{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"arrhenius-equation"},"result":{"entity_type":"calculator","id":"arrhenius-equation","calculator_id":"arrhenius-equation","canonical_url":"https://tttkmbb.com/chemistry/arrhenius-equation","name":"Arrhenius Equation Calculator","title":"Arrhenius Equation Calculator – Rate Constant at a New Temperature or Activation Energy from Two Rate Constants","category":"chemistry","category_name":"Chemistry","tool_name":"solve_arrhenius_equation","featured_mcp_tool":false,"description":"Uses the two-point Arrhenius equation ln(k2/k1) = (Ea/R)(1/T1 − 1/T2) to find the rate constant at a second temperature and the rate ratio from the activation energy, or the activation energy from rate constants measured at two temperatures, plus the pre-exponential factor A.","use_when":"You need how much faster a reaction runs at a different temperature, the rate constant at a new temperature, or the activation energy from two measured rate constants.","do_not_use_when":"The rate constant is unknown at both temperatures and only Ea and A are known (compute k = A·exp(−Ea/RT) directly), or the reaction changes mechanism between the temperatures.","inputs":[{"name":"activation_energy_kj_per_mol","label":"Activation energy Ea","type":"number","unit":"kJ/mol","required":false,"max":5000,"exclusive_min":0,"description":"Activation energy in kJ/mol. Leave empty to solve it from two rate constants.","example":50},{"name":"temperature_1","label":"Temperature 1","type":"number","required":true,"description":"First temperature in temperature_unit.","example":25},{"name":"temperature_2","label":"Temperature 2","type":"number","required":true,"description":"Second temperature in temperature_unit.","example":35},{"name":"temperature_unit","label":"Temperature unit","type":"enum","required":false,"default":"celsius","values":[{"value":"celsius","label":"°C","aliases":["c"]},{"value":"kelvin","label":"K","aliases":["k"]}],"description":"Unit of the temperature inputs; the calculation uses kelvin (K = °C + 273.15).","example":"celsius"},{"name":"rate_constant_1","label":"Rate constant k1 at T1","type":"number","required":false,"exclusive_min":0,"description":"Rate constant at temperature_1 in any unit (s⁻¹, L/(mol·s)); k2 is returned in the same unit.","example":0.001},{"name":"rate_constant_2","label":"Rate constant k2 at T2","type":"number","required":false,"exclusive_min":0,"description":"Rate constant at temperature_2 (same unit as k1); needed to solve Ea."}],"outputs":[{"name":"activation_energy_kj_per_mol","label":"Activation energy","type":"number","unit":"kJ/mol","decimals":4,"description":"Ea, given or solved."},{"name":"activation_energy_j_per_mol","label":"Activation energy","type":"number","unit":"J/mol","decimals":1,"description":"Ea in joules per mole."},{"name":"rate_ratio_k2_over_k1","label":"Rate ratio k2 / k1","type":"number","decimals":6,"description":"exp((Ea/R)(1/T1 − 1/T2)): factor by which the rate changes from T1 to T2."},{"name":"rate_constant_1","label":"k1","type":"number","decimals":8,"description":"Rate constant at T1 (unit of the input)."},{"name":"rate_constant_2","label":"k2","type":"number","decimals":8,"description":"Rate constant at T2 (unit of the input)."},{"name":"pre_exponential_factor","label":"Pre-exponential factor A","type":"number","decimals":4,"description":"A = k1 × exp(Ea / (R·T1)), same unit as k (only when k1 is known)."},{"name":"temperature_1_k","label":"T1","type":"number","unit":"K","decimals":2,"description":"First temperature in kelvin."},{"name":"temperature_2_k","label":"T2","type":"number","unit":"K","decimals":2,"description":"Second temperature in kelvin."},{"name":"solved_for","label":"Solved for","type":"string","decimals":4,"description":"rate_constant_2, rate_constant_1, rate_ratio_k2_over_k1 or activation_energy_kj_per_mol."}],"input_schema":{"type":"object","properties":{"activation_energy_kj_per_mol":{"description":"Activation energy in kJ/mol. Leave empty to solve it from two rate constants. Unit: kJ/mol.","type":"number","maximum":5000,"exclusiveMinimum":0,"examples":[50],"x-unit":"kJ/mol"},"temperature_1":{"description":"First temperature in temperature_unit.","type":"number","examples":[25]},"temperature_2":{"description":"Second temperature in temperature_unit.","type":"number","examples":[35]},"temperature_unit":{"description":"Unit of the temperature inputs; the calculation uses kelvin (K = °C + 273.15).","type":"string","enum":["celsius","kelvin"],"default":"celsius","examples":["celsius"]},"rate_constant_1":{"description":"Rate constant at temperature_1 in any unit (s⁻¹, L/(mol·s)); k2 is returned in the same unit.","type":"number","exclusiveMinimum":0,"examples":[0.001]},"rate_constant_2":{"description":"Rate constant at temperature_2 (same unit as k1); needed to solve Ea.","type":"number","exclusiveMinimum":0}},"additionalProperties":false,"required":["temperature_1","temperature_2"]},"output_schema":{"type":"object","properties":{"activation_energy_kj_per_mol":{"description":"Ea, given or solved. Unit: kJ/mol.","type":"number","x-unit":"kJ/mol"},"activation_energy_j_per_mol":{"description":"Ea in joules per mole. Unit: J/mol.","type":"number","x-unit":"J/mol"},"rate_ratio_k2_over_k1":{"description":"exp((Ea/R)(1/T1 − 1/T2)): factor by which the rate changes from T1 to T2.","type":"number"},"rate_constant_1":{"description":"Rate constant at T1 (unit of the input).","type":"number"},"rate_constant_2":{"description":"Rate constant at T2 (unit of the input).","type":"number"},"pre_exponential_factor":{"description":"A = k1 × exp(Ea / (R·T1)), same unit as k (only when k1 is known).","type":"number"},"temperature_1_k":{"description":"First temperature in kelvin. Unit: K.","type":"number","x-unit":"K"},"temperature_2_k":{"description":"Second temperature in kelvin. Unit: K.","type":"number","x-unit":"K"},"solved_for":{"description":"rate_constant_2, rate_constant_1, rate_ratio_k2_over_k1 or activation_energy_kj_per_mol.","type":"string"}}},"formula":"k = A × exp(−Ea / (R × T)); ln(k2 / k1) = (Ea / R) × (1/T1 − 1/T2) with R = 8.314462618 J/(mol·K) and T in kelvin; Ea = R × ln(k2 / k1) / (1/T1 − 1/T2); A = k1 × exp(Ea / (R × T1))","method":"Assumes Ea and A are independent of temperature over the range considered, which holds well for intervals of a few tens of kelvin.","sources":[{"name":"OpenStax Chemistry 2e – 12.5 Collision Theory","url":"https://openstax.org/books/chemistry-2e/pages/12-5-collision-theory","type":"textbook","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Arrhenius equation","url":"https://en.wikipedia.org/wiki/Arrhenius_equation","type":"reference","retrieved_at":"2026-09-24"},{"name":"NIST – CODATA internationally recommended values of the fundamental physical constants (molar gas constant R = 8.314462618 J/(mol·K), exact)","url":"https://physics.nist.gov/cuu/Constants/index.html","type":"standard","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"Ea 50 kJ/mol, k1 = 0.001 at 25 °C, k2 at 35 °C","inputs":{"activation_energy_kj_per_mol":50,"temperature_1":25,"temperature_2":35,"rate_constant_1":0.001},"expected":{"rate_ratio_k2_over_k1":1.92426,"rate_constant_2":0.00192426,"pre_exponential_factor":574938,"temperature_1_k":298.15,"temperature_2_k":308.15,"solved_for":"rate_constant_2"},"url":"https://tttkmbb.com/api/v1/calculate/arrhenius-equation?activation_energy_kj_per_mol=50&temperature_1=25&temperature_2=35&rate_constant_1=0.001"},{"name":"Ea from k = 2.5e-3 at 20 °C and 1.0e-2 at 40 °C","inputs":{"temperature_1":20,"temperature_2":40,"rate_constant_1":0.0025,"rate_constant_2":0.01},"expected":{"activation_energy_kj_per_mol":52.9056,"activation_energy_j_per_mol":52905.6,"rate_ratio_k2_over_k1":4,"solved_for":"activation_energy_kj_per_mol"},"url":"https://tttkmbb.com/api/v1/calculate/arrhenius-equation?temperature_1=20&temperature_2=40&rate_constant_1=0.0025&rate_constant_2=0.01"}],"faq":[{"q":"Does a reaction really double every 10 °C?","a":"Only for Ea around 50 kJ/mol near room temperature (this example gives ×1.92). Reactions with larger Ea are more temperature-sensitive, smaller Ea less so."},{"q":"Which unit does k need?","a":"Any: only the ratio k2/k1 enters the equation, so the solved rate constant and A come out in the unit you used for k1."}],"tags":["arrhenius equation","activation energy","rate constant temperature","reaction rate","pre-exponential factor","kinetics"],"related":[{"calculator_id":"half-life","reason":"Convert a first-order rate constant to a half-life (t½ = ln 2 / k)."},{"calculator_id":"temperature","reason":"Convert temperatures between °C, °F and K."},{"calculator_id":"logarithm","reason":"Natural logarithm arithmetic used in the two-point form."}],"links":{"html":"https://tttkmbb.com/chemistry/arrhenius-equation","markdown":"https://tttkmbb.com/chemistry/arrhenius-equation.md","json":"https://tttkmbb.com/chemistry/arrhenius-equation.json","api":"https://tttkmbb.com/api/v1/calculate/arrhenius-equation","schema":"https://tttkmbb.com/api/v1/calculators/arrhenius-equation","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:48:19Z"}