# Radioactive Dating > Radioactive dating for JEE Physics: carbon-14 half-life 5730 years, activity ratios, K-Ar and U-Pb methods, dating limits and sample calculations. - Canonical URL: https://prepelephant.com/topics/jee/physics/radioactive-dating-jee - Exam / course: JEE · Subject: Physics - Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team - First published: 2026-10-02 - Last updated: 2026-10-02 - How to cite: "Radioactive Dating", PrepElephant, https://prepelephant.com/topics/jee/physics/radioactive-dating-jee ## Direct answer While you are alive, your carbon keeps pace with the atmosphere: cosmic-ray neutrons convert nitrogen-14 to carbon-14 (n + ¹⁴N → ¹⁴C + p), the isotope decays with half-life 5730 years, and continuous exchange keeps living tissue at a steady activity near 15 disintegrations per minute per gram of carbon. Death freezes the exchange, and from that moment the carbon-14 decays without replenishment — so a wooden artefact showing one-eighth of the living activity is three half-lives, about 17,200 years, old. The working equation is t = (1/λ) ln(A₀/A) with λ = ln2/5730 yr⁻¹, applied only while enough activity remains to measure: beyond roughly 50,000 years the method expires. Older material turns to slower clocks — potassium-40 (half-life 1.25 billion years) trapping argon-40 in minerals, and uranium-238/235 chains closing on lead-206/207 — the systems that dated the Earth itself at about 4.5 billion years. ## What you must remember - **Production:** cosmic-ray neutrons on atmospheric nitrogen, n + ¹⁴N → ¹⁴C + p; the carbon-14 oxidises to CO₂ and enters the biosphere through photosynthesis and food. - **Half-life and activity:** t½ = 5730 years (the Cambridge value used in Indian textbooks — not Libby's original 5568); living carbon activity ≈ 15 counts per minute per gram. - **Dating equation:** N = N₀e^−λt, so t = (1/λ)ln(N₀/N) = (t½/ln2) × ln(A₀/A); activity ratios work because activity ∝ number of nuclei. - **Fraction arithmetic:** ½ remaining = one half-life, ⅛ = three, 1/2ⁿ = n half-lives; t = n × 5730 years — most numericals are solved this way without logarithms. - **Method limit:** usable to about 50,000 years (roughly nine half-lives), after which the activity sinks into background; older samples need slower clocks. - **K-Ar method:** ⁴⁰K (t½ = 1.25 × 10⁹ yr) decays to ⁴⁰Ar, a gas trapped only after the mineral crystallises; accumulated argon measures the rock's age. - **U-Pb method:** dual chains ²³⁸U → ²⁰⁶Pb and ²³⁵U → ²⁰⁷Pb in zircon give cross-checking ages; the oldest terrestrial zircons reach about 4.4 billion years. ## Carbon dating worked through An excavated cloth yields 3.75 disintegrations per minute per gram of carbon, one quarter of the living value of 15. Two half-lives have passed — a quarter remains — so the cloth is 2 × 5730 = 11,460 years old. A bone at 1/8 activity: three half-lives, 17,190 years. The logarithm form only enters for awkward fractions: an activity 60 per cent of living gives t = (5730/0.693) × ln(1/0.6) ≈ 4225 years. The habit: convert activity ratios to remaining fractions first, then either count half-lives (clean powers of two) or reach for t = (t½/ln2)ln(A₀/A). The method's assumptions each earn a JEE statement. Living activity A₀ is taken constant over archaeological time — calibrated against tree-ring chronologies (dendrochronology), since industrial fossil-fuel dilution and atmospheric nuclear tests have both perturbed modern carbon-14 levels. The sample must have exchanged carbon until death and stayed chemically uncontaminated since. And the clock reads the death of the tissue, not the building of the monument: old timber reused in a young structure dates its own felling, not the construction. For rocks, the same logic with slower isotopes: molten material loses its argon gas, crystallisation restarts the clock at zero, and the present ⁴⁰Ar/⁴⁰K ratio converts directly to age — how lunar samples and Earth's oldest minerals were placed at over four billion years. ## Dating questions in JEE The fraction trap leads: "a sample decays to 1/16 — how much has decayed?" The answer is 15/16, not 1/16; remaining versus decayed is the single most common one-mark loss in this chapter. Second, half-life value: use 5730 years, the value in Indian texts and data tables; older sources citing 5568 (Libby's) produce subtly wrong options. Third, the activity-versus-number relation: dating uses activity ratios because counting decays is what laboratories do, and A ∝ N makes the substitution legal — a justification asked directly in statement form. Fourth, the method's range: carbon-14 cannot date dinosaur bones (too old) or the age of the Earth (much too old); matching clock to timescale — C-14 for archaeology, K-Ar and U-Pb for geology — is a recurring match-the-column item. Fifth, the production equation (n + ¹⁴N → ¹⁴C + p) appears as a fill-in-the-particle question. Main tests the arithmetic; Advanced couples dating with decay-series logic — for instance, how much ²⁰⁶Pb has accumulated in a uranium ore of known mass and age. ## Frequently asked questions ### How does carbon-14 enter living organisms? Cosmic rays convert atmospheric nitrogen-14 into carbon-14, which becomes CO₂, enters plants by photosynthesis and animals by feeding, maintaining equilibrium activity until death stops the exchange. ### What is the half-life of carbon-14 and the standard living activity? 5730 years, and living carbon shows about 15 disintegrations per minute per gram — the reference activity from which sample ages are reckoned. ### A specimen shows one-eighth of living carbon-14 activity — how old is it? Three half-lives, 3 × 5730 = 17,190 years, since ⅛ = (½)³. ### Why is carbon dating unreliable beyond about 50,000 years? After roughly nine half-lives the residual activity is too faint to distinguish from background radiation, so the method's precision collapses. ### Which isotopes date the age of rocks instead of organic remains? Potassium-40 (decaying to trapped argon-40, half-life 1.25 billion years) and uranium-238/235 (ending in lead-206/207), whose slow clocks span the Earth's 4.5-billion-year history.