{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"carbon-dating"},"result":{"entity_type":"calculator","id":"carbon-dating","calculator_id":"carbon-dating","canonical_url":"https://tttkmbb.com/science/carbon-dating","name":"Carbon Dating Calculator","title":"Carbon-14 Dating Calculator – Age from Remaining ¹⁴C (Half-Life 5730 Years) and Libby Radiocarbon Age","category":"science","category_name":"Biology, Earth & Space","tool_name":"calculate_carbon_dating_age","featured_mcp_tool":false,"description":"Computes the age of a sample from the percentage of carbon-14 remaining by first-order decay with the Cambridge half-life of 5730 years, or the remaining percentage from a known age, and also reports the conventional radiocarbon age based on the Libby half-life of 5568 years.","use_when":"You know the fraction of ¹⁴C left relative to modern carbon and want an uncalibrated age, or want the ¹⁴C remaining after a given number of years.","do_not_use_when":"You need calibrated calendar dates (that requires a calibration curve such as IntCal20), the sample is older than about 50,000 years, or you are working with another isotope or drug (use half-life).","inputs":[{"name":"percent_c14_remaining","label":"¹⁴C remaining","type":"number","unit":"%","required":false,"max":100,"exclusive_min":0,"description":"Carbon-14 activity of the sample as a percentage of the modern reference (100 = modern). Give this or age_years.","example":10},{"name":"age_years","label":"Age","type":"number","unit":"years","required":false,"min":0,"max":1000000,"description":"Known age; when given, the remaining percentage is computed instead. Give this or percent_c14_remaining."},{"name":"half_life_years","label":"Half-life","type":"number","unit":"years","required":false,"default":5730,"max":1000000000,"exclusive_min":0,"description":"Half-life used for the age: default is the Cambridge value 5730 ± 40 years for ¹⁴C; the Libby value 5568 is reported separately.","example":5730}],"outputs":[{"name":"age_years","label":"Age","type":"number","unit":"years","decimals":0,"description":"t = −(t½ / ln 2) × ln(fraction remaining), uncalibrated."},{"name":"radiocarbon_age_libby_years","label":"Conventional radiocarbon age (Libby)","type":"number","unit":"years BP","decimals":0,"description":"Same fraction with the Libby half-life 5568 years, as used for conventional radiocarbon ages (before calibration)."},{"name":"percent_remaining","label":"¹⁴C remaining","type":"number","unit":"%","decimals":4,"description":"100 × 2^(−t / t½)."},{"name":"fraction_remaining","label":"Fraction remaining","type":"number","decimals":6,"description":"N / N0."},{"name":"half_lives_elapsed","label":"Half-lives elapsed","type":"number","decimals":4,"description":"t / t½ = log2(N0 / N)."},{"name":"decay_constant_per_year","label":"Decay constant λ","type":"number","unit":"1/year","decimals":8,"description":"ln 2 / t½."},{"name":"mean_lifetime_years","label":"Mean lifetime τ","type":"number","unit":"years","decimals":1,"description":"t½ / ln 2 (8267 years for 5730)."},{"name":"solved_for","label":"Solved for","type":"string","decimals":4,"description":"age_years or percent_remaining."}],"input_schema":{"type":"object","properties":{"percent_c14_remaining":{"description":"Carbon-14 activity of the sample as a percentage of the modern reference (100 = modern). Give this or age_years. Unit: %.","type":"number","maximum":100,"exclusiveMinimum":0,"examples":[10],"x-unit":"%"},"age_years":{"description":"Known age; when given, the remaining percentage is computed instead. Give this or percent_c14_remaining. Unit: years.","type":"number","minimum":0,"maximum":1000000,"x-unit":"years"},"half_life_years":{"description":"Half-life used for the age: default is the Cambridge value 5730 ± 40 years for ¹⁴C; the Libby value 5568 is reported separately. Unit: years.","type":"number","maximum":1000000000,"exclusiveMinimum":0,"default":5730,"examples":[5730],"x-unit":"years"}},"additionalProperties":false},"output_schema":{"type":"object","properties":{"age_years":{"description":"t = −(t½ / ln 2) × ln(fraction remaining), uncalibrated. Unit: years.","type":"number","x-unit":"years"},"radiocarbon_age_libby_years":{"description":"Same fraction with the Libby half-life 5568 years, as used for conventional radiocarbon ages (before calibration). Unit: years BP.","type":"number","x-unit":"years BP"},"percent_remaining":{"description":"100 × 2^(−t / t½). Unit: %.","type":"number","x-unit":"%"},"fraction_remaining":{"description":"N / N0.","type":"number"},"half_lives_elapsed":{"description":"t / t½ = log2(N0 / N).","type":"number"},"decay_constant_per_year":{"description":"ln 2 / t½. Unit: 1/year.","type":"number","x-unit":"1/year"},"mean_lifetime_years":{"description":"t½ / ln 2 (8267 years for 5730). Unit: years.","type":"number","x-unit":"years"},"solved_for":{"description":"age_years or percent_remaining.","type":"string"}}},"formula":"t = −(t½ / ln 2) × ln(fraction_remaining); fraction_remaining = 2^(−t / t½); λ = ln 2 / t½; τ = t½ / ln 2; Libby age = −(5568 / ln 2) × ln(fraction_remaining)","method":"First-order decay with the Cambridge half-life 5730 ± 40 years. Laboratories report conventional radiocarbon ages with the Libby half-life (5568 years, mean life 8033 years) per Stuiver & Polach (1977); both are uncalibrated and must be converted to calendar years with a calibration curve. Practical limit about 50,000 years (≈ 0.2 % remaining).","sources":[{"name":"Wikipedia – Radiocarbon dating","url":"https://en.wikipedia.org/wiki/Radiocarbon_dating","type":"reference","retrieved_at":"2026-09-24"},{"name":"Stuiver M & Polach HA (1977) Discussion: Reporting of 14C data, Radiocarbon 19(3):355-363","url":"https://doi.org/10.1017/S0033822200003672","type":"peer_reviewed","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":"10 % of ¹⁴C remaining","inputs":{"percent_c14_remaining":10},"expected":{"age_years":19035,"radiocarbon_age_libby_years":18496,"half_lives_elapsed":3.3219,"decay_constant_per_year":0.00012097,"mean_lifetime_years":8266.6,"solved_for":"age_years"},"url":"https://tttkmbb.com/api/v1/calculate/carbon-dating?percent_c14_remaining=10"},{"name":"Sample 2,000 years old","inputs":{"age_years":2000},"expected":{"percent_remaining":78.5106,"fraction_remaining":0.785106,"half_lives_elapsed":0.349,"radiocarbon_age_libby_years":1943,"solved_for":"percent_remaining"},"url":"https://tttkmbb.com/api/v1/calculate/carbon-dating?age_years=2000"}],"faq":[{"q":"Why two half-lives?","a":"Libby's original 5568 years is kept by convention for reported radiocarbon ages (years BP) so that all dates stay comparable; the more accurate Cambridge value 5730 gives the physically better age. Calibration curves correct the Libby-based age anyway."},{"q":"Is the age a calendar date?","a":"No. Atmospheric ¹⁴C varied over time, so an uncalibrated age must be calibrated (IntCal20, SHCal20, Marine20) to get a calendar range; differences reach several thousand years for old samples."},{"q":"How is percent_c14_remaining measured?","a":"As the sample's ¹⁴C/¹²C ratio (or activity) divided by that of the 1950 modern standard, after correcting for isotopic fractionation (δ¹³C); 100 % means modern carbon."}],"tags":["carbon dating","radiocarbon","carbon 14","half life","age of sample","libby"],"related":[{"calculator_id":"half-life","reason":"General first-order decay for any isotope or elimination half-life."},{"calculator_id":"logarithm","reason":"Natural logarithm arithmetic used to solve for the age."}],"links":{"html":"https://tttkmbb.com/science/carbon-dating","markdown":"https://tttkmbb.com/science/carbon-dating.md","json":"https://tttkmbb.com/science/carbon-dating.json","api":"https://tttkmbb.com/api/v1/calculate/carbon-dating","schema":"https://tttkmbb.com/api/v1/calculators/carbon-dating","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:46:20Z","next_actions":[{"tool":"run_calculator","calculator_id":"carbon-dating","reason":"Run Carbon Dating Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/science/carbon-dating.md"}}