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How to Calculate What Remains After Decay
Heshan Fernando
Co-founder & COO
After one half-life, half remains. After two, a quarter. After ten, about a thousandth.
The arithmetic is simple. What catches people is that decay never reaches zero, and that the average atom lives considerably longer than the half-life suggests.
The two equivalent forms
N = N₀ × (1/2)^(t / half-life)
N = N₀ × e^(−λt)
These describe the same thing. The first uses half-life directly, which is how decay is usually quoted. The second uses the decay constant λ, which is how it appears in most physics.
They relate by:
λ = ln(2) / half-life
Both are worth having, because literature uses whichever suits the context and converting between them is a common source of factor-of-0.693 errors.
| Half-lives elapsed | Remaining |
|---|---|
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 7 | ~0.8% |
| 10 | ~0.1% |
Mean lifetime is longer than half-life
The result people find counter-intuitive.
The mean lifetime — the average time an individual atom survives — is:
τ = 1/λ = half-life / ln(2) ≈ half-life × 1.443
For carbon-14 with a half-life of 5,730 years, the mean lifetime is about 8,267 years.
The reason is the long tail. Half the atoms decay within one half-life, but the survivors continue decaying over an unbounded period, and those long-lived atoms pull the average up. A distribution with a long tail has a mean above its median, and the half-life is the median.
Which quantity is relevant depends on the question. Half-life answers “when will half be gone”; mean lifetime appears in calculations involving the total decay integrated over time.
Half-life is essentially fixed
Unlike most rate processes, radioactive decay is almost entirely insensitive to external conditions.
Temperature, pressure, chemical state and physical form make no practical difference. The decay is a nuclear process, and chemistry happens in the electron shells far outside the nucleus.
There are measurable exceptions in decay modes involving electron capture, where the electron density at the nucleus matters slightly and can be altered by chemical environment or extreme ionisation. Those effects are tiny and specialised.
For practical purposes, a half-life is a constant of the isotope. That reliability is exactly what makes radiometric dating possible.
Dating is more than the exponential
The equation is the easy part of radiocarbon dating. The complications are:
Atmospheric variation. Carbon-14 production has varied over time, so raw radiocarbon years must be calibrated against tree ring and other records to give calendar dates.
Contamination. Modern carbon in an old sample skews results dramatically, and older carbon skews them the other way. Sample preparation is most of the work.
The reservoir effect. Marine samples draw carbon from water with a different carbon-14 concentration than the atmosphere, requiring correction.
Range limits. After around ten half-lives, so little remains that measurement becomes unreliable — for carbon-14 that is roughly 50,000 years.
The exponential gets quoted; the calibration and sample chemistry are what make a date trustworthy.
Activity and quantity are different
Two related measurements that are frequently confused.
Quantity is how many atoms remain, or their mass. Activity is how many decays occur per second, measured in becquerels.
They are proportional — activity equals the decay constant times the number of atoms — but they answer different questions. A sample of a long-lived isotope can contain an enormous number of atoms and have low activity, because each one decays rarely. A short-lived isotope has high activity from very little material.
That matters for anything about radiation exposure, which depends on activity and the type of emission rather than on mass. It is why a barely detectable mass of a short-lived isotope can be hazardous and a large mass of a long-lived one may not be.
Common mistakes to avoid
- Confusing half-life with mean lifetime, a factor of about 1.44.
- Dropping the ln(2) when converting between half-life and decay constant.
- Assuming decay reaches zero — it approaches it asymptotically.
- Applying the exponential to dating without calibration.
- Expecting half-life to change with temperature or chemical state.
How to do it with Radioactive Decay Calculator
The Radioactive Decay Calculator solves in any direction.
- Enter the initial quantity and the half-life in matching units.
- Enter elapsed time, or solve for time from a measured remaining fraction.
- Read the decay constant and mean lifetime alongside, since literature uses both.
- For dating work, treat the result as radiocarbon years requiring calibration.
Other physics tools are in the tools directory.
Frequently asked questions
Why is mean lifetime longer than half-life?
Because the distribution has a long tail. Half the atoms decay within one half-life and the survivors keep decaying over a much longer period, pulling the average up by a factor of about 1.44.
Does half-life change with conditions?
For practical purposes no — it is a nuclear property unaffected by temperature, pressure or chemistry. A few electron-capture decay modes show tiny chemical effects, which are negligible in ordinary use.
Is this how carbon dating works?
It is the arithmetic behind it. Real dating also requires calibration for atmospheric variation, contamination control and reservoir corrections, all of which matter more than the exponential.
Final thought
Note whether a source is quoting half-life or mean lifetime. They differ by 44% and both appear in the literature without always being labelled.