Last updated: July 3, 2026
Lattice Energy Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The lattice energy calculator estimates ionic-crystal formation energy using the Born–Landé equation, U = −(NA M z⁺z⁻ e²)/(4πε₀ r₀)(1 − 1/n), with r₀ entered in picometres and output in kJ/mol. It also provides a simple Coulomb/proportionality mode showing that higher ion charges and smaller ion-center distances produce larger lattice-energy magnitudes.
Lattice energy by the Born Lande model equals minus Avogadro's number times the Madelung constant times the ion charge product times elementary charge squared, divided by four pi epsilon naught r zero, times one minus one over the Born exponent. More negative values mean stronger ionic lattices.
Key Takeaways
- Born–Landé combines Coulomb attraction, Madelung geometry, ion separation, and Born repulsion.
- This calculator reports lattice formation energy as negative U; |U| is the lattice-strength magnitude.
- NaCl with M = 1.7476, r₀ = 282 pm, and n = 8 gives about −753 kJ/mol with the constants used here.
- Changing from 1+/1− to 2+/2− ions roughly quadruples |U| at fixed distance and structure.
- Shorter ion-center distance increases |U|, explaining the strong lattice energies of small highly charged oxides.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
U = −(NA·M·z+·z−·e²)/(4πε0·r0) · (1 − 1/n); simple view U ∝ (z+·z−)/r0
Where:
- U=Lattice energy for formation of one mole of ionic crystal(kJ mol⁻¹)
- NA=Avogadro constant(mol⁻¹)
- M=Madelung constant for the crystal structure(dimensionless)
- z⁺, z⁻=Cation and anion charge magnitudes(dimensionless)
- e=Elementary charge(C)
- ε0=Vacuum permittivity(F m⁻¹)
- r0=Distance between nearest ion centers(pm (converted to m))
- n=Born exponent for repulsive interaction(dimensionless)
Worked Examples
NaCl Born–Landé estimate
Sodium chloride in the rock-salt structure using M = 1.7476, r₀ = 282 pm, and n = 8.
- 1Convert r₀ = 282 pm to 2.82 × 10⁻¹⁰ m.
- 2Use U = −(NA·M·z⁺z⁻·e²)/(4πε₀r₀) × (1 − 1/n).
- 3Substitute M = 1.7476, z⁺z⁻ = 1, n = 8.
- 4The calculated formation lattice energy is about −753 kJ/mol, consistent with the expected −740 to −770 kJ/mol band.
MgO charge and distance effect
Magnesium oxide has 2+/2− ions and a shorter ion-center distance, so |U| is several times larger than NaCl.
- 1Use z⁺z⁻ = 4, r₀ = 212 pm, and n = 7.
- 2The charge product alone gives roughly a fourfold increase versus a 1+/1− salt at equal distance.
- 3The smaller r₀ further increases electrostatic attraction.
- 4Born–Landé gives a value near −3926 kJ/mol, close to the experimental magnitude of MgO.
CsCl with larger ion separation
Cesium chloride has M = 1.7627 but a larger r₀ = 356 pm, reducing the magnitude relative to NaCl.
- 1Use the CsCl Madelung constant M = 1.7627.
- 2Enter z⁺ = z⁻ = 1, r₀ = 356 pm, and n = 10.5.
- 3The larger ion-center distance weakens the attraction even with a slightly larger M.
- 4The result is about −622 kJ/mol, close to the expected ~−630 kJ/mol.
Introduction
The lattice energy calculator estimates the energy released when gaseous ions assemble into an ionic crystal. The primary sign convention here is formation of the lattice, so stronger crystals have a more negative U and a larger |U|. Use the Born–Landé mode for quantitative teaching calculations, or switch to the simple Coulomb mode to see the core trend U ∝ z⁺z⁻/r₀. This topic sits between ionic bonding, crystal structure, and electrostatic energy; for electrochemical context compare the cell EMF calculator and Nernst equation calculator.
Born–Landé equation and sign convention
The Born–Landé equation models one mole of an ionic crystal as long-range Coulomb attraction corrected for short-range repulsion: U = −(NA·M·z⁺z⁻·e²)/(4πε₀r₀)·(1 − 1/n). The calculator reports U in kJ/mol for lattice formation, so NaCl is approximately −753 kJ/mol with the constants used here. Some textbooks tabulate lattice enthalpy for lattice dissociation as a positive number; that is the same magnitude with the opposite sign.
Choosing Madelung constant, charges, distance, and Born exponent
Use a Madelung constant matching the crystal structure: rock salt NaCl ≈ 1.7476, CsCl ≈ 1.7627, and zinc blende ZnS ≈ 1.6381. Enter charge magnitudes (1 for Na⁺, 2 for Mg²⁺) and r₀ in picometres; r₀ is the nearest ion-center distance, often approximated from ionic radii. Typical Born exponents are 5–12, with 8–9 common in introductory problems. Authoritative definitions of lattice energy appear in the IUPAC Gold Book.
Simple Coulomb proportionality mode
The secondary strategy intentionally omits Madelung geometry and the Born repulsion term to emphasize the dominant trend: U becomes more negative as z⁺z⁻ increases and as r₀ decreases. This is a Kapustinskii-like teaching view, not a substitute for crystal-specific thermochemistry. It is especially useful for comparing hypothetical pairs, then returning to Born–Landé mode for a structured ionic solid.
Worked examples: NaCl, MgO, and CsCl
NaCl with M = 1.7476, z⁺z⁻ = 1, r₀ = 282 pm, and n = 8 gives about −753 kJ/mol, close to the expected −756 kJ/mol textbook estimate. MgO with 2+/2− charges and r₀ = 212 pm gives about −3926 kJ/mol, showing why high-charge oxides have very large lattice energies. CsCl has a slightly larger Madelung constant than NaCl but a larger ion separation, so the magnitude falls to about −622 kJ/mol.
How to interpret trends
Lattice-energy magnitude rises roughly with charge product and falls with ion-center distance. Doubling both ion charges from 1+/1− to 2+/2− roughly quadruples |U| when structure and distance are held constant. Smaller ions also increase |U|, which is why MgO is much stronger than NaCl. The same ion data may help with stoichiometry or molar quantities in the molar mass calculator and ionic strength calculator.
Limitations and thermochemical context
Born–Landé is an electrostatic model. It assumes a simple ionic solid, point charges, a known crystal structure, and a repulsive exponent. Real lattice enthalpies include polarization, covalent character, zero-point effects, thermal corrections, and experimental definitions. For broader background see LibreTexts on lattice energy/Descriptive_Chemistry/Main_Group_Reactions/Lattice_Energies) and NIST CODATA constants.
Quick Reference Card
Lattice Energy — Quick Reference
Quick reference • Lattice Energy Calculator
Born–Landé: U = −(NA·M·z⁺z⁻·e²)/(4πε₀r₀)·(1 − 1/n); simple trend: U ∝ z⁺z⁻/r₀Valid range: Use positive charge magnitudes, M > 0, r₀ > 0 pm, and n > 1. Typical n values are about 5–12.
Common Values
⚠ Watch Out
- •Do not mix sign conventions: formation energy is negative, dissociation enthalpy is positive.
- •Enter r₀ in picometres; the formula internally converts pm to metres.
- •Use charge magnitudes as positive inputs; the negative sign is applied by the formation-energy convention.
- •Choose a Madelung constant matching the actual crystal structure, not just the formula unit.
- •Born–Landé does not capture covalent character or polarization in strongly distorted lattices.
Pro Tips
- →For a quick r₀ estimate, add cation and anion ionic radii from the same coordination-number table.
- →Compare trends with |U|, because a more negative U means a stronger lattice.
- →If charges double on both ions, expect roughly four times the magnitude before distance and structure corrections.
- →Use the simple Coulomb mode for teaching trends, then Born–Landé for numerical examples.
- →Keep at least four significant figures for Madelung constants when reproducing textbook examples.
FAQs
What sign should lattice energy have?
This calculator uses the lattice-formation convention: separated gaseous ions form a crystal and release energy, so U is negative. If your textbook defines lattice dissociation enthalpy, use the positive magnitude |U| instead.
Why is MgO so much larger in magnitude than NaCl?
MgO combines 2+ and 2− charges, giving a charge product of 4 instead of 1, and it has a shorter ion-center distance. Both factors make the Coulomb attraction much stronger.
What is the Madelung constant?
The Madelung constant accounts for the three-dimensional arrangement of all ions in a crystal, not just one nearest-neighbor pair. It depends on structure: NaCl, CsCl, and ZnS have different values.
Can I use ionic radii for r₀?
Yes for many classroom estimates. Add the cation and anion radii to approximate r₀, then enter the sum in picometres. Experimental nearest-neighbor distances are better when available.
What is the Born exponent n?
The Born exponent represents short-range repulsion between overlapping electron clouds. Values around 7–10 are common for many salts; using a source-specific n improves estimates.
Is the simple Coulomb mode a real lattice-energy formula?
It is a teaching approximation that preserves the charge-distance trend. It omits Madelung geometry and repulsion, so use Born–Landé or thermochemical data for quantitative values.