Half-Life Calculator

Anything that decays at a rate proportional to how much is left can be described with a single number: its half-life. Tritium in groundwater, a drug clearing from the bloodstream, a pollutant breaking down in soil — they all follow the same curve, and the half-life says how fast.

This calculator works that curve in both directions. Give it a starting amount, a half-life and a span of time, and it returns what is left. Give it a pair of measured amounts and the time between them, and it returns the half-life that fits. It also carries the two other ways of writing the same rate — the decay constant $\lambda$ and the mean lifetime $\tau$ — so you can move between a handbook value, a measurement and a rate constant without leaving the page.

The definition behind all of it is the one the US EPA gives for radioactive half-life: the time required for half of the radioactive atoms present to decay. Half of what is there now, not half of what you started with — which is why a substance's half-life does not depend on how much of it the sample holds.

Worked example: two half-lives of tritium

Tritium has a half-life of 12.3 years, so a 24.6-year span covers exactly two of them. Start with 8 grams of tritiated water, give the half-life and the span in years, and let the remaining amount be the answer:

One equation carries the whole page. The amount left is the starting amount scaled by a power of two:

Written with the rate instead of the half-life, the same curve is an exponential in $\lambda$, which is how it usually appears in physics and kinetics:

The constant is a probability, not a countdown

$\lambda$ is the rate at which any one nucleus is expected to decay, so a sample of $N$ nuclei produces activity $A = \lambda N$. That is why the same number appears both in counting experiments and, for a first-order chemical reaction, as the rate constant $k = \ln 2 / t_{1/2}$; the rate constant calculator covers reaction orders beyond the first.

Every direction the page solves

• Forward: a starting amount, a half-life and a span give the amount left.
• Back to the starting amount: $N(0) = N(t)\,2^{\,t/T}$ — what a sample must have contained to read the way it does today.
• Back to the half-life: $T = \dfrac{t\,\ln 2}{\ln\!\left(N(0)/N(t)\right)}$ — the measurement direction, and the one a lab uses to characterise an unknown nuclide.
• Back to the elapsed time: $t = \dfrac{T\,\ln\!\left(N(0)/N(t)\right)}{\ln 2}$ — how long the sample has been decaying.
• Between the three rate descriptions: half-life, decay constant and mean lifetime are filled in from each other in both directions.

Half-lives span fourteen orders of magnitude

The scale of $T$ is what makes the seven time scales useful. Tritium sits at 12.3 years, while uranium-238 is measured in billions of them — the EPA lists 4.47 billion years for U-238, 700 million for U-235 and 244,000 for U-234. Enter either value in the half-life box, and give the elapsed time on the same scale or on another one: the calculator normalises both before it evaluates the power, so mixing a half-life in years with a span in seconds is safe.

Units and scales

The time boxes share one set of seven scales with exact definitions: 1 minute = 60 s, 1 hour = 3,600 s, 1 day = 86,400 s, 1 week = 604,800 s, 1 month = 2,629,800 s and 1 year = 31,557,600 s, where the year is the Julian year of 365.25 days and a month is a twelfth of it. The decay constant is offered on the reciprocal of those scales, because a rate is per unit time: a half-life of 12.3 years is $\lambda = 0.05635\ \mathrm{yr^{-1}} = 1.786 \times 10^{-9}\ \mathrm{s^{-1}}$, the same physical rate written two ways.

Three situations that use different parts of the page. Every result is the value the calculator returns for those inputs.

Measuring a half-life from two readings

A lab counts a tritiated water sample twice: 8.0 units of activity on the first day, 6.0 units five years later. The half-life is the unknown, so leave it empty:

The answer comes back as 380,177,695 seconds when the half-life box is on the seconds scale, which is the same 12.05 years. It also lands about 2% below tritium's accepted 12.3 years — close enough to confirm the sample, not precise enough to redefine the nuclide, which is exactly how a two-point measurement behaves. With the half-life known, the page reports the matching rate on the same screen: $\lambda = 1.82 \times 10^{-9}\ \mathrm{s^{-1}}$ and $\tau = 17.38\ \mathrm{years}$.

Reading the rate instead of the half-life

Enter tritium's 12.3-year half-life on its own and the two rate boxes fill themselves in:

The mean lifetime is longer than the half-life — always, by a factor of $1/\ln 2 \approx 1.443$. Half of a sample is gone by 12.3 years, but the atoms that survive keep surviving, and averaging over that long tail pushes the mean out to 17.75 years. Both numbers describe the same sample; use the half-life for "when is half of it gone" and the mean lifetime when a model expects an exponential time constant.

A half-life longer than the age of the Earth

Uranium-238 has a half-life of 4.47 billion years. Give it 1 kilogram and a span of one million years, and almost nothing appears to happen:

About 0.16 grams of that kilogram has decayed. The elapsed time is only $0.00022$ half-lives, so the ratio barely moves — which is why long-lived nuclides survive in the crust at all, and why a meaningful measurement on one needs a span comparable to its half-life rather than a human one. The same page reports its rate as $\lambda = 4.9 \times 10^{-18}\ \mathrm{s^{-1}}$, a reminder that a small rate constant and a long half-life are two readings of the same fact.