Radioactive Dating
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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.