Half-Life Decay Calculator
Half-life decay calculator. Find the remaining amount after a time, or the half-life from an observed decay, using exponential decay. Educational only.
Half-Life Decay Calculator
Use the same time unit for half-life and elapsed time (the result carries that unit).
How to use the half-life calculator
Choose what to solve for
Pick "Remaining amount" to find how much is left after a time, or "Half-life" to work out the half-life from an observed before-and-after amount.
Enter the starting amount
Type the initial quantity N₀ — it can be a mass, a count, an activity, or a concentration; only the ratio matters.
Enter the half-life or remaining amount
For remaining-amount mode, give the half-life and the elapsed time in the same unit. For half-life mode, give the remaining amount and the elapsed time.
Read the result
You get the answer plus the number of half-lives elapsed, the fraction remaining, the decay constant, and the mean lifetime. Use consistent time units throughout.
Half-life and exponential decay
A constant proportion, not a constant amount
Many natural processes decay exponentially, meaning a fixed proportion of what remains disappears in each equal slice of time, rather than a fixed amount. The half-life is the time it takes for the quantity to fall to half its value, and because the proportion is constant, every half-life halves the amount again: after one half-life, half remains; after two, a quarter; after three, an eighth; and so on. This is the signature of first-order kinetics, and it shows up across science — the radioactive decay of unstable isotopes, the elimination of a drug from the bloodstream, the discharge of a capacitor, the cooling of an object, and the decline of a chemical reactant in a first-order reaction. The governing equation is N = N₀ × (1/2)^(t / half-life): the amount remaining equals the starting amount multiplied by one-half raised to the power of the number of half-lives that have passed. Equivalently, decay can be written with the natural exponential, N = N₀ × e^(−λt), where λ is the decay constant, related to the half-life by λ = ln2 / half-life ≈ 0.693 / half-life.
That relationship means the half-life and the decay constant are just two ways of describing the same rate. A third, the mean lifetime τ, is the average time a particle survives and equals 1/λ, or the half-life divided by 0.693 — so the mean lifetime is always a bit longer than the half-life. Running the equation backward lets you find a half-life from measurements: if you know the starting amount, the amount remaining, and the elapsed time, the half-life is the elapsed time times ln2 divided by the natural log of the ratio of starting to remaining amount. This is exactly how an unknown isotope's half-life, or a drug's elimination half-life, is determined experimentally.
"Exponential decay removes a constant fraction, not a constant amount — so the substance never quite vanishes, it just halves and halves again. One half-life, and the clock resets on whatever is left."
Reading half-lives correctly
A few ideas prevent common misunderstandings. Because decay is exponential, the quantity approaches zero but never mathematically reaches it; after ten half-lives only about a thousandth remains, which is why radioactive waste with a long half-life stays hazardous for a very long time. The half-life is independent of the starting amount — a gram and a kilogram of the same isotope have the same half-life — and, for radioactive decay, it is essentially unaffected by temperature, pressure, or chemical state, which is what makes radiometric dating reliable. In pharmacology, the elimination half-life sets how long a drug lingers and underpins dosing intervals; it takes about four to five half-lives for a drug to be effectively cleared, or to reach steady state on repeated dosing. The model assumes a single, constant decay rate (true first-order behaviour); processes with multiple pathways or changing rates need more elaborate treatment. This calculator handles the standard exponential-decay relationships in both directions and reports the related quantities. Use it to learn and to estimate; for dosing, dating, or safety decisions, rely on validated methods and professional guidance.
10 Facts About Half-Life
Half-life = time for a quantity to halve.
N = N₀ · (1/2)^(t/half-life).
Decay removes a constant fraction, not amount.
Decay constant λ = ln2 / half-life ≈ 0.693/HL.
Mean lifetime τ = 1/λ, longer than the half-life.
After 10 half-lives, ~0.1% remains.
Half-life is independent of the starting amount.
Radioactive half-lives barely change with temperature.
A drug clears in about 4–5 half-lives.
It underpins radiometric dating like carbon-14.
Frequently asked questions
A half-life is the time it takes for a quantity that decays exponentially to fall to half its starting value. Because a constant fraction (not a constant amount) is lost in each equal time interval, the quantity halves again every half-life: after one half-life half remains, after two a quarter, after three an eighth, and so on. It applies to radioactive decay, drug elimination, capacitor discharge, and any first-order process.
Use N = N₀ × (1/2)^(t / half-life). Divide the elapsed time by the half-life to get the number of half-lives, then multiply the starting amount by one-half raised to that power. For example, after 30 units of time with a 10-unit half-life, that's 3 half-lives, so 100 becomes 100 × (1/2)³ = 12.5. The calculator does this and also reports the fraction remaining and the percent decayed.
If you know the starting amount, the amount remaining, and the elapsed time, the half-life equals the elapsed time times ln2 divided by the natural log of the ratio of starting to remaining amount: half-life = t × ln2 / ln(N₀/N). This is how an unknown isotope's half-life or a drug's elimination half-life is determined from a decay measurement. Select "Half-life" mode in the calculator to compute it this way.
The decay constant λ is the fraction of the quantity that decays per unit time, used in the form N = N₀e^(−λt). It's directly related to the half-life: λ = ln2 / half-life, about 0.693 divided by the half-life. A larger decay constant means faster decay and a shorter half-life. The mean lifetime, the average time before a particle decays, is its reciprocal (1/λ) and is always a little longer than the half-life.
No. The half-life is a property of the decaying substance or process, not of how much you start with. A gram and a tonne of the same isotope both have the same half-life — they just contain proportionally different numbers of atoms decaying at the same rate. This independence is what makes the half-life such a robust, characteristic quantity and is fundamental to techniques like radiometric dating.
Because exponential decay removes a fraction of what's left each interval, the amount approaches zero but never mathematically reaches it. After 7 half-lives under 1% remains, after 10 about a thousandth, after 20 about a millionth. In practice a substance becomes negligible after several half-lives, but the mathematics has no end point. For radioactive materials with long half-lives, this is why even small residual amounts persist for very long times.
A drug's elimination half-life is how long the body takes to clear half of it, and it largely sets the dosing interval. As a rule of thumb, it takes about four to five half-lives for a drug to be effectively eliminated, and the same number to reach steady-state levels when dosing repeatedly. Drugs with short half-lives need frequent dosing; long half-lives allow once-daily or less. Actual dosing also depends on the therapeutic range, metabolism, and many clinical factors, so follow professional guidance.
For radioactive decay, essentially not at all — the nuclear decay rate is independent of temperature, pressure, and chemical state (with only tiny exceptions for certain decay modes). That invariance is what makes radiometric dating trustworthy. For chemical or biological first-order processes, however, the rate (and thus the effective half-life) can depend strongly on temperature, catalysts, and conditions. So whether a half-life is environment-independent depends on the kind of decay involved.
Use it to learn the mathematics and make estimates. Drug dosing, radiometric dating, and radiation-safety decisions involve many factors beyond the simple exponential model and must be handled with validated methods and professional expertise. This tool runs the standard half-life and exponential-decay relationships and is educational; it is not a substitute for clinical, scientific, or safety guidance.
No. The values you enter are processed entirely in your browser. Nothing is sent to a server, stored, or shared, and no account is required. The calculation runs on your device only.
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