Lattice Energy Calculator

Lattice energy is the work needed to pull an ionic crystal apart into its gaseous ions. It is the quantity that decides how high a salt melts, how hard it is, and whether a compound can be made at all, and it is the payoff line of a Born-Haber cycle. This page estimates it from the ions alone, using four classical electrostatic approximations at once, so you can read a number and see how much the choice of model moved it.

Two of the four models need nothing but a formula and two ionic radii. The other three add the geometry of the crystal through a Madelung constant. Working under All shows all four answers side by side, which is the honest way to use a method like this: the spread between them is the uncertainty the model itself carries. For sodium chloride the four land between 745.9 and 857.9 kJ/mol — a span of about 110 kJ/mol that has nothing to do with your arithmetic and everything to do with which approximation you picked.

This is a working tool for chemistry students comparing an estimate against a cycle, for teachers showing why the repulsion correction matters, and for anyone who has radii and a crystal structure but no measured lattice energy. It is not a substitute for a Born-Haber cycle, which uses real thermochemical data rather than a fitted repulsion term.

A complete run-through on sodium chloride

The page opens on sodium chloride in the rock salt structure, so nothing has to be typed: the Na⁺ ion contributes 102 pm and the Cl⁻ ion 181 pm, and the distance the equations use is their sum.

With $\nu=2$, $z^{+}=1$, $z^{-}=-1$, $M=1.74756$, $n=8$ and $\rho=30\ \mathrm{pm}$, the four models return:

The hard-sphere figure is the ceiling: the other three multiply it by a repulsion factor just below one, so they must come out lower. The Born-Haber cycle value ordinarily quoted for sodium chloride is about 787 kJ/mol, which puts the three corrected models within a few per cent below it and the bare hard-sphere sum about 9% above it. That pattern — corrected models slightly low, the uncorrected one high — is what you should expect on any ionic solid, and if one of your numbers sits far outside it, the input is usually the problem rather than the model.

All four approximations start from the same idea: a lattice is held together by the electrostatic attraction of point charges, and the energy per mole is that attraction summed over the whole crystal.

The sum over the crystal is where the Madelung constant comes from. It is a purely geometric number: it counts how much stronger the attraction is in an infinite lattice than between one isolated ion pair. Because it is geometry and not chemistry, chemically different crystals with the same arrangement share it — rock salt and many other 1:1 salts all sit at 1.74756, and fluorite and anti-fluorite share 2.51939.

Kapustinskii: no crystal structure required

Two empirical constants do all the work here: $K$ converts charges and a distance in picometres into kilojoules per mole, and $d=34.5\ \mathrm{pm}$ stands in for the repulsion that the other models compute separately. Because it needs no Madelung constant, it works for any compound whose formula and radii you know — including awkward structures you cannot assign — at the cost of being the least precise of the four.

The three structure-aware models

Born-Landé, Born-Mayer and the hard sphere share one electrostatic core and differ only in how they pay for repulsion:

The two repulsion factors are not independent, and there are crystals where they coincide: with the page’s defaults $n=8$ and $\rho=30\ \mathrm{pm}$, the Born-Landé and Born-Mayer corrections agree whenever $r=n\rho=240\ \mathrm{pm}$. Calcium oxide has exactly that distance with the tabulated radii, which is why its Born-Landé and Born-Mayer answers are both 3,540.805 kJ/mol. Treat that as a reminder that agreement between two models is not always confirmation.

Sensitivity is worth internalising before you tune anything. Every model scales with the product of the charges, so going from a 1:1 salt to a 2:2 oxide quadruples the energy, and all of them divide by the interatomic distance, so a 10% error in a radius moves the answer by roughly 10%. The fitted parameters matter far less: the Born-Landé correction runs from $1-1/5=0.800$ at $n=5$ to $1-1/12=0.917$ at $n=12$, so moving $n$ across its whole defensible range changes the estimate by under 15% — far less than a radius taken at the wrong coordination number.