The lattice energy calculator above estimates the energy released when gaseous ions come together to form one mole of an ionic crystal, using three different routes. The Born-Landé equation builds it from electrostatics and a repulsion term. The Kapustinskii equation is a simplified form that needs only charges, radii and an ion count. The Born-Haber cycle takes it from measured thermochemical data instead, which is the closest thing to an experimental value available.
Arb Digital publishes free calculators that make the model visible rather than presenting a single number as fact. Lattice energy is a good example of why that matters: the theoretical and experimental routes disagree, and the size of the disagreement is itself informative. A compound where they agree closely is behaving ionically. One where they diverge has significant covalent character, and this page is set up so you can see that.
What This Lattice Energy Calculator Does
Lattice energy is defined as the enthalpy change when one mole of a solid ionic compound is formed from its constituent gaseous ions. Written that way it is a large negative number, because forming the crystal releases energy. Some textbooks define it in the opposite direction, as the energy needed to pull the crystal apart into gaseous ions, and then quote it as positive. This page uses the formation convention and reports a negative value, stating the magnitude separately so there is no ambiguity.
The Born-Landé mode needs a Madelung constant for the structure type, the two ion charges, the interionic distance and the Born exponent. The Kapustinskii mode replaces the Madelung constant with an ion count, which is what makes it useful for compounds whose structure you do not know. The Born-Haber mode takes five thermochemical quantities and returns the lattice energy that closes the cycle.
The neighbouring tools cover different ground. The enthalpy calculator handles general constant-pressure enthalpy changes and does not touch solid-state ionic thermodynamics. The Coulomb's law calculator gives the force and energy between two isolated point charges, which is one term in the sum this page performs over an entire lattice. And the unit cell calculator describes what is inside a repeating cell, whereas this page is about the energy holding it together.
How to Use It
- Pick the method. Born-Landé if you know the structure type, Kapustinskii if you do not, Born-Haber if you have thermochemical data.
- Choose a structure type to load its Madelung constant, or select Custom and type your own.
- Enter the interionic distance in picometres. This is the nearest-neighbour cation to anion distance, not a unit cell edge.
- Set the Born exponent from the noble gas configurations of your two ions, averaging them when they differ.
- Compare the routes. Run the same compound through Born-Landé and Born-Haber and read the difference as a covalency indicator.
The Formulas and How They Are Calculated
The Born-Landé equation is U = −(NA M z+ z− e²) / (4πε0 r0) × (1 − 1/n). The first factor is the Madelung sum of all the electrostatic attractions and repulsions in the lattice; the bracketed term reduces it to account for short-range repulsion between the electron clouds. The combination NAe²/4πε0 evaluates to 1.38935 × 10−4 J·m per mole using the defined SI values of the Avogadro constant and the elementary charge.
Sodium chloride is the standard worked example. With M = 1.74756, both charges 1, r0 = 282 pm and n = 8, the equation returns about −753 kJ/mol. The Kapustinskii route uses U = −120200 ν |z+z−| / (r+ + r−) × (1 − 34.5 / (r+ + r−)) with radii in picometres; for sodium chloride with ν = 2 and a radius sum of 283 pm it gives about −746 kJ/mol.
The Born-Haber cycle applies Hess's law around a closed loop, so the lattice energy is the enthalpy of formation minus every other step: U = ΔHf − ΔHsub − IE − D − EA. For sodium chloride, −411 − 107 − 496 − 122 − (−349) = −787 kJ/mol. That the three routes give −753, −746 and −787 for the same compound is the honest picture: the purely ionic models underestimate the binding by about five percent, and the shortfall is real chemistry rather than arithmetic error. The LibreTexts treatment of the Born-Landé equation sets out the derivation, and its companion page on Madelung constants is the source of the structure values offered in the dropdown.
What the Madelung Constant Actually Is
Left as a bare number in a dropdown, the Madelung constant looks arbitrary. It is not. It is the sum of an infinite series. Take one ion in the lattice and count its neighbours in shells: six oppositely charged ions at distance r, then twelve like-charged at r√2, then eight opposite at r√3, and so on outwards. Add the attractions, subtract the repulsions, and the series converges to 1.74756 for rock salt.
Two things follow. First, the constant depends only on geometry, not on which ions occupy the sites, which is why the same value covers every compound with the rock salt structure from lithium fluoride to magnesium oxide. Second, its value tells you how efficiently a structure packs opposite charges together, and the spread is narrow for simple 1:1 structures: caesium chloride at 1.76 sits barely above rock salt at 1.75 despite having eight nearest neighbours instead of six, because the extra neighbours are further away.
The large values, like 2.52 for fluorite and 4.17 for corundum, look dramatic but are partly a bookkeeping artefact. They are defined per formula unit, and a formula unit of corundum contains five ions carrying charges of 3 and 2. Comparing Madelung constants across different stoichiometries without accounting for that is meaningless. The series itself is famously slow and conditionally convergent, which is why the values are quoted from published summations rather than computed here.
Why Theory and Experiment Disagree
The gap between a Born-Landé estimate and a Born-Haber value is the most useful output on this page, and it has a clear interpretation. Both purely ionic models assume the ions are hard spheres carrying whole-number charges with no electron sharing at all. When that picture is close to true, the two routes agree to within a percent or two: the alkali halides mostly do.
When they diverge, the Born-Haber value is almost always larger in magnitude, and the excess is attributed to covalent contribution. Silver iodide is the classic case, with a discrepancy of well over a hundred kilojoules per mole, because the polarisable iodide anion and the polarising silver cation share electron density that the ionic model cannot represent. Fajans' rules predict exactly where this happens: small highly charged cations paired with large soft anions.
There is a second, less interesting source of disagreement worth ruling out first. Born-Haber values inherit the errors of five separate measured quantities, and electron affinities in particular are hard to measure and were historically revised. A cycle assembled from inconsistent data sources produces a discrepancy that means nothing at all. Check the provenance of your thermochemical numbers before drawing a bonding conclusion from them; the NIST Chemistry WebBook is the usual reference point.
What Lattice Energy Predicts, and What It Does Not
High lattice energy explains high melting points, high hardness and low volatility, because all three require breaking the ionic framework apart. Magnesium oxide and sodium fluoride have almost identical interionic distances, but doubling both charges multiplies the lattice energy roughly fourfold, and the melting point rises from around 990 to around 2,850 degrees Celsius.
What lattice energy does not predict on its own is solubility, and this trips people up constantly. Dissolving an ionic solid requires overcoming the lattice energy, but the energy comes back from hydrating the separated ions, and hydration enthalpies also scale with charge and inverse radius. The two large terms largely cancel, leaving a small difference that can fall either way, and the entropy term then decides the outcome. Lithium fluoride has a very high lattice energy and is poorly soluble; sodium chloride has a lower one and dissolves readily; but plenty of counterexamples exist because the cancellation is not systematic.
This is why arguing from lattice energy alone to a solubility prediction is unsound. If you are working with a real solution, the molarity calculator and solution concentration calculator deal in measured quantities, and the Gibbs free energy calculator is where the enthalpy and entropy terms are properly combined.
Choosing Between the Three Methods
Use Born-Landé when the structure type is known and you want a defensible theoretical estimate. It is the most physically transparent of the three and it lets you see how each factor contributes.
Use Kapustinskii when you do not know the structure, which is common for less studied compounds. It works by folding a typical Madelung constant per ion into the numerical coefficient, so it needs no structural information at all. That convenience costs accuracy: it is generally within a few percent for simple binary salts and drifts further for complex ions and layered structures.
Use Born-Haber when you have reliable thermochemical data and want the closest available approach to a measured value. It is not model-free, since it still assumes the compound is fully ionic when it assigns the residual to lattice energy, but it carries no assumption about geometry or repulsion. For the reverse problem, where the lattice energy is known and one cycle term is not, rearranging the same equation gives the missing quantity.
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Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Using a unit cell edge as r0 — the interionic distance is the nearest cation to anion separation, which for rock salt is half the cell edge.
- Mixing sign conventions — formation from gaseous ions is negative, dissociation into gaseous ions is positive, and quoting one as the other doubles the apparent error.
- Entering electron affinity with the wrong sign — as an enthalpy change a favourable electron gain is negative, so chlorine is about −349 kJ/mol.
- Forgetting the bond dissociation is per half a mole of diatomic — forming one mole of NaCl uses half a mole of Cl₂, so the term is half the bond enthalpy.
- Reading a large theory-experiment gap as an arithmetic mistake — for polarisable ions it is genuine covalent character, but check your data provenance before concluding that.
Related Free Tools From Arb Digital
The Coulomb's law calculator covers the single-pair electrostatics behind the Madelung sum, and the unit cell calculator describes the repeating structure itself. Use the enthalpy calculator and Gibbs free energy calculator for the surrounding thermodynamics, the molar mass calculator to keep quantities per mole straight, and the Miller indices calculator for the plane geometry of the same crystals. The full free online tools hub lists everything else.
Frequently Asked Questions
It is the enthalpy change when one mole of an ionic solid forms from its separated gaseous ions. Defined that way it is negative, because energy is released. The opposite convention, the energy to break the crystal into gaseous ions, gives the same magnitude with a positive sign.
It calculates lattice energy from the Madelung constant, the ion charges, the interionic distance and the Born exponent. The first part is the electrostatic attraction summed over the whole lattice; the Born exponent term subtracts the short-range repulsion between electron clouds.
It measures how strongly the ions resist being pushed closer together and falls between 5 and 12, set by the noble gas configuration of each ion. When the two ions differ, the average of their individual values is used.
Born-Lande assumes a purely ionic crystal of hard spheres. Born-Haber uses measured thermochemistry and therefore captures any covalent contribution as well. The gap between them is a rough measure of covalent character, largest for polarisable ions such as silver iodide.
When the structure type and therefore the Madelung constant are unknown. It substitutes an ion count for the structural information, so it needs only charges, radii and stoichiometry. Accuracy is a few percent for simple binary salts and worse for complex ions.
Not reliably. Dissolving must overcome the lattice energy, but hydration of the separated ions returns energy that also scales with charge and inverse radius. The two large terms largely cancel and entropy decides the outcome, so lattice energy alone is a poor solubility predictor.
Because lattice energy scales with the product of the ion charges. Both compounds have similar interionic distances and the same rock salt structure, but doubling both charges multiplies the energy roughly fourfold, which is why magnesium oxide melts far higher.
The nearest-neighbour distance between a cation and an anion, in picometres. For a rock salt structure that is half the cubic unit cell edge, not the edge itself. Using the cell edge halves the calculated lattice energy.
This calculator is provided for education and general reference. It describes how lattice energy models are evaluated and is not laboratory, materials selection or safety guidance; follow the methods and risk assessments issued by your own institution.
