Half-life is the time it takes for half of a radioactive sample to decay. What makes it useful is that it does not depend on how much you started with — one gram or one kilogram of carbon-14, half is gone after 5,730 years either way.
That constancy is what turns decay into a clock.
The two forms of the equation
The same relationship gets written two ways, and knowing why saves confusion.
Base-2 form, best for mental arithmetic:
N(t) = N₀ × (1/2)^(t / t½)Read literally: every time `t` reaches another multiple of the half-life, the exponent gains 1 and the amount halves again. The powers of one-half are easy to remember — 50%, 25%, 12.5%, 6.25%.
Base-e form, used in physics and pharmacokinetics:
N(t) = N₀ × e^(−λt) where λ = ln(2) / t½ ≈ 0.693 / t½λ (lambda) is the decay constant — the probability that any single atom decays per unit time. The two forms give identical answers; base-e is preferred where the equation needs differentiating.
To solve for elapsed time, invert the base-2 form:
t = t½ × log₂(N₀ / N)Worked example: dating a piece of charcoal
A charcoal fragment from a hearth measures 12.5 disintegrations per minute per gram of carbon. Living wood measures 15.3 dpm/g. Carbon-14's half-life is 5,730 years. How old is it?
Step 1 — set up the ratio. N₀ = 15.3, N = 12.5. Units cancel, so dpm/g is fine; you never need to convert to atoms.
Step 2 — apply the formula.
t = 5730 × log₂(15.3 / 12.5)
= 5730 × log₂(1.224)
= 5730 × 0.2916
= 1,671 yearsStep 3 — sanity-check it. 0.29 half-lives is well under one, and the sample retains 82% of its original activity. A young sample. That is consistent.
The half-life calculator does this in any direction — give it any three of N₀, N, t and t½ and it returns the fourth.
Here is the full decay curve for the same isotope:
| Half-lives | Years | Fraction left | dpm/g |
|---|---|---|---|
| 0 | 0 | 100% | 15.30 |
| 0.29 | 1,671 | 81.7% | 12.50 |
| 1 | 5,730 | 50% | 7.65 |
| 2 | 11,460 | 25% | 3.83 |
| 3 | 17,190 | 12.5% | 1.91 |
| 5 | 28,650 | 3.13% | 0.48 |
| 10 | 57,300 | 0.098% | 0.015 |
That last row is why radiocarbon dating runs out around 50,000 years. After ten half-lives there is so little carbon-14 left that the measurement is swamped by background, and the error bars swallow the answer.
Why a raw radiocarbon date is not a calendar date
The 1,671 years above is a radiocarbon year, and archaeologists never publish that number directly.
The method assumes atmospheric carbon-14 has been constant. It has not. Solar activity, geomagnetic changes, and — dramatically — the industrial revolution and atmospheric nuclear testing have all shifted it. Burning fossil fuels released carbon with no C-14 at all, diluting the atmosphere; bomb testing in the 1950s and 60s nearly doubled it.
Real dates are corrected against a calibration curve (IntCal20) built from tree rings, corals and cave deposits, where the true age is known independently. The correction can reach several hundred years in some periods.
So: this calculation gives you a correct radiocarbon age. Converting that to a calendar date needs the curve.
Common mistakes
- Mixing units between half-life and elapsed time. Iodine-131's half-life is 8.02 days. Entering an elapsed time of 30 while thinking in years is wrong by a factor of 365.
- Confusing half-life with mean lifetime. Mean lifetime τ = 1/λ = t½ / ln(2) ≈ 1.44 × t½. After one mean lifetime, 36.8% remains, not 50%.
- Entering N larger than N₀. Decay only decreases the amount. If your measured value exceeds the baseline, the fields are swapped or the baseline is wrong.
- Treating a decay chain as one isotope. Uranium-238 passes through fourteen radioactive intermediates before reaching stable lead-206. A single-isotope calculation describes the first step only.
- Applying physical half-life to a drug. A radiopharmaceutical is also cleared biologically. The effective half-life is shorter than either process: 1/t_eff = 1/t_physical + 1/t_biological.
- Expecting exactness for few atoms. Decay is statistical. The smooth curve is an excellent average for the ~10²⁰ atoms in a lab sample and meaningless for a handful.
Half-lives worth knowing
| Isotope | Half-life | Used for |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeology, up to ~50,000 years |
| Iodine-131 | 8.02 days | Thyroid treatment and imaging |
| Technetium-99m | 6.01 hours | Most common medical imaging tracer |
| Cobalt-60 | 5.27 years | Sterilisation, radiotherapy |
| Uranium-238 | 4.47 billion years | Dating rocks, Earth's age |
| Potassium-40 | 1.25 billion years | Dating volcanic rock |
Technetium-99m's six hours is deliberate: long enough to image a patient, short enough that the dose clears quickly.
Frequently asked questions
Why does half-life not depend on the starting amount?
Because decay is first-order — each atom has a fixed probability of decaying per unit time, independent of its neighbours. Double the atoms and you double the decays, so the *fraction* remaining after a given time is unchanged.
Can half-life be changed by temperature or pressure?
Essentially no. Nuclear decay is unaffected by chemical or physical conditions, with tiny exceptions for electron-capture isotopes.
What is the difference between activity and amount?
Activity is decays per second; amount is number of atoms. They are proportional (A = λN), so the ratio N/N₀ equals A/A₀ — which is why you can use dpm directly.
How do I go from a mass to a number of atoms?
Divide the mass by the molar mass, then multiply by Avogadro's number. The [molar mass calculator](/molar-mass-calculator) handles the first step for any formula.
Does this work for non-radioactive decay?
The same maths describes any first-order process — drug clearance, capacitor discharge, cooling. It does not describe zero-order or second-order kinetics.
Related reading
For lab work, the molar mass calculator converts sample masses to moles. The exponential here is the same one that governs compound growth with the sign flipped — the compound interest calculator runs the identical curve upward.