Carbon 14 Half Life Calculator & Formula

This calculator works out the remaining amount of carbon-14 in a sample after a given time, or the time elapsed since death based on the remaining fraction. You enter the initial amount (or fraction) and the elapsed time, and you get back the remaining mass and the number of half-lives passed. The standard half-life of carbon-14 is 5,730 years, which this tool uses for all calculations.

Formula

The decay of carbon-14 follows the exponential decay law:

N(t) = N₀ × (1/2)^(t / T½)

Where:

• N(t) = remaining amount of carbon-14 after time t (grams or any mass unit)
• Nâ‚€ = initial amount of carbon-14 (same unit as N(t))
• t = elapsed time (years)
• T½ = half-life of carbon-14 = 5,730 years

To find the age from a measured remaining fraction, rearrange:

t = T½ × log₂(N₀ / N(t)) = 5,730 × ln(N₀/N(t)) / ln(2)

The calculator uses the first form directly: you input the remaining fraction (N(t)/Nâ‚€) and the elapsed time, and it outputs the remaining amount (as a fraction of 1 gram) plus the number of half-lives.

Worked Example

Using the default values: time elapsed = 5,730 years, remaining fraction = 0.5.

1. Half-lives elapsed = 5,730 / 5,730 = 1
2. Remaining amount = 0.5 × 1 = 0.5 grams (if you started with 1 gram)

The calculator shows: remaining = 0.5000 g, half-lives = 1.00, age = 5,730 years. This matches the definition: after one half-life, exactly half remains.

How Carbon-14 Dating Works

Carbon-14 (¹�C) is a radioactive isotope produced in the upper atmosphere when cosmic rays convert nitrogen-14 into carbon-14. Plants absorb it as carbon dioxide, animals eat plants, so every living thing has a constant ratio of ¹�C to stable carbon-12 while alive. When an organism dies, it stops exchanging carbon with the atmosphere, and the ¹�C decays without replenishment. By measuring the remaining ¹�C/¹²C ratio and comparing it to the atmospheric baseline, scientists estimate how long ago the organism died.

The half-life of 5,730 years means that after 5,730 years, half the original ¹�C is gone; after 11,460 years, a quarter remains; after 17,190 years, an eighth remains. This exponential decay continues until the amount is too small to measure reliably.

When to Use This Calculator

Use this tool for educational purposes, rough estimates, or checking textbook problems. It is useful for:

• Understanding how radioactive decay reduces a sample over time
• Estimating the age of archaeological organic material (wood, bone, charcoal) up to about 50,000 years
• Converting between elapsed time and remaining fraction
• Verifying homework problems involving half-life calculations

For real dating, laboratories use accelerator mass spectrometry (AMS) and calibrate results against tree-ring data, because atmospheric ¹�C levels have varied over time. This calculator gives the raw radiocarbon age, not the calibrated calendar age.

How to Read the Results

The first output, remaining amount, is the mass of carbon-14 left if you started with 1 gram. If you enter a different initial amount, multiply the output by that amount. For example, if you start with 2 grams and enter a remaining fraction of 0.25, the remaining mass is 0.25 × 2 = 0.5 grams.

The half-lives elapsed is simply the time divided by 5,730. It tells you how many half-life periods have passed. The age output is just the time you entered, provided for clarity.

If you want to find the age from a measured fraction, set the time input to a guess and adjust until the remaining fraction matches your measurement. Alternatively, use the rearranged formula: age = 5,730 × ln(1/fraction) / ln(2).

Common Mistakes and Limitations

• Unit mix-ups: Time must be in years. If you have dates in calendar years, convert to years elapsed before entering.
• Fraction vs. percentage: Enter 0.5 for 50%, not 50. The calculator has no percent sign.
• Assuming constant atmospheric ¹â�´C: The ratio has changed over time due to solar activity, fossil fuel burning, and nuclear testing. Calibration curves (like IntCal) correct for this.
• Contamination: Any modern carbon in the sample skews the result. Samples must be clean.
• Age limit: Beyond ~50,000 years, so little ¹â�´C remains that measurement errors become huge. Other methods (like potassium-argon) are used for older materials.

Also note that the half-life is sometimes quoted as 5,730 ± 40 years. The calculator uses the standard value of 5,730, which is accurate enough for most purposes.

Alternatives to Carbon-14 Dating

For materials older than 50,000 years, scientists use other isotopes with longer half-lives:

• Potassium-40 (half-life 1.25 billion years) – for volcanic rock and minerals
• Uranium-238 (half-life 4.5 billion years) – for zircon crystals and ancient rocks
• Rubidium-87 (half-life 49 billion years) – for very old rocks

For relatively recent organic materials (last few centuries), tree-ring counting and historical records provide more precise dates than radiocarbon.

Frequently Asked Questions

How accurate is carbon-14 dating?

With careful calibration, radiocarbon dating can be accurate to within a few decades for samples up to 10,000 years old, and within a few hundred years for older samples. The uncertainty comes from measurement error, contamination, and variations in atmospheric ¹�C. Laboratories report a standard deviation, typically ±30–50 years for well-preserved samples.

Can carbon-14 dating be used on anything?

No. Only materials that were once alive and contain carbon can be dated: wood, charcoal, bone, shell, peat, cloth, and similar organic remains. Inorganic materials like stone, metal, or pottery cannot be directly dated unless they contain organic residues (e.g., food crusts or charcoal in the clay).

Why is the half-life 5,730 years and not a round number?

The half-life is a measured physical constant, not chosen. It was first determined in the 1940s by measuring the decay rate of purified carbon-14. The value has been refined over decades; the currently accepted figure is 5,730 ± 40 years. It is not a round number because it arises from the quantum probability of nuclear decay.

What does a “remaining fraction� of 0.1 mean?

It means 10% of the original carbon-14 is still present. To find the age, use the formula: age = 5,730 × ln(1/0.1) / ln(2) ≈ 5,730 × 3.3219 ≈ 19,035 years. So a sample with 10% remaining is about 19,000 years old.

Does the calculator account for calibration?

No. It gives the raw radiocarbon age based on the simple decay equation. Real-world dating requires calibration against tree-ring or other records to convert the radiocarbon age into a calendar age. This calculator is for educational and estimation purposes only.

Can I date a living organism?

No. Living organisms maintain a constant ¹�C level through exchange with the atmosphere and food chain. Only after death does the level start to decrease. Therefore, carbon-14 dating only works for dead organic matter.

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in t | Time elapsed | years | 5730 | 0
in fraction | Remaining fraction | - | 0.5 | 0
out remaining | Remaining amount | g | fraction * 1 | 4
out half_lives | Half-lives elapsed | - | t / 5730 | 2
out age | Age of sample | years | t | 0