Lattice Energy Calculator | Free Born-Landé Equation Tool

What Is Lattice Energy?

Lattice energy is the energy released when separate gas-phase ions come together to form one mole of a solid ionic compound. A lattice energy calculator uses the Born-Landé equation to estimate this value from ion charges, ion sizes, and one adjustable parameter called the Born exponent.

The process lattice energy describes is:

$M^{n+}(g) + X^{n-}(g) \rightarrow MX(s)$

Here $M^{n+}$ is a metal cation (a positively charged ion) and $X^{n-}$ is a non-metal anion (a negatively charged ion). Lattice energy is always negative, because energy flows out of the system as the ions lock into a crystal. Chemists call a process that releases energy "exothermic."

Three things control the size of lattice energy:

1. Ion charge. Ions with bigger charges pull on each other more strongly.
2. Ion size. Smaller ions can get closer together, which strengthens the pull.
3. Crystal structure. The geometric pattern the ions form changes how many neighbors each ion has, which is captured by a number called the Madelung constant.

Lattice Energy Formula (Born-Landé Equation)

The Born-Landé equation, published by physicists Max Born and Alfred Landé in 1918, is:

$U = -\frac{N_0 A |z_1 z_2| e^2}{4\pi\varepsilon_0 r_0} \left(1-\frac{1}{n}\right)$

Where:

• $U$ = lattice energy, in kilojoules per mole (kJ/mol)
• $N_0$ = Avogadro's number, 6.022 × 10²³ per mole
• $A$ = Madelung constant (this calculator uses 1.7476, the value for the rock-salt structure that NaCl, MgO, and many other simple salts share)
• $z_1$, $z_2$ = the charges on the cation and anion
• $e$ = the charge of one electron, 1.602 × 10⁻¹⁹ coulombs
• $\varepsilon_0$ = the permittivity of vacuum, 8.854 × 10⁻¹² F/m
• $r_0$ = the interionic distance, the distance between the centers of the two ions
• $n$ = the Born exponent, a number that accounts for how strongly the ions resist being squeezed together (usually between 5 and 12)

The first part of the equation calculates the pull between two opposite charges. The term $(1 - 1/n)$ subtracts off the repulsion that appears once the ions' electron clouds start to overlap.

Interionic Distance

The distance between the ion centers, $r_0$, is simply the sum of the two ionic radii:

$r_0 = r_{cation} + r_{anion}$

Ionic radii are usually given in picometers (pm), where one picometer is one trillionth of a meter.

How to Calculate Lattice Energy

1. Enter the cation charge as a positive whole number (1 for Na⁺, 2 for Mg²⁺).
2. Enter the anion charge as a negative whole number (-1 for Cl⁻, -2 for O²⁻).
3. Enter the cation radius in picometers.
4. Enter the anion radius in picometers.
5. Enter a Born exponent. A value of 9 works for many compounds; the exponent cannot be exactly 1.
6. Read the interionic distance and lattice energy in the results panel.

Example: Lattice Energy of Sodium Chloride (NaCl)

Sodium chloride, common table salt, is a useful first example because its ions carry single charges.

• Cation charge: +1 (Na⁺)
• Anion charge: -1 (Cl⁻)
• Cation radius: 102 pm
• Anion radius: 181 pm
• Born exponent: 9

Step 1: Find the interionic distance.

$r_0 = 102\text{ pm} + 181\text{ pm} = 283\text{ pm}$

Step 2: Apply the Born-Landé equation.

$U = -\frac{(6.022\times10^{23})(1.7476)(1)(1.602\times10^{-19})^2}{4\pi(8.854\times10^{-12})(2.83\times10^{-10})}\left(1-\frac{1}{9}\right)$

$U \approx -762\text{ kJ/mol}$

This is close to, but not the same as, the value usually quoted in textbooks, about -787 kJ/mol, which comes from experimental thermodynamic data measured through a Born-Haber cycle rather than from this equation. The gap exists because the Born-Landé equation treats ions as perfect rigid spheres and uses one repulsion exponent for the whole crystal, so it is an approximation rather than an exact match to reality.

Example: Lattice Energy of Magnesium Oxide (MgO)

Magnesium oxide has doubly charged ions, so it makes a good comparison.

• Cation charge: +2 (Mg²⁺), radius 72 pm
• Anion charge: -2 (O²⁻), radius 140 pm
• Born exponent: 9
• Interionic distance: 72 pm + 140 pm = 212 pm

Putting these numbers into the same equation gives a lattice energy of about -4071 kJ/mol, more than five times more negative than NaCl's. The experimentally measured value is about -3795 kJ/mol. Doubling both ion charges roughly quadruples the electrostatic pull, since the equation multiplies $z_1$ by $z_2$, and the shorter interionic distance in MgO strengthens the attraction further.

Common Ionic Radii and Born Exponents

Selected Cation Radii

Selected Anion Radii

Typical Born Exponents

These are starting points. Published sources sometimes give slightly different numbers.

What Affects Lattice Energy

Lattice energy connects to several properties chemists can measure directly.

• Melting point. Stronger ionic attraction usually means a higher melting point. Magnesium oxide's ions carry twice the charge of sodium chloride's ions and sit closer together, giving MgO a far more negative lattice energy. That is part of why MgO melts at about 2852°C, while NaCl melts at about 801°C.
• Hardness. Crystals with higher lattice energy tend to resist scratching and deformation better.
• Solubility. Compounds with very high lattice energy are often harder to dissolve in water, because breaking the crystal apart takes more energy than the water molecules can supply by surrounding the ions.

Lattice energy is also one term in the Born-Haber cycle, a bookkeeping method chemists use to work out energy changes during the formation of an ionic compound from its elements.

Limits of the Born-Landé Equation and This Calculator

This calculator always uses the Madelung constant for the rock-salt structure (1.7476). That value is correct for compounds that crystallize the way NaCl and MgO do, but it is not correct for compounds with other structures, such as cesium chloride or calcium fluoride, which have different Madelung constants. The Born-Landé equation also assumes the ions are perfect spheres and ignores effects like covalent bonding character, so its results are close estimates rather than exact experimental values.

Frequently Asked Questions

What is lattice energy in simple terms? It is the energy given off when charged particles called ions come together and lock into a solid crystal. A more negative number means the ions are held together more tightly.

Why is lattice energy always negative? Because forming the solid releases energy rather than absorbing it. Some textbooks instead define "lattice enthalpy" as the positive energy needed to break the crystal apart; both describe the same bond strength, just with opposite signs.

What Born exponent should I use? Nine is a common default that works reasonably well for many ionic compounds. For a specific compound, a chemistry reference or crystallography database will list a more precise value based on the electron configurations of its ions.

Why doesn't the calculator match the number in my textbook exactly? Textbook values often come from experiments (a Born-Haber cycle), while this calculator uses the theoretical Born-Landé equation. The two methods usually agree within a few percent but rarely match exactly.

Does a bigger ion always mean lower lattice energy? Usually, yes. Larger ions increase the interionic distance $r_0$, which sits in the denominator of the equation, so lattice energy becomes less negative as ions get bigger, all else being equal.

Can this calculator handle any ionic compound? It works best for simple compounds that share the rock-salt crystal structure, such as NaCl, MgO, and CaO. It is less reliable for compounds with different crystal structures or significant covalent bonding.

References

1. Atkins, P. W., & De Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.
2. Housecroft, C. E., & Sharpe, A. G. (2018). Inorganic Chemistry (5th ed.). Pearson.
3. Born, M., & Landé, A. (1918). Über die Berechnung der Kompressibilität regulärer Kristalle aus der Gittertheorie. Verhandlungen der Deutschen Physikalischen Gesellschaft, 20, 210-216.
4. Shannon, R. D. (1976). Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallographica Section A, 32(5), 751-767.