{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"grahams-law"},"result":{"entity_type":"calculator","id":"grahams-law","calculator_id":"grahams-law","canonical_url":"https://tttkmbb.com/chemistry/grahams-law","name":"Graham's Law Calculator","title":"Graham's Law Calculator – Effusion and Diffusion Rate Ratio of Two Gases from Molar Mass","category":"chemistry","category_name":"Chemistry","tool_name":"calculate_grahams_law","featured_mcp_tool":false,"description":"Applies Graham's law, rate1/rate2 = √(M2/M1), to two gases given by formula or molar mass, returning the effusion-rate ratio, the time ratio and the root-mean-square molecular speeds; alternatively solves the unknown molar mass of gas 2 from a measured rate ratio.","use_when":"You need how much faster one gas effuses or diffuses than another, the relative effusion times, or the molar mass of an unknown gas from its effusion rate relative to a known gas.","do_not_use_when":"You need the absolute effusion rate through a specific orifice (depends on geometry and pressure), or the gases are at high pressure where mean free paths make diffusion non-ideal.","inputs":[{"name":"gas_1_formula","label":"Gas 1 formula","type":"string","required":false,"description":"Formula of gas 1, e.g. He, H2, O2, CO2, UF6 (case-sensitive symbols). Alternative to gas_1_molar_mass_g_mol.","example":"He"},{"name":"gas_1_molar_mass_g_mol","label":"Gas 1 molar mass","type":"number","unit":"g/mol","required":false,"max":100000,"exclusive_min":0,"description":"Molar mass of gas 1, if no formula is given."},{"name":"gas_2_formula","label":"Gas 2 formula","type":"string","required":false,"description":"Formula of gas 2. Leave both gas 2 inputs empty and give rate_ratio_1_to_2 to solve its molar mass.","example":"O2"},{"name":"gas_2_molar_mass_g_mol","label":"Gas 2 molar mass","type":"number","unit":"g/mol","required":false,"max":100000,"exclusive_min":0,"description":"Molar mass of gas 2, if no formula is given."},{"name":"rate_ratio_1_to_2","label":"Measured rate ratio rate1 / rate2","type":"number","required":false,"exclusive_min":0,"description":"Measured effusion-rate ratio of gas 1 to gas 2 (equal to the time ratio t2/t1); used only to solve the molar mass of an unknown gas 2."},{"name":"temperature","label":"Temperature","type":"number","required":false,"default":25,"description":"Temperature for the root-mean-square speeds, in temperature_unit.","example":25},{"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"}],"outputs":[{"name":"molar_mass_1_g_mol","label":"Molar mass of gas 1","type":"number","unit":"g/mol","decimals":3,"description":"M1 from the formula or as given."},{"name":"molar_mass_2_g_mol","label":"Molar mass of gas 2","type":"number","unit":"g/mol","decimals":3,"description":"M2 from the formula, as given, or solved from the rate ratio."},{"name":"rate_ratio_1_to_2","label":"Rate ratio rate1 / rate2","type":"number","decimals":4,"description":"√(M2 / M1): how many times faster gas 1 effuses than gas 2."},{"name":"time_ratio_1_to_2","label":"Time ratio t1 / t2","type":"number","decimals":4,"description":"√(M1 / M2): time for gas 1 to effuse a given amount relative to gas 2."},{"name":"faster_gas","label":"Faster gas","type":"string","decimals":4,"description":"The lighter gas effuses faster."},{"name":"rms_speed_1_m_s","label":"RMS speed of gas 1","type":"number","unit":"m/s","decimals":1,"description":"√(3RT/M1) at the given temperature."},{"name":"rms_speed_2_m_s","label":"RMS speed of gas 2","type":"number","unit":"m/s","decimals":1,"description":"√(3RT/M2) at the given temperature."},{"name":"temperature_k","label":"Temperature","type":"number","unit":"K","decimals":2,"description":"Temperature used for the speeds."},{"name":"solved_for","label":"Solved for","type":"string","decimals":4,"description":"rate_ratio_1_to_2 or molar_mass_2_g_mol."}],"input_schema":{"type":"object","properties":{"gas_1_formula":{"description":"Formula of gas 1, e.g. He, H2, O2, CO2, UF6 (case-sensitive symbols). Alternative to gas_1_molar_mass_g_mol.","type":"string","examples":["He"]},"gas_1_molar_mass_g_mol":{"description":"Molar mass of gas 1, if no formula is given. Unit: g/mol.","type":"number","maximum":100000,"exclusiveMinimum":0,"x-unit":"g/mol"},"gas_2_formula":{"description":"Formula of gas 2. Leave both gas 2 inputs empty and give rate_ratio_1_to_2 to solve its molar mass.","type":"string","examples":["O2"]},"gas_2_molar_mass_g_mol":{"description":"Molar mass of gas 2, if no formula is given. Unit: g/mol.","type":"number","maximum":100000,"exclusiveMinimum":0,"x-unit":"g/mol"},"rate_ratio_1_to_2":{"description":"Measured effusion-rate ratio of gas 1 to gas 2 (equal to the time ratio t2/t1); used only to solve the molar mass of an unknown gas 2.","type":"number","exclusiveMinimum":0},"temperature":{"description":"Temperature for the root-mean-square speeds, in temperature_unit.","type":"number","default":25,"examples":[25]},"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"]}},"additionalProperties":false},"output_schema":{"type":"object","properties":{"molar_mass_1_g_mol":{"description":"M1 from the formula or as given. Unit: g/mol.","type":"number","x-unit":"g/mol"},"molar_mass_2_g_mol":{"description":"M2 from the formula, as given, or solved from the rate ratio. Unit: g/mol.","type":"number","x-unit":"g/mol"},"rate_ratio_1_to_2":{"description":"√(M2 / M1): how many times faster gas 1 effuses than gas 2.","type":"number"},"time_ratio_1_to_2":{"description":"√(M1 / M2): time for gas 1 to effuse a given amount relative to gas 2.","type":"number"},"faster_gas":{"description":"The lighter gas effuses faster.","type":"string"},"rms_speed_1_m_s":{"description":"√(3RT/M1) at the given temperature. Unit: m/s.","type":"number","x-unit":"m/s"},"rms_speed_2_m_s":{"description":"√(3RT/M2) at the given temperature. Unit: m/s.","type":"number","x-unit":"m/s"},"temperature_k":{"description":"Temperature used for the speeds. Unit: K.","type":"number","x-unit":"K"},"solved_for":{"description":"rate_ratio_1_to_2 or molar_mass_2_g_mol.","type":"string"}}},"formula":"rate_1 / rate_2 = √(M2 / M1); t_1 / t_2 = √(M1 / M2); M2 = M1 × (rate_1 / rate_2)²; v_rms = √(3RT / M) with M in kg/mol and R = 8.314462618 J/(mol·K)","method":"Graham's law follows from equal average kinetic energy of ideal gases at the same temperature; it applies to effusion through a small hole and approximately to diffusion. Molar masses use IUPAC 2021 abridged atomic weights.","sources":[{"name":"OpenStax Chemistry 2e – 9.4 Effusion and Diffusion of Gases","url":"https://openstax.org/books/chemistry-2e/pages/9-4-effusion-and-diffusion-of-gases","type":"textbook","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Graham's law","url":"https://en.wikipedia.org/wiki/Graham%27s_law","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":"Helium vs oxygen at 25 °C","inputs":{"gas_1_formula":"He","gas_2_formula":"O2"},"expected":{"molar_mass_1_g_mol":4.003,"molar_mass_2_g_mol":31.998,"rate_ratio_1_to_2":2.8274,"time_ratio_1_to_2":0.3537,"faster_gas":"He","rms_speed_1_m_s":1363.1,"rms_speed_2_m_s":482.1,"solved_for":"rate_ratio_1_to_2"},"url":"https://tttkmbb.com/api/v1/calculate/grahams-law?gas_1_formula=He&gas_2_formula=O2"},{"name":"Unknown gas effusing 4 times slower than H2","inputs":{"gas_1_formula":"H2","rate_ratio_1_to_2":4},"expected":{"molar_mass_2_g_mol":32.256,"rate_ratio_1_to_2":4,"time_ratio_1_to_2":0.25,"solved_for":"molar_mass_2_g_mol"},"url":"https://tttkmbb.com/api/v1/calculate/grahams-law?gas_1_formula=H2&rate_ratio_1_to_2=4"}],"faq":[{"q":"Is the ratio the same for diffusion?","a":"Approximately. Graham's law is exact for effusion through a pinhole into vacuum; for diffusion through another gas it gives the right trend but collisions make real rates lower."},{"q":"I measured times, not rates. What do I enter?","a":"A gas that takes 3 times longer effuses 3 times slower, so rate_ratio_1_to_2 = t2 / t1. The output time_ratio_1_to_2 is the inverse of the rate ratio."}],"tags":["graham's law","effusion","diffusion rate","rate of effusion","molar mass from effusion","rms speed"],"related":[{"calculator_id":"molar-mass","reason":"Molar mass of each gas from its formula."},{"calculator_id":"ideal-gas-law","reason":"Pressure, volume and moles of the gases."}],"links":{"html":"https://tttkmbb.com/chemistry/grahams-law","markdown":"https://tttkmbb.com/chemistry/grahams-law.md","json":"https://tttkmbb.com/chemistry/grahams-law.json","api":"https://tttkmbb.com/api/v1/calculate/grahams-law","schema":"https://tttkmbb.com/api/v1/calculators/grahams-law","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T02:02:35Z"}