 181 = 283 pm, z⁺ = z⁻ = 1, ν = 2, M = 1.74756, n = (7 + 9)/2 = 8.
Kapustinskii: U = 120 250 × 2 × 1 / 283 × (1 − 34.5/283) = 849.8 × 0.8781 = 746.2 kJ/mol.
Born–Landé: U = 138 935 × 1.74756 / 283 × (1 − 1/8) = 857.9 × 0.875 = 750.7 kJ/mol.
Born–Mayer: 857.9 × (1 − 34.5/283) = 753.4 kJ/mol. The Born–Haber value is about 787 kJ/mol.
Ionic radii and Born exponents used
| Ion | Charge | Radius (pm, CN 6) | Born exponent |
|---|---|---|---|
| Li⁺ | +1 | 76 | 5 |
| Be²⁺ | +2 | 45 | 5 |
| Na⁺ | +1 | 102 | 7 |
| Mg²⁺ | +2 | 72 | 7 |
| K⁺ | +1 | 138 | 9 |
| Ca²⁺ | +2 | 100 | 9 |
| Rb⁺ | +1 | 152 | 10 |
| Sr²⁺ | +2 | 118 | 10 |
| Cs⁺ | +1 | 167 | 12 |
| Ba²⁺ | +2 | 135 | 12 |
| O²⁻ | −2 | 140 | 7 |
| F⁻ | −1 | 133 | 7 |
| S²⁻ | −2 | 184 | 9 |
| Cl⁻ | −1 | 181 | 9 |
| Se²⁻ | −2 | 198 | 10 |
| Br⁻ | −1 | 196 | 10 |
| Te²⁻ | −2 | 221 | 12 |
| I⁻ | −1 | 220 | 12 |
Tips
- Standard potentials are reduction potentials at 25 °C, 1 M and 1 bar.
- n is the number of electrons transferred in the balanced reaction.
- Faraday constant: 96 485 C per mole of electrons.
- Everything is calculated in your browser.
Common mistakes to avoid
- Flipping the sign of a potential and also subtracting it (counting the reversal twice).
- Multiplying a standard potential by the coefficient of the half-reaction – potentials are intensive.
- Using log base e where the Nernst equation uses log₁₀ (or the other way round).
Inputs explained
| Field | What to enter |
|---|---|
| Method | Choose one: Kapustinskii (no structure needed), Born–Landé, Born–Mayer, Hard sphere (Coulomb only) … |
| Cation | Choose one: Li⁺, Be²⁺, Na⁺, Mg²⁺ … |
| Cation charge | A number |
| Cation radius | Units: pm, Å, nm |
| Cation Born exponent | A number. He-like 5, Ne 7, Ar 9, Kr 10, Xe 12 |
| Anion | Choose one: O²⁻, F⁻, S²⁻, Cl⁻ … |
| Anion charge (magnitude) | A number |
| Anion radius | Units: pm, Å, nm |
| Anion Born exponent | A number |
| Crystal structure (Madelung constant) | Choose one: Rock salt (NaCl type, 6:6), Caesium chloride (CsCl type, 8:8), Zinc blende (ZnS, 4:4), Wurtzite (ZnS, 4:4) … |
| Madelung constant | A number |
What each result means
- Ions per formula unit ν = [Anion charge |z⁻|]÷gcd([Cation charge z⁺], [Anion charge |z⁻|]) + [Cation charge z⁺]÷gcd([Cation charge z⁺], [Anion charge |z⁻|])
- Lattice energy – Kapustinskii = 120250.0×[Ions per formula unit ν]×[Cation charge z⁺]×[Anion charge |z⁻|]÷[Interionic distance r₀ = r⁺ + r⁻]×(1 – 34.5÷[Interionic distance r₀ = r⁺ + r⁻])
- Lattice energy – Born–Landé = 138935.4×[Madelung constant M]×[Cation charge z⁺]×[Anion charge |z⁻|]÷[Interionic distance r₀ = r⁺ + r⁻]×(1 – 1÷[Born exponent n (mean of the two ions)])
- Lattice energy – Born–Mayer = 138935.4×[Madelung constant M]×[Cation charge z⁺]×[Anion charge |z⁻|]÷[Interionic distance r₀ = r⁺ + r⁻]×(1 – 34.5÷[Interionic distance r₀ = r⁺ + r⁻])
- Lattice energy – hard sphere (Coulomb only) = 138935.4×[Madelung constant M]×[Cation charge z⁺]×[Anion charge |z⁻|]÷[Interionic distance r₀ = r⁺ + r⁻]
Formula
Born–Landé: U = N_A·M·|z⁺z⁻|·e²/(4πε₀r₀) · (1 − 1/n) · Born–Mayer: U = N_A·M·|z⁺z⁻|·e²/(4πε₀r₀) · (1 − ρ/r₀), ρ = 34.5 pm · Kapustinskii: U = K·ν·|z⁺z⁻|/r₀ · (1 − d/r₀), K = 1.2025 × 10⁻⁴ J·m/mol, d = 34.5 pm · r₀ = r⁺ + r⁻Worked example
Inputs — Method: All methods; Cation: Na⁺; Anion: Cl⁻; Crystal structure (Madelung constant): Rock salt (NaCl type, 6:6).
Results — Interionic distance r₀ = r⁺ + r⁻ = 283 pm; Born exponent n (mean of the two ions) = 8; Ions per formula unit ν = 2; Madelung constant M = 1.74756; Lattice energy – Kapustinskii = 746.2 kJ/mol; Lattice energy – Born–Landé = 750.7 kJ/mol; Lattice energy – Born–Mayer = 753.4 kJ/mol; Lattice energy – hard sphere (Coulomb only) = 857.9 kJ/mol; Lattice energy (selected method) = 746.2 kJ/mol.
Key terms
- Lattice energy
- Energy to break one mole of an ionic crystal into gaseous ions.
- Madelung constant
- Geometric factor that sums all Coulomb interactions in a given crystal structure.
- Born exponent
- Exponent of the repulsion term in the Born–Landé equation.
- Born–Haber cycle
- Hess's-law cycle that obtains lattice energy from measurable enthalpies.
Frequently asked questions
What is lattice energy?
The energy needed to separate one mole of an ionic solid into its gaseous ions (positive by this convention); the same amount is released when the lattice forms.
What is the lattice energy of NaCl?
The Born–Haber cycle gives about 787 kJ/mol. With Shannon radii (r₀ = 283 pm) this calculator gives 746 kJ/mol (Kapustinskii) and 751 kJ/mol (Born–Landé).
Why does MgO have a much larger lattice energy than NaCl?
Both ions carry two charges, so z⁺z⁻ = 4, and the ions are smaller (r₀ = 212 pm). Kapustinskii gives about 3800 kJ/mol – almost five times NaCl.
Which has the higher lattice energy, NaF or NaCl?
NaF, because F⁻ (133 pm) is smaller than Cl⁻ (181 pm) – the ions sit closer, so the attraction is stronger.
When should I use Kapustinskii instead of Born–Landé?
When you do not know the crystal structure (no Madelung constant), e.g. for a new compound or a complex ion. It typically lands within a few percent of the Born–Landé value.
What is the Born exponent?
A number from 5 to 12 describing how steeply the repulsion between electron clouds rises; it grows with the size of the noble-gas core (He 5, Ne 7, Ar 9, Kr 10, Xe 12).
Why do the theoretical values fall below the Born–Haber values?
The simple models ignore van der Waals attraction, zero-point energy and partial covalency; for silver and copper halides the gap is large because their bonding is partly covalent.