{
  "id": "lattice-energy",
  "version": "b7246e1659d7",
  "status": "published",
  "name": "Lattice Energy Calculator",
  "question": "What is the lattice energy?",
  "summary": "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.",
  "category": "chemistry",
  "subcategory": "atoms",
  "url": "https://www.acalculator.org/chemistry/lattice-energy-calculator",
  "markdown": "https://www.acalculator.org/chemistry/lattice-energy-calculator.md",
  "kind": "function",
  "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).",
  "assumptions": [
    "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."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "The lattice energy by the Born–Haber cycle, ΔH°f or the electron affinity from it, or an estimate from a known salt.",
        "type": "string",
        "enum": [
          "u",
          "f",
          "ea",
          "scale"
        ]
      },
      "a": {
        "title": "Metal atoms per formula unit (a)",
        "description": "How many metal ions one formula unit holds: 1 in CsF, 2 in Na₂O.",
        "type": "integer",
        "minimum": 1,
        "maximum": 6
      },
      "b": {
        "title": "Nonmetal atoms per formula unit (b)",
        "description": "How many nonmetal ions one formula unit holds: 1 in CsF, 2 in MgCl₂.",
        "type": "integer",
        "minimum": 1,
        "maximum": 6
      },
      "hf": {
        "title": "ΔH°f of the compound (kJ/mol)",
        "description": "The standard enthalpy of formation of the solid, usually negative.",
        "type": "number",
        "minimum": -100000,
        "maximum": 100000
      },
      "hs": {
        "title": "ΔH°s of the metal (kJ/mol)",
        "description": "The enthalpy to turn the metal into gaseous atoms (sublimation), per mole of atoms.",
        "type": "number",
        "minimum": -100000,
        "maximum": 100000
      },
      "ie": {
        "title": "Ionization energy (kJ/mol)",
        "description": "The energy to remove the cation’s electrons from a mole of gaseous metal atoms (IE₁ + IE₂ + …).",
        "type": "number",
        "minimum": -100000,
        "maximum": 100000
      },
      "hd": {
        "title": "½D: atomize the nonmetal (kJ/mol)",
        "description": "The enthalpy to make a mole of gaseous nonmetal atoms: half the X₂ bond energy for a gas such as F₂.",
        "type": "number",
        "minimum": -100000,
        "maximum": 100000
      },
      "ea": {
        "title": "Electron affinity (kJ/mol)",
        "description": "The energy change when a mole of gaseous nonmetal atoms gains the anion’s electrons; negative when energy is released (F: −328.2).",
        "type": "number",
        "minimum": -100000,
        "maximum": 100000
      },
      "u": {
        "title": "Lattice energy (kJ/mol)",
        "description": "The energy to separate a mole of the solid into gaseous ions: positive.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 100000
      },
      "u1": {
        "title": "Known lattice energy (kJ/mol)",
        "description": "The lattice energy of a salt with the same crystal structure, such as 1,023 kJ/mol for LiF.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 100000
      },
      "z1c": {
        "title": "Known salt: cation charge",
        "description": "The size of the known salt’s cation charge: 1 for Li⁺.",
        "type": "integer",
        "minimum": 1,
        "maximum": 4
      },
      "z1a": {
        "title": "Known salt: anion charge",
        "description": "The size of the known salt’s anion charge: 1 for F⁻.",
        "type": "integer",
        "minimum": 1,
        "maximum": 4
      },
      "r1": {
        "title": "Known salt: interionic distance (pm)",
        "description": "The known salt’s cation–anion distance in pm, such as 200.8 for LiF.",
        "type": "number",
        "minimum": 1,
        "maximum": 10000
      },
      "z2c": {
        "title": "New salt: cation charge",
        "description": "The size of the new salt’s cation charge: 2 for Mg²⁺.",
        "type": "integer",
        "minimum": 1,
        "maximum": 4
      },
      "z2a": {
        "title": "New salt: anion charge",
        "description": "The size of the new salt’s anion charge: 2 for O²⁻.",
        "type": "integer",
        "minimum": 1,
        "maximum": 4
      },
      "r2": {
        "title": "New salt: interionic distance (pm)",
        "description": "The new salt’s cation–anion distance in pm, such as 231 for NaF.",
        "type": "number",
        "minimum": 1,
        "maximum": 10000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The lattice energy, ΔH°f or the electron affinity, in kJ/mol.",
      "format": "text"
    },
    "lattice": {
      "label": "Lattice energy (kJ/mol)",
      "description": "The energy to separate a mole of the solid into gaseous ions.",
      "format": "number"
    },
    "formation": {
      "label": "ΔH°f (kJ/mol)",
      "description": "The standard enthalpy of formation of the compound.",
      "format": "number"
    },
    "affinity": {
      "label": "Electron affinity (kJ/mol)",
      "description": "The energy change per mole of nonmetal atoms gaining the anion’s electrons.",
      "format": "number"
    },
    "metal": {
      "label": "Metal steps (kJ/mol)",
      "description": "a × (ΔH°s + IE): making the gaseous cations.",
      "format": "number"
    },
    "nonmetal": {
      "label": "Nonmetal steps (kJ/mol)",
      "description": "b × (½D + EA): making the gaseous anions.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "u",
      "a": 1,
      "b": 1,
      "hf": -553.5,
      "hs": 76.5,
      "ie": 375.7,
      "hd": 79.4,
      "ea": -328.2,
      "u": 756.9,
      "u1": 1023,
      "z1c": 1,
      "z1a": 1,
      "r1": 200.8,
      "z2c": 1,
      "z2a": 1,
      "r2": 231
    },
    "outputs": {
      "result": "ΔH_lattice = 756.9 kJ/mol",
      "lattice": 756.9,
      "formation": -553.5,
      "affinity": -328.2,
      "metal": 452.2,
      "nonmetal": -248.8
    },
    "text": "ΔH_lattice = 756.9 kJ/mol."
  },
  "examples": [
    {
      "given": {
        "find": "u",
        "a": 1,
        "b": 1,
        "hf": -553.5,
        "hs": 76.5,
        "ie": 375.7,
        "hd": 79.4,
        "ea": -328.2
      },
      "expect": {
        "lattice": 756.9,
        "metal": 452.2,
        "nonmetal": -248.8,
        "result": "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; Python 3"
    },
    {
      "given": {
        "find": "f",
        "a": 1,
        "b": 1,
        "u": 756.9,
        "hs": 76.5,
        "ie": 375.7,
        "hd": 79.4,
        "ea": -328.2
      },
      "expect": {
        "formation": -553.5,
        "result": "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; Python 3"
    },
    {
      "given": {
        "find": "ea",
        "a": 1,
        "b": 1,
        "u": 756.9,
        "hf": -553.5,
        "hs": 76.5,
        "ie": 375.7,
        "hd": 79.4
      },
      "expect": {
        "affinity": -328.2,
        "result": "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; Python 3"
    },
    {
      "given": {
        "find": "scale",
        "u1": 1023,
        "z1c": 1,
        "z1a": 1,
        "r1": 200.8,
        "z2c": 1,
        "z2a": 1,
        "r2": 231
      },
      "expect": {
        "lattice": 889.2571428571429,
        "result": "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; Python 3"
    },
    {
      "given": {
        "find": "scale",
        "u1": 1023,
        "z1c": 1,
        "z1a": 1,
        "r1": 201,
        "z2c": 2,
        "z2a": 2,
        "r2": 205
      },
      "expect": {
        "lattice": 4012.1560975609755,
        "result": "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; Python 3"
    },
    {
      "given": {
        "find": "scale",
        "u1": 794,
        "z1c": 1,
        "z1a": 1,
        "r1": 269,
        "z2c": 1,
        "z2a": 1,
        "r2": 231
      },
      "expect": {
        "lattice": 924.6147186147186
      },
      "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; Python 3"
    }
  ],
  "sources": [
    "OpenStax, Chemistry 2e, §7.5 Strengths of Ionic and Covalent Bonds: lattice energy (positive, for MX(s) → M⁺(g) + X⁻(g)), ΔH_lattice = C(Z⁺)(Z⁻) ÷ R₀, the Born–Haber cycle for CsF in Table 7.4 (ΔH°s = 76.5, ½D = 79.4, IE = 375.7, EA = −328.2, ΔH°f = −553.5, ΔH_lattice = 756.9 kJ/mol), LiF 1,023 and MgO 3,900 kJ/mol. https://openstax.org/books/chemistry-2e/pages/7-5-strengths-of-ionic-and-covalent-bonds (retrieved 2026-10-19)",
    "OpenStax, Chemistry 2e, Chapter 7 Exercises, §7.5: Exercise 76 (NaF from LiF, choose 890 kJ/mol), Exercise 79 (MgO from LiF, choose 4,008 kJ/mol), Exercise 84 (NaF from KF, choose 924 kJ/mol). https://openstax.org/books/chemistry-2e/pages/7-exercises (retrieved 2026-10-19)"
  ],
  "related": [
    "enthalpy",
    "electronegativity",
    "gibbs-free-energy",
    "effective-nuclear-charge"
  ],
  "changelog": []
}