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What is the lattice energy?

Enter the energies of the Born–Haber cycle for an ionic compound, and the calculator gives its lattice energy, or solves the cycle for the enthalpy of formation or the electron affinity. It also estimates a lattice energy from a salt with the same crystal structure, using the ion charges and distances.

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Answer
ΔH_lattice = 756.9 kJ/mol

ΔH_lattice = 756.9 kJ/mol.

Lattice energy (kJ/mol)
756.9
ΔH°f (kJ/mol)
−553.5
Electron affinity (kJ/mol)
−328.2
Metal steps (kJ/mol)
452.2
Nonmetal steps (kJ/mol)
−248.8

Answer: ΔH_lattice = 756.9 kJ/mol. ΔH_lattice = 756.9 kJ/mol.

How to calculate

Finds the lattice energy of an ionic compound with the Born–Haber cycle from the enthalpy of formation, sublimation, ionization, bond dissociation and electron affinity, solves the cycle for ΔH°f or the electron affinity, or estimates a lattice energy from a salt of the same structure.

Example with the default inputs (Find Lattice energy, Metal atoms per formula unit (a) 1, Nonmetal atoms per formula unit (b) 1, ΔH°f of the compound (kJ/mol) −553.5, ΔH°s of the metal (kJ/mol) 76.5, Ionization energy (kJ/mol) 375.7, ½D: atomize the nonmetal (kJ/mol) 79.4, Electron affinity (kJ/mol) −328.2): ΔH_lattice = 756.9 kJ/mol.

Method: ΔH_lattice = a × (ΔH°s + IE) + b × (½D + EA) − ΔH°f; ΔH°f = a × (ΔH°s + IE) + b × (½D + EA) − ΔH_lattice; EA = (ΔH_lattice + ΔH°f − a × (ΔH°s + IE)) ÷ b − ½D; U₂ = U₁ × (Z₂⁺Z₂⁻ ÷ Z₁⁺Z₁⁻) × (R₁ ÷ R₂).

  • Lattice energy is positive: the energy to separate the solid into gaseous ions (OpenStax’s sign). Some books use the negative of it.
  • ΔH°s, IE, ½D and EA are per mole of atoms; IE is the sum of the ionization energies the cation needs, and EA the sum of the electron affinities the anion needs.
  • The estimate from a known salt holds only for the same crystal structure, where the constant C is the same.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Find Lattice energy, Metal atoms per formula unit (a) 1, Nonmetal atoms per formula unit (b) 1, ΔH°f of the compound (kJ/mol) -553.5, ΔH°s of the metal (kJ/mol) 76.5, Ionization energy (kJ/mol) 375.7, ½D: atomize the nonmetal (kJ/mol) 79.4, Electron affinity (kJ/mol) -328.2 gives Lattice energy (kJ/mol) 756.9, Metal steps (kJ/mol) 452.2, Nonmetal steps (kJ/mol) −248.8, Answer ΔH_lattice = 756.9 kJ/mol.Source: OpenStax, Chemistry 2e, §7.5 Strengths of Ionic and Covalent Bonds (Born–Haber cycle for CsF: 756.9 kJ/mol; ΔH_lattice = C(Z⁺)(Z⁻) ÷ R₀), https://openstax.org/books/chemistry-2e/pages/7-5-strengths-of-ionic-and-covalent-bonds, the Born–Haber cycle for CsF (Table 7.4): 756.9 kJ/mol
  2. Find ΔH°f, Metal atoms per formula unit (a) 1, Nonmetal atoms per formula unit (b) 1, Lattice energy (kJ/mol) 756.9, ΔH°s of the metal (kJ/mol) 76.5, Ionization energy (kJ/mol) 375.7, ½D: atomize the nonmetal (kJ/mol) 79.4, Electron affinity (kJ/mol) -328.2 gives ΔH°f (kJ/mol) −553.5, Answer ΔH°f = −553.5 kJ/mol.Source: OpenStax, Chemistry 2e, §7.5 Strengths of Ionic and Covalent Bonds (Born–Haber cycle for CsF: 756.9 kJ/mol; ΔH_lattice = C(Z⁺)(Z⁻) ÷ R₀), https://openstax.org/books/chemistry-2e/pages/7-5-strengths-of-ionic-and-covalent-bonds: the CsF cycle solved for ΔH°f
  3. Find Electron affinity, Metal atoms per formula unit (a) 1, Nonmetal atoms per formula unit (b) 1, Lattice energy (kJ/mol) 756.9, ΔH°f of the compound (kJ/mol) -553.5, ΔH°s of the metal (kJ/mol) 76.5, Ionization energy (kJ/mol) 375.7, ½D: atomize the nonmetal (kJ/mol) 79.4 gives Electron affinity (kJ/mol) −328.2, Answer EA = −328.2 kJ/mol.Source: OpenStax, Chemistry 2e, §7.5 Strengths of Ionic and Covalent Bonds (Born–Haber cycle for CsF: 756.9 kJ/mol; ΔH_lattice = C(Z⁺)(Z⁻) ÷ R₀), https://openstax.org/books/chemistry-2e/pages/7-5-strengths-of-ionic-and-covalent-bonds: the CsF cycle solved for the electron affinity of F
  4. Find From a known salt, Known lattice energy (kJ/mol) 1,023, Known salt: cation charge 1, Known salt: anion charge 1, Known salt: interionic distance (pm) 200.8, New salt: cation charge 1, New salt: anion charge 1, New salt: interionic distance (pm) 231 gives Lattice energy (kJ/mol) 889.257, Answer ΔH_lattice ≈ 889.257 kJ/mol.Source: OpenStax, Chemistry 2e, Chapter 7 Exercises, §7.5, https://openstax.org/books/chemistry-2e/pages/7-exercises, Exercise 76: the closest choice is 890 kJ/mol
  5. Find From a known salt, Known lattice energy (kJ/mol) 1,023, Known salt: cation charge 1, Known salt: anion charge 1, Known salt: interionic distance (pm) 201, New salt: cation charge 2, New salt: anion charge 2, New salt: interionic distance (pm) 205 gives Lattice energy (kJ/mol) 4,012.16, Answer ΔH_lattice ≈ 4,012.16 kJ/mol.Source: OpenStax, Chemistry 2e, Chapter 7 Exercises, §7.5, https://openstax.org/books/chemistry-2e/pages/7-exercises, Exercise 79: the closest choice is 4,008 kJ/mol
  6. Find From a known salt, Known lattice energy (kJ/mol) 794, Known salt: cation charge 1, Known salt: anion charge 1, Known salt: interionic distance (pm) 269, New salt: cation charge 1, New salt: anion charge 1, New salt: interionic distance (pm) 231 gives Lattice energy (kJ/mol) 924.615.Source: OpenStax, Chemistry 2e, Chapter 7 Exercises, §7.5, https://openstax.org/books/chemistry-2e/pages/7-exercises, Exercise 84: the closest choice is 924 kJ/mol

How it works

Born–Haber cycle (OpenStax §7.5). For a compound MaXb made from its elements:

ΔH_lattice = a × (ΔH°s + IE) + b × (½D + EA) − ΔH°f

  • ΔH°s: the enthalpy to make a mole of gaseous metal atoms (sublimation).
  • IE: the total ionization energy to make a mole of the cations from gaseous atoms (IE₁ + IE₂ + …).
  • ½D: the enthalpy to make a mole of gaseous nonmetal atoms (half the bond energy of a diatomic gas such as F₂).
  • EA: the energy change when a mole of gaseous nonmetal atoms gains the anion’s electrons, negative when energy is released (F: −328.2 kJ/mol).
  • ΔH°f: the standard enthalpy of formation of the solid.

a and b are the metal and nonmetal ions per formula unit (whole numbers from 1 to 6). The lattice energy is positive: the energy to separate a mole of the solid into its gaseous ions. Rearranged:

  • ΔH°f = a × (ΔH°s + IE) + b × (½D + EA) − ΔH_lattice
  • EA = (ΔH_lattice + ΔH°f − a × (ΔH°s + IE)) ÷ b − ½D

The page also shows the metal steps, a × (ΔH°s + IE), and the nonmetal steps, b × (½D + EA).

Estimate from a known salt (OpenStax §7.5, ΔH_lattice = C × Z⁺ × Z⁻ ÷ R₀). For two salts with the same crystal structure (the same C):

U₂ = U₁ × (Z₂⁺ × Z₂⁻) ÷ (Z₁⁺ × Z₁⁻) × R₁ ÷ R₂

with Z the sizes of the ion charges (whole numbers from 1 to 4) and R the interionic distances in pm.

Rules the page checks. Energies from −100,000 to 100,000 kJ/mol; a typed lattice energy from 0.001 to 100,000 kJ/mol; distances from 1 to 10,000 pm. A Born–Haber lattice energy of 0 or less has no answer (no ionic solid has one; the signs are usually wrong). In the cycle modes, a result or step sum that is not 0 but smaller in size than 2⁻¹⁰²² kJ/mol (about 2.2 × 10⁻³⁰⁸) has no answer, because a number cannot keep its digits.

How results are written. Every result is worked exactly from the typed decimals and rounded once: 6 significant figures, trailing zeros dropped, thousands commas, the minus sign −, and m × 10ⁿ below 0.000001 or from 10²¹. The sentence reads “ΔH_lattice = 756.9 kJ/mol”, “ΔH°f = −553.5 kJ/mol”, “EA = −328.2 kJ/mol”, or for the estimate “ΔH_lattice ≈ 889.257 kJ/mol”.

Assumptions. Values per mole of atoms, as above; the estimate needs the same crystal structure.

Worked examples by hand

CsF (OpenStax Table 7.4). ΔH_lattice = 76.5 + 375.7 + 79.4 + (−328.2) − (−553.5) = 756.9 kJ/mol. Metal steps 76.5 + 375.7 = 452.2; nonmetal steps 79.4 − 328.2 = −248.8.

ΔH°f of CsF from the lattice energy. 452.2 + (−248.8) − 756.9 = −553.5 kJ/mol.

Electron affinity of F. (756.9 + (−553.5) − 452.2) ÷ 1 − 79.4 = −328.2 kJ/mol.

NaF from LiF (Exercise 76). 1,023 × 200.8 ÷ 231 = 889.257 kJ/mol (closest choice: 890).

MgO from LiF (Exercise 79). 1,023 × (2 × 2) ÷ (1 × 1) × 201 ÷ 205 = 4,012.16 kJ/mol (closest choice: 4,008).

NaF from KF (Exercise 84). 794 × 269 ÷ 231 = 924.615 kJ/mol (closest choice: 924).

Other questions people ask

How do I calculate lattice energy with the Born–Haber cycle?

Add the energies to make the gaseous ions from the elements, then subtract the enthalpy of formation: ΔH_lattice = ΔH°s + IE + ½D + EA − ΔH°f. For CsF (OpenStax): 76.5 + 375.7 + 79.4 + (−328.2) − (−553.5) = 756.9 kJ/mol.

What sign does lattice energy have?

OpenStax defines it as the energy to separate a mole of the solid into gaseous ions, which is positive (endothermic). Some books use the reverse step, forming the solid from the ions, and give a negative number of the same size.

What do I enter for a compound such as MgCl₂ or Na₂O?

Set a and b to the numbers of metal and nonmetal ions per formula unit (MgCl₂: a = 1, b = 2). Enter IE as the total to make the cation (IE₁ + IE₂ for Mg²⁺) and EA as the total to make the anion (EA₁ + EA₂ for O²⁻), each per mole of atoms; the page multiplies the metal steps by a and the nonmetal steps by b.

Why does lattice energy depend on ion charge and size?

It follows Coulomb’s law: ΔH_lattice = C × Z⁺ × Z⁻ ÷ R₀, with Z the ion charges and R₀ the distance between ion centers. Doubling both charges makes it about four times as large: LiF is 1,023 kJ/mol and MgO, with nearly the same distance, 3,900 kJ/mol.

How can I estimate a lattice energy without the full cycle?

Scale from a salt with the same crystal structure: U₂ = U₁ × (Z₂⁺Z₂⁻ ÷ Z₁⁺Z₁⁻) × (R₁ ÷ R₂). NaF from LiF: 1,023 × 200.8 ÷ 231 = 889 kJ/mol, near OpenStax’s answer of 890 kJ/mol. The structures must match, because the constant C depends on it.

How do I find the electron affinity from the lattice energy?

Solve the cycle for EA: EA = (ΔH_lattice + ΔH°f − ΔH°s − IE) ÷ b − ½D. With CsF’s values and a lattice energy of 756.9 kJ/mol, fluorine’s electron affinity comes out at −328.2 kJ/mol.