Radioactive Decay Calculator
Calculate remaining quantity, half-life, decay constant, activity, and time elapsed for radioactive isotopes with visual decay curves and dose estimation.
Carbon Dating Calculator
Calculate the age of organic samples using radiocarbon (¹⁴C) dating with half-life decay, percent modern carbon, and calibrated BP age ranges.
Radiocarbon Age⧉
—
Conventional radiocarbon age (Before Present = 1950)
Calendar Year (approx.)⧉
—
Uncalibrated estimate: 1950 − age (calibration curves needed for accuracy)
pMC⧉
0.00%
Percent Modern Carbon relative to 1950 standard
Fraction Remaining⧉
0.0000
N/N₀ — ratio of current to original ¹⁴C
Half-Lives Elapsed⧉
0.00
Using t₁/₂ = 5,568 years
Specific Activity⧉
0.000 dpm/g
Decays per minute per gram of carbon
Decay Constant (λ)⧉
1.2449e-4 yr⁻¹
λ = ln(2) / t₁/₂
Mean Lifetime (τ)⧉
8,033 years
τ = 1/λ — average time before a ¹⁴C atom decays
¹⁴C Decay Curve
| Age (years) | % Modern Carbon | Remaining |
|---|---|---|
| 0 | 100.00% | 100.0% |
| 1,000 | 88.29% | 88.3% |
| 2,000 | 77.96% | 78.0% |
| 5,000 | 53.66% | 53.7% |
| 5,730 | 49.00% | 49.0% |
| 10,000 | 28.80% | 28.8% |
| 15,000 | 15.45% | 15.5% |
| 20,000 | 8.29% | 8.3% |
| 30,000 | 2.39% | 2.4% |
| 40,000 | 0.69% | 0.7% |
| 50,000 | 0.20% | 0.2% |
Famous Radiocarbon-Dated Samples
| Sample | ~Age (BP) | ~pMC | Significance |
|---|---|---|---|
| Turin Shroud | 691 | 91.7% | Medieval linen (1260-1390 CE) |
| Dead Sea Scrolls | 2,050 | 77.7% | Oldest biblical manuscripts |
| Ötzi the Iceman | 5,300 | 52.7% | Copper Age mummy (Alps) |
| Lascaux paintings | 17,000 | 12.1% | Paleolithic cave art (France) |
| Woolly mammoth | 25,000 | 4.8% | Last glacial period remains |
| Oldest AMS dates | 50,000 | 0.2% | Near detection limit |
Half-Life Convention Comparison
| Convention | t₁/₂ (years) | Age for pMC=50 | Age for pMC=25 | Usage |
|---|---|---|---|---|
| Libby | 5,568 | 5,568 yr | 11,136 yr | Standard lab reporting |
| Cambridge | 5,730 | 5,730 yr | 11,460 yr | More accurate value |
| Difference | 162 yr (2.9%) | 162 yr | 324 yr | Calibration corrects for this |
Planning notes, formulas, and examples
About the Carbon Dating Calculator
Radiocarbon dating is the most widely used method for determining the age of organic materials up to about 50,000 years old. Living organisms continuously exchange carbon with the atmosphere, maintaining a roughly constant ratio of radioactive ¹⁴C to stable ¹²C. When an organism dies, ¹⁴C intake stops and the existing ¹⁴C decays with a half-life of 5,730 years (Cambridge half-life; the conventional Libby half-life of 5,568 years is still used for reporting by convention).
By measuring the remaining ¹⁴C activity compared to the initial activity, we can calculate the time since death: t = −(t₁/₂ / ln 2) × ln(N/N₀). This calculation gives a "conventional radiocarbon age" in years Before Present (BP, where present = 1950 CE). Because atmospheric ¹⁴C levels have varied over time due to solar activity, ocean circulation, volcanic eruptions, and nuclear testing, raw radiocarbon ages must be calibrated using tree-ring and other records to produce calendar ages.
This calculator computes ages from measured ¹⁴C activity or percent modern carbon, converts between Libby and Cambridge ages, and provides reference data for famous archaeological and paleontological samples. It demonstrates the exponential decay curve and shows the practical dating range limitations.
When This Page Helps
This calculator quickly converts measured ¹⁴C data into ages, compares half-life conventions, and shows the decay curve. It's perfect for archaeology students, lab workers processing radiocarbon results, and anyone curious about how we date the past.
How to Use the Inputs
- Select the input mode: percent modern carbon (pMC), measured activity (dpm/g), or remaining fraction (N/N₀).
- Enter the measured ¹⁴C value for your sample.
- Choose the half-life convention (Libby: 5,568 yr or Cambridge: 5,730 yr).
- The calculator computes the conventional radiocarbon age in years BP.
- Review the remaining ¹⁴C percentage, number of half-lives elapsed, and decay rate.
- Use presets for famous archaeological samples to explore the dating range.
- Check the decay curve and reference table for calibration context.
Formula used
Radiocarbon Age: t = −(t₁/₂ / ln 2) × ln(N/N₀) = −(t₁/₂ / ln 2) × ln(pMC / 100), where t₁/₂ = 5,568 years (Libby) or 5,730 years (Cambridge), N = remaining ¹⁴C, N₀ = initial ¹⁴C. Fraction remaining = (1/2)^(t/t₁/₂).Example Calculation
Result: Age = 5,568 years BP
At 50% modern carbon (pMC = 50), exactly one half-life has elapsed. Using the Libby half-life: t = −(5568/ln 2) × ln(0.5) = 5,568 years BP.
Tips & Best Practices
- Always report radiocarbon ages as "X years BP" (Before Present = 1950 CE) using the Libby half-life by convention.
- pMC values above 100% indicate the sample contains bomb carbon (post-1950).
- Subtracting 13C fractionation correction is essential for accurate dates — most labs do this automatically.
- Marine organisms appear older than they are (marine reservoir effect: ~400 years) due to deep ocean water mixing.
- Contamination with modern or ancient carbon dramatically skews results — sample preparation is critical.
- The IntCal20 calibration curve is the current standard for converting radiocarbon years to calendar years.
History of Radiocarbon Dating
Willard Libby developed radiocarbon dating in 1949, earning the Nobel Prize in Chemistry in 1960. He initially tested the method on Egyptian artifacts of known age, confirming that ¹⁴C decay faithfully records time since death. The technique revolutionized archaeology, geology, and paleoclimatology by providing an absolute dating method independent of cultural context.
Calibration: From Radiocarbon Years to Calendar Years
The assumption of constant atmospheric ¹⁴C is only approximately true. Calibration curves constructed from tree rings (dendrochronology) going back 14,000 years, and from coral/speleothem records beyond that, convert "radiocarbon years BP" to true calendar years. The current international calibration curve, IntCal20, extends to 55,000 years. Calendar ages can differ from radiocarbon ages by centuries — the difference is especially large around 10,000-11,000 BP due to rapid ¹⁴C fluctuations.
Modern Applications and Limitations
Beyond archaeology, ¹⁴C dating is used in forensic science (determining if remains are modern), environmental science (tracing carbon in the food web), and atmospheric science (measuring fossil fuel CO₂ dilution of atmospheric ¹⁴C, the Suess effect). The bomb peak from nuclear testing provides a time marker more precise than any archaeological application — it can date wine vintages, identify art forgeries, and track cell turnover in the human body.
Sources & Methodology
Last updated:
Frequently Asked Questions
pMC compares ¹⁴C activity in a sample to the 1950 standard (corrected for isotope fractionation). A living organism has ~100 pMC; a 5,730-year-old sample has ~50 pMC.
Libby originally measured t₁/₂ = 5,568 years. The more accurate Cambridge value is 5,730 years. By convention, radiocarbon labs still report ages using the Libby half-life for consistency.
Practical limit is ~50,000 years (about 9 half-lives), where ¹⁴C activity drops below reliable detection limits. AMS dating pushes this slightly further than traditional counting methods.
Atmospheric ¹⁴C hasn't been constant — solar cycles, ocean mixing, volcanic CO₂, and nuclear testing cause fluctuations. Calibration curves (IntCal20) convert radiocarbon years to calendar years using tree rings, corals, and lake sediments.
No. ¹⁴C dating works only on materials that were once living and exchanging carbon with the atmosphere: wood, bone, shell, charcoal, seeds, peat, and similar organic materials.
Nuclear testing (1950s-60s) nearly doubled atmospheric ¹⁴C. Samples from that era have >100 pMC. This "bomb carbon" is now useful for dating recent biological materials and studying carbon cycling.
Accelerator Mass Spectrometry (AMS) directly counts ¹⁴C atoms rather than waiting for decay events. It needs much smaller samples (~1 mg vs grams) and is more sensitive for old samples.