{
  "id": "activation-energy",
  "version": "1cbdea238fe8",
  "status": "published",
  "name": "Activation Energy Calculator",
  "question": "What is activation energy?",
  "summary": "Finds the activation energy Ea of a reaction from rate constants at two temperatures with the Arrhenius equation, the rate constant at a new temperature from Ea, or Ea from one rate constant and the pre-exponential factor A.",
  "category": "chemistry",
  "subcategory": "stoichiometry",
  "url": "https://www.acalculator.org/chemistry/activation-energy-calculator",
  "markdown": "https://www.acalculator.org/chemistry/activation-energy-calculator.md",
  "kind": "function",
  "method": "Ea = R × ln(k₂ ÷ k₁) ÷ (1 ÷ T₁ − 1 ÷ T₂); k₂ = k₁ × e^((Ea ÷ R)(1 ÷ T₁ − 1 ÷ T₂)); Ea = R × T × ln(A ÷ k); A = k₁ × e^(Ea ÷ (R × T₁)); R = 8.31446261815324 J/(mol·K).",
  "assumptions": [
    "Ea and A do not change with temperature over the range used.",
    "k₁, k₂, k and A are in the same unit; temperatures are absolute (the page converts °C and °F to kelvins)."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "Ea from two rate constants, k at a new temperature, or Ea from one rate constant and A.",
        "type": "string",
        "enum": [
          "ea",
          "k2",
          "ea1"
        ]
      },
      "t1": {
        "title": "Temperature T₁",
        "description": "The first temperature.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "temperature",
        "minimum": 1,
        "maximum": 100000
      },
      "k1": {
        "title": "Rate constant k₁",
        "description": "The rate constant at T₁, in any unit (the same for k₂).",
        "type": "number",
        "minimum": 1e-100,
        "maximum": 1e+100
      },
      "t2": {
        "title": "Temperature T₂",
        "description": "The second temperature.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "temperature",
        "minimum": 1,
        "maximum": 100000
      },
      "k2": {
        "title": "Rate constant k₂",
        "description": "The rate constant at T₂, in the same unit as k₁.",
        "type": "number",
        "minimum": 1e-100,
        "maximum": 1e+100
      },
      "ea": {
        "title": "Activation energy Ea (kJ/mol)",
        "description": "The activation energy in kilojoules per mole.",
        "type": "number",
        "minimum": -10000,
        "maximum": 10000
      },
      "a": {
        "title": "Pre-exponential factor A",
        "description": "The frequency factor A, in the unit of k.",
        "type": "number",
        "minimum": 1e-100,
        "maximum": 1e+100
      },
      "k": {
        "title": "Rate constant k",
        "description": "The rate constant at temperature T, in the unit of A.",
        "type": "number",
        "minimum": 1e-100,
        "maximum": 1e+100
      },
      "t": {
        "title": "Temperature T",
        "description": "The temperature of the rate constant.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "temperature",
        "minimum": 1,
        "maximum": 100000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The activation energy in kJ/mol, or the rate constant at T₂.",
      "format": "text"
    },
    "eaKj": {
      "label": "Activation energy (kJ/mol)",
      "description": "Ea in kilojoules per mole.",
      "format": "number"
    },
    "eaJ": {
      "label": "Activation energy (J/mol)",
      "description": "Ea in joules per mole.",
      "format": "number"
    },
    "kNew": {
      "label": "Rate constant k₂",
      "description": "The rate constant at T₂, in the unit of k₁.",
      "format": "number"
    },
    "factor": {
      "label": "k₂ ÷ k₁",
      "description": "How many times faster the reaction is at T₂ than at T₁.",
      "format": "number"
    },
    "aValue": {
      "label": "Pre-exponential factor A",
      "description": "k₁ × e^(Ea ÷ RT₁), in the unit of k; left out beyond 10^±300.",
      "format": "number"
    },
    "lnA": {
      "label": "ln A",
      "description": "The natural logarithm of A.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "ea",
      "t1": "555 K",
      "k1": 3.52e-7,
      "t2": "781 K",
      "k2": 0.0395,
      "ea": 180,
      "a": 10000000000000,
      "k": 0.001,
      "t": "300 K"
    },
    "outputs": {
      "result": "Ea = 185.431 kJ/mol",
      "eaKj": 185.43064579187495,
      "eaJ": 185430.64579187497,
      "factor": 112215.90909090909,
      "aValue": 99604610165.55698,
      "lnA": 25.324474287268544
    },
    "text": "Ea = 185.431 kJ/mol."
  },
  "examples": [
    {
      "given": {
        "find": "ea",
        "t1": 555,
        "k1": 3.52e-7,
        "t2": 781,
        "k2": 0.0395
      },
      "expect": {
        "eaKj": 185.43064579187495,
        "lnA": 25.324474287268547,
        "result": "Ea = 185.431 kJ/mol"
      },
      "source": "OpenStax, Chemistry 2e, §12.5 Collision Theory (k = Ae^(−Ea/RT); ln(k₁/k₂) = (Ea/R)(1/T₂ − 1/T₁)), https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory, Example 12.13 Determination of Ea: 1.8 × 10⁵ J/mol (180 kJ/mol); NIST, CODATA value: molar gas constant R = 8.314 462 618… J/(mol·K), exact, https://physics.nist.gov/cgi-bin/cuu/Value?r; Python 3",
      "tolerance": 1e-12
    },
    {
      "given": {
        "find": "ea",
        "t1": 650,
        "k1": 1.66,
        "t2": 700,
        "k2": 7.39
      },
      "expect": {
        "eaKj": 112.98624860015951,
        "result": "Ea = 112.986 kJ/mol"
      },
      "source": "OpenStax, Chemistry 2e, §12.5 Collision Theory (k = Ae^(−Ea/RT); ln(k₁/k₂) = (Ea/R)(1/T₂ − 1/T₁)), https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory, Example 12.13 Check Your Learning: 1.1 × 10⁵ J/mol; Python 3",
      "tolerance": 1e-12
    },
    {
      "given": {
        "find": "k2",
        "ea": 50,
        "t1": 298.15,
        "k1": 1,
        "t2": 308.15
      },
      "expect": {
        "kNew": 1.9242648158276077,
        "result": "k₂ = 1.92426"
      },
      "source": "OpenStax, Chemistry 2e, §12.5 Collision Theory (k = Ae^(−Ea/RT); ln(k₁/k₂) = (Ea/R)(1/T₂ − 1/T₁)), https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory: k₂ = k₁ × e^((Ea/R)(1/T₁ − 1/T₂)); hand calculation in content.mdx; Python 3",
      "tolerance": 1e-12
    },
    {
      "given": {
        "find": "ea1",
        "a": 10000000000000,
        "k": 0.001,
        "t": 300
      },
      "expect": {
        "eaKj": 91.8948368679163,
        "result": "Ea = 91.8948 kJ/mol"
      },
      "source": "OpenStax, Chemistry 2e, §12.5 Collision Theory (k = Ae^(−Ea/RT); ln(k₁/k₂) = (Ea/R)(1/T₂ − 1/T₁)), https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory: Ea = RT ln(A/k) from k = Ae^(−Ea/RT); hand calculation in content.mdx; Python 3",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, Chemistry 2e, §12.5 Collision Theory: the Arrhenius equation k = Ae^(−Ea/RT), ln k = (−Ea/R)(1/T) + ln A, the two-point form ln(k₁/k₂) = (Ea/R)(1/T₂ − 1/T₁), Example 12.13 (HI: rate constants from 555 K to 781 K, Ea = 1.8 × 10⁵ J/mol) and its Check Your Learning (N₂O₅: 1.66 at 650 K, 7.39 at 700 K, 1.1 × 10⁵ J/mol). https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory (retrieved 2026-10-19)",
    "NIST, CODATA value: molar gas constant R = 8.314 462 618… J/(mol·K), exact (N_A × k). https://physics.nist.gov/cgi-bin/cuu/Value?r (retrieved 2026-10-19)"
  ],
  "related": [
    "half-life",
    "gibbs-free-energy",
    "enthalpy",
    "equilibrium-constant"
  ],
  "changelog": []
}