Lattice Energy & Born-Haber Cycle Calculator

Free Lattice Energy & Born-Haber Cycle Calculator

This tool solves two of the most common lattice-energy problems in general and inorganic chemistry. The Born-Haber Cycle Solver takes the five other energy changes involved in forming an ionic solid — sublimation, ionization energy, bond dissociation, electron affinity, and enthalpy of formation — and works out whichever one you're missing, most often the lattice energy itself. The Born-Landé Equation tool goes a different route: instead of starting from measured enthalpies, it predicts the lattice energy straight from theory, using nothing but the ionic charges, the distance between the ions, and the crystal's Madelung constant.

Both tools show every line of working, not just a final number, and both come with a diagram — a step-by-step waterfall chart for the Born-Haber cycle, and a side-by-side comparison chart showing how the theoretical Born-Landé number stacks up against the real experimental value for six common compounds. Everything runs in your browser and updates as you type, with no sign-up and no downloads.

What Is Lattice Energy?

Lattice energy is the amount of energy tied up in holding an ionic solid together. More precisely, it's the energy needed to pull one mole of an ionic compound completely apart into separate gaseous ions — for table salt, that means splitting solid NaCl into individual Na+ and Cl- ions floating freely in a gas, far enough apart that they no longer attract each other. Because you're fighting against a strong electrostatic pull to do that, lattice energy is always a large positive number when it's defined this way.

The reverse process — gaseous ions coming together to form the solid crystal — releases that same amount of energy, so it's strongly exothermic. You'll sometimes see this reverse value called the lattice formation enthalpy, and it's written as a negative number (energy given off) rather than a positive one (energy required). This calculator shows both: the signed lattice formation enthalpy and the positive lattice energy magnitude, so you can match whichever convention your textbook or course uses.

What Is the Born-Haber Cycle?

You can't measure lattice energy directly in a lab. There's no experiment where you watch a crystal of salt fly apart into separate gas-phase ions and measure the energy that took. Instead, chemists use an indirect route based on Hess's Law: since enthalpy only depends on the starting point and the ending point, not the path between them, you can build an imaginary multi-step path between the elements and the finished ionic solid, measure the energy of every other step along that path, and let the lattice energy be whatever number makes the whole cycle add up correctly.

That imaginary path is the Born-Haber cycle, named after the German physicists Max Born and Fritz Haber. It breaks the formation of an ionic compound from its elements into five separate, individually measurable steps, and because Hess's Law guarantees the total energy change is the same regardless of route, the sum of those five steps has to equal the compound's standard enthalpy of formation. Rearranging that sum is how you back out the one step nobody can measure directly.

The Born-Haber Cycle Equation, Step by Step

For a simple ionic compound MX made from a metal M and a nonmetal X, the cycle equation is: ΔHf° = ΔHsub + ΣIE + ΔHdiss + ΣEA + ΔHlattice. Each term stands for a real physical step, always starting from the elements in their everyday, standard-state form and ending at the solid ionic compound.

• ΔHsub — Sublimation (or atomization): turning the solid metal into separate gas-phase atoms. Almost always endothermic (positive), since you're breaking metallic bonds.
• ΣIE — Ionization energy: stripping one or more electrons off the gaseous metal atom to make the cation. Always endothermic, since pulling an electron away from a positively charged nucleus takes energy. For a 2+ ion like Mg2+, add the first and second ionization energies together.
• ΔHdiss — Bond dissociation / atomization of the nonmetal: breaking the nonmetal's molecular bond (like Cl2 or O2) into individual gas-phase atoms, scaled to however many atoms the formula actually needs. Endothermic.
• ΣEA — Electron affinity: the nonmetal atom picking up one or more electrons to become the anion. Usually exothermic (negative) for the first electron, but adding a second electron to an already-negative ion (like O- becoming O2-) is normally endothermic, which is why oxide and sulfide compounds often have a small positive electron affinity term overall.
• ΔHlattice — Lattice formation: the gaseous cation and anion coming together to build the solid crystal. This step is strongly exothermic (negative) and is usually the single biggest number in the whole cycle.

Worked Example: The Born-Haber Cycle for Sodium Chloride

Sodium chloride is the textbook example, and it's the calculator's default preset. The known values are: ΔHf°(NaCl) = -411 kJ/mol, ΔHsub(Na) = +108 kJ/mol, IE1(Na) = +496 kJ/mol, ½D(Cl2) = +121 kJ/mol, and EA(Cl) = -349 kJ/mol.

Rearranging the cycle equation to isolate the lattice term: ΔHlattice = ΔHf° - ΔHsub - ΣIE - ΔHdiss - ΣEA = -411 - 108 - 496 - 121 - (-349) = -787 kJ/mol. That's a large negative number, meaning the lattice forms with a huge release of energy, which matches what you'd expect — ionic solids like salt are extremely stable compared to their separated gas-phase ions. Reported as a positive lattice energy, U = 787 kJ/mol.

Worked Example: Why Magnesium Oxide Has a Much Bigger Lattice Energy

Switch the preset to magnesium oxide and the numbers look completely different: ΔHf°(MgO) = -602 kJ/mol, ΔHsub(Mg) = +148 kJ/mol, ΣIE(Mg2+) = +2,189 kJ/mol (the first and second ionization energies added together), ½D(O2) = +249 kJ/mol, and ΣEA(O2-) = +657 kJ/mol (the first electron affinity is favorable, but the second is strongly unfavorable and dominates the sum).

Solving the same way gives ΔHlattice = -602 - 148 - 2,189 - 249 - 657 = -3,845 kJ/mol, a lattice energy of roughly 3,845 kJ/mol — about five times bigger than sodium chloride's. That jump is exactly what the trends further down this page predict: doubling both ionic charges (1+/1- to 2+/2-) roughly quadruples the electrostatic attraction, and MgO's ions also sit closer together than NaCl's do, pushing the lattice energy even higher.

The Born-Landé Equation: Predicting Lattice Energy From Theory

The Born-Haber cycle tells you the lattice energy after the fact, built from measured enthalpies. The Born-Landé equation does the opposite — it predicts what the lattice energy should be, based purely on a simple physical model of the crystal as a regular, repeating array of point charges. The formula is: U = (NA × M × |z+| × |z-| × e²) / (4πε0 × r0) × (1 - 1/n).

The first part of that expression, without the (1 - 1/n) correction, is just Coulomb's Law scaled up to a whole crystal lattice instead of a single ion pair — it's the pure electrostatic attraction between every ion and every other ion in the structure. The (1 - 1/n) term is a small correction for short-range repulsion: real ions aren't point charges, and once their electron clouds get close enough to overlap, they push back against each other. Without that correction, the equation would predict a lattice energy that's slightly too large.

Understanding the Madelung Constant

The Madelung constant, M, accounts for the fact that every ion in a crystal is attracted to every oppositely-charged ion around it and repelled by every same-charged ion further out — not just its single nearest neighbour. Adding up that entire infinite series of attractions and repulsions for a given crystal geometry produces one fixed number, and that number depends only on how the ions are arranged, never on which specific ions are involved or how far apart they sit.

Because it depends only on geometry, every compound that crystallizes in the same structure type shares the same Madelung constant. Rock-salt-structured compounds (which is most of the 1:1 alkali halides, plus oxides like MgO and CaO) all use M = 1.748, caesium-chloride-structured compounds use M = 1.763, and so on. This calculator's quick-pick list covers the six most commonly taught structures.

What Is the Born Exponent?

The Born exponent, n, sets how sharply the short-range repulsion grows as two ions get closer together. It isn't arbitrary — it's tied to how many electrons each ion has, since ions with electron configurations matching bigger, more diffuse noble gases resist compression more than ions matching smaller ones. Common textbook values are n = 5 for a helium-like ion, 7 for neon-like, 9 for argon-like, 10 for krypton-like, and 12 for xenon-like.

When the cation and anion have different electron configurations, which is the usual case, you take the average of their two individual n values. Sodium chloride is the standard example: Na+ has the same 10-electron configuration as neon (n = 7), while Cl- has the same 18-electron configuration as argon (n = 9), giving an average Born exponent of 8 — exactly the value used in this calculator's NaCl preset.

Born-Landé vs Born-Haber: Why the Numbers Don't Always Match

Run the same compound through both tools and you'll usually see the Born-Landé prediction come out a little lower than the Born-Haber experimental value — typically within a few percent for compounds like sodium chloride or potassium bromide. That gap exists because the Born-Landé model treats every ion as a perfect point charge with a purely electrostatic, purely ionic bond, and real chemical bonds are never quite that simple.

The size of that gap actually tells you something useful. For genuinely ionic compounds — the alkali halides, and oxides of the alkaline earth metals — the gap stays small, confirming the point-charge picture is a good approximation. For compounds where the bonding has real covalent character, most famously silver halides like AgCl and AgI, the experimental lattice energy runs noticeably higher than the Born-Landé prediction, because extra covalent bonding is adding stabilization the simple electrostatic model never accounted for. Comparing the two numbers is a standard way chemistry courses illustrate the ionic-covalent spectrum, and the built-in comparison chart on this page lets you see that pattern across six compounds at a glance.

How to Use This Calculator

On the Born-Haber Cycle Solver, start by loading a compound preset or building your own from scratch. Pick which one of the five values you want to solve for from the Solve For dropdown — usually the lattice energy, since that's the one you can't measure directly — and fill in the other four fields with kJ/mol values from your textbook or lecture notes. The result, the full cycle diagram, and the step-by-step working update immediately.

On the Born-Landé Equation tool, either load a compound preset to see typical values filled in automatically, or build your own using the Madelung constant and Born exponent quick-pick menus if you're not sure what numbers to use for your compound's crystal structure and ion sizes. The interionic distance, r0, should be entered in picometres and is simply the sum of the cation and anion radii.

Factors That Affect Lattice Energy

Two variables drive almost every trend in lattice energy, and both fall straight out of the Born-Landé equation.

• Ionic charge has the biggest effect, since the electrostatic attraction is proportional to the product of the two charge magnitudes. Going from 1+/1- ions to 2+/2- ions roughly quadruples the attraction if the ions stay about the same size, which is why oxides and sulfides of small, highly-charged metal ions have lattice energies several times larger than the alkali halides.
• Ionic radius matters because the attraction gets weaker as the distance between ion centres grows — smaller ions can pack closer together, so smaller ions generally mean a higher lattice energy for compounds with the same charges. That's why lithium fluoride, built from two very small ions, has a noticeably higher lattice energy than potassium bromide, even though both are simple 1:1 alkali halides.

Why Lattice Energy Matters

Lattice energy isn't just an abstract number chemistry courses ask you to calculate — it explains a lot of the everyday behaviour of ionic solids. Compounds with very high lattice energies, like magnesium oxide, tend to have extremely high melting points and real physical hardness, because pulling the ions apart to melt or scratch the solid takes serious energy. It's also a major factor in solubility: dissolving an ionic solid in water means breaking the lattice apart, so a compound with an unusually high lattice energy compared to its hydration energy often turns out to be poorly soluble.

Lattice energy also shows up constantly in materials science and battery chemistry, where researchers care a great deal about how tightly ions are held in a crystal structure, since that directly affects how easily ions can move through a solid electrolyte or how stable a compound will be under real-world conditions.

Common Mistakes When Working With the Born-Haber Cycle

A handful of errors come up again and again in Born-Haber cycle problems.

• Forgetting to scale the bond dissociation term to match the formula — for a compound like NaCl that only needs one Cl atom per formula unit, use ½D(Cl2), not the full D(Cl2) value.
• Adding both ionization energies for a 2+ ion but only the first electron affinity for a 2- ion (or the other way around) — every electron transferred needs its own term added to the running total.
• Mixing up the sign of the lattice term — lattice formation (gaseous ions to solid) is exothermic and negative, while lattice energy reported as a magnitude is written as a positive number. Know which one your course wants before comparing your answer to a textbook value.
• Using enthalpy of formation values for the wrong physical state — always double-check that ΔHf° refers to the solid ionic compound, not a hydrated or aqueous form.

Common Mistakes When Working With the Born-Landé Equation

The Born-Landé tool has its own set of easy slip-ups.

• Entering r0 in the wrong units — this calculator expects picometres (pm), and 1 pm = 10⁻¹² m, not nanometres or angstroms.
• Using the wrong Madelung constant for the crystal structure — the value only depends on geometry, so double-check whether your compound is rock-salt, caesium-chloride, or another structure type before picking a preset.
• Forgetting that z+ and z- should be entered as positive magnitudes, not with their actual positive/negative sign, since the equation already accounts for the attraction between opposite charges.
• Expecting an exact match to the experimental value — the Born-Landé equation is a model, not a measurement, so a small gap from the Born-Haber value is expected, not a sign something went wrong.

Quick Reference: Formulas on This Page

ΔHf° = ΔHsub + ΣIE + ΔHdiss + ΣEA + ΔHlattice — the Born-Haber cycle, built from Hess's Law. U = (NA × M × |z+| × |z-| × e²) / (4πε0 × r0) × (1 - 1/n) — the Born-Landé equation for theoretical lattice energy. Lattice energy (U, positive) = -ΔHlattice, since lattice formation is exothermic while lattice energy is defined as the energy needed to reverse it.