{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"half-life"},"result":{"entity_type":"calculator","id":"half-life","calculator_id":"half-life","canonical_url":"https://tttkmbb.com/chemistry/half-life","name":"Half-Life Calculator","title":"Half-Life Calculator – Exponential Decay, Remaining Amount, Decay Constant and Elapsed Time","category":"chemistry","category_name":"Chemistry","tool_name":"calculate_half_life_decay","featured_mcp_tool":false,"description":"Computes first-order exponential decay: given two of half-life, elapsed time and remaining amount (with an initial amount) it solves the third and reports the remaining fraction, number of half-lives, decay constant and mean lifetime.","use_when":"You need how much of a radioactive isotope, drug or other first-order decaying quantity remains after a time, or the half-life or elapsed time implied by a measured remaining amount.","do_not_use_when":"The process is not first-order (e.g. zero-order elimination, growth, or compound interest – use compound-interest for money) or several isotopes decay in a chain.","inputs":[{"name":"initial_amount","label":"Initial amount","type":"number","required":false,"default":100,"exclusive_min":0,"description":"Starting quantity in any unit (g, Bq, atoms, mg/L). Default 100 makes remaining_amount a percentage.","example":100},{"name":"half_life","label":"Half-life","type":"number","required":false,"exclusive_min":0,"description":"Half-life in any time unit (same unit as elapsed_time). Leave empty to solve for it.","example":5730},{"name":"elapsed_time","label":"Elapsed time","type":"number","required":false,"min":0,"description":"Time elapsed, same unit as half_life. Leave empty to solve for it.","example":11460},{"name":"remaining_amount","label":"Remaining amount","type":"number","required":false,"exclusive_min":0,"description":"Quantity left after elapsed_time, same unit as initial_amount. Leave empty to solve for it."}],"outputs":[{"name":"remaining_amount","label":"Remaining amount","type":"number","decimals":6,"description":"N(t) = N0 × (1/2)^(t / half_life)."},{"name":"remaining_fraction","label":"Remaining fraction","type":"number","decimals":6,"description":"N(t) / N0."},{"name":"remaining_percent","label":"Remaining","type":"number","unit":"%","decimals":4,"description":"100 × N(t) / N0."},{"name":"decayed_amount","label":"Decayed amount","type":"number","decimals":6,"description":"N0 − N(t)."},{"name":"number_of_half_lives","label":"Half-lives elapsed","type":"number","decimals":4,"description":"elapsed_time / half_life."},{"name":"half_life","label":"Half-life","type":"number","decimals":6,"description":"Half-life, given or solved (time unit of the inputs)."},{"name":"elapsed_time","label":"Elapsed time","type":"number","decimals":6,"description":"Elapsed time, given or solved (time unit of the inputs)."},{"name":"decay_constant","label":"Decay constant λ","type":"number","decimals":8,"description":"ln 2 / half_life, per time unit of the inputs."},{"name":"mean_lifetime","label":"Mean lifetime τ","type":"number","decimals":4,"description":"1 / λ = half_life / ln 2."},{"name":"solved_for","label":"Solved for","type":"string","decimals":4,"description":"remaining_amount, half_life or elapsed_time."}],"input_schema":{"type":"object","properties":{"initial_amount":{"description":"Starting quantity in any unit (g, Bq, atoms, mg/L). Default 100 makes remaining_amount a percentage.","type":"number","exclusiveMinimum":0,"default":100,"examples":[100]},"half_life":{"description":"Half-life in any time unit (same unit as elapsed_time). Leave empty to solve for it.","type":"number","exclusiveMinimum":0,"examples":[5730]},"elapsed_time":{"description":"Time elapsed, same unit as half_life. Leave empty to solve for it.","type":"number","minimum":0,"examples":[11460]},"remaining_amount":{"description":"Quantity left after elapsed_time, same unit as initial_amount. Leave empty to solve for it.","type":"number","exclusiveMinimum":0}},"additionalProperties":false},"output_schema":{"type":"object","properties":{"remaining_amount":{"description":"N(t) = N0 × (1/2)^(t / half_life).","type":"number"},"remaining_fraction":{"description":"N(t) / N0.","type":"number"},"remaining_percent":{"description":"100 × N(t) / N0. Unit: %.","type":"number","x-unit":"%"},"decayed_amount":{"description":"N0 − N(t).","type":"number"},"number_of_half_lives":{"description":"elapsed_time / half_life.","type":"number"},"half_life":{"description":"Half-life, given or solved (time unit of the inputs).","type":"number"},"elapsed_time":{"description":"Elapsed time, given or solved (time unit of the inputs).","type":"number"},"decay_constant":{"description":"ln 2 / half_life, per time unit of the inputs.","type":"number"},"mean_lifetime":{"description":"1 / λ = half_life / ln 2.","type":"number"},"solved_for":{"description":"remaining_amount, half_life or elapsed_time.","type":"string"}}},"formula":"N(t) = N0 × (1/2)^(t / t½) = N0 × e^(−λt); λ = ln 2 / t½; τ = 1 / λ; t½ = t / log2(N0 / N); t = t½ × log2(N0 / N)","method":"First-order kinetics: the decay rate is proportional to the amount present, so the same fraction decays in every half-life regardless of the starting amount.","sources":[{"name":"OpenStax Chemistry 2e – 21.3 Radioactive Decay","url":"https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay","type":"textbook","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Half-life","url":"https://en.wikipedia.org/wiki/Half-life","type":"reference","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":"Carbon-14 after 11,460 years","inputs":{"initial_amount":100,"half_life":5730,"elapsed_time":11460},"expected":{"remaining_amount":25,"remaining_fraction":0.25,"number_of_half_lives":2,"decay_constant":0.00012097,"mean_lifetime":8266.6,"solved_for":"remaining_amount"},"url":"https://tttkmbb.com/api/v1/calculate/half-life?initial_amount=100&half_life=5730&elapsed_time=11460"},{"name":"80 mg of iodine-131 falls to 10 mg in 24.06 days","inputs":{"initial_amount":80,"remaining_amount":10,"elapsed_time":24.06},"expected":{"half_life":8.02,"number_of_half_lives":3,"remaining_percent":12.5,"decay_constant":0.086427,"solved_for":"half_life"},"url":"https://tttkmbb.com/api/v1/calculate/half-life?initial_amount=80&remaining_amount=10&elapsed_time=24.06"}],"faq":[{"q":"Which time unit should I use?","a":"Any, as long as half_life and elapsed_time share it; the decay constant and mean lifetime are then per that unit (e.g. per year for a half-life in years)."},{"q":"Does this work for drug elimination?","a":"Yes for drugs with first-order (linear) elimination, using the plasma half-life; about 97% is eliminated after 5 half-lives. Zero-order elimination (e.g. ethanol at high doses) is not modelled."}],"tags":["half life","radioactive decay","exponential decay","decay constant","carbon dating","remaining amount"],"related":[{"calculator_id":"moles-mass","reason":"Convert the remaining mass of an isotope to a number of atoms."},{"calculator_id":"logarithm","reason":"Log2 and natural log arithmetic used in solving for time."}],"links":{"html":"https://tttkmbb.com/chemistry/half-life","markdown":"https://tttkmbb.com/chemistry/half-life.md","json":"https://tttkmbb.com/chemistry/half-life.json","api":"https://tttkmbb.com/api/v1/calculate/half-life","schema":"https://tttkmbb.com/api/v1/calculators/half-life","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-23T23:33:25Z"}