Half Life Calculator

Imagine pouring sand through an hourglass where each grain has an equal, random chance of falling every second. You cannot predict when any single grain will fall, but you can predict with precision that exactly half the grains remaining at any moment will have fallen after one fixed interval — the half-life. This probabilistic symmetry is what makes the exponential decay law both elegant and exact.

The mathematics works because each atom or molecule in a decaying substance has no memory of how long it has existed. A Carbon-14 atom that has survived for 10,000 years has exactly the same probability of decaying in the next second as a brand-new Carbon-14 atom. This property, called memorylessness, is what forces the decay curve to be exponential rather than linear. If decay were linear, older atoms would decay faster — but they do not.

The decay constant lambda (the probability of decay per unit time) is directly related to half-life by the equation lambda = ln(2) / t½. This constant appears naturally in the exponent of the decay equation: N(t) = N0 x e^(-lambda x t). The version using 0.5^(t/t½) is mathematically identical but easier to interpret — every full half-life period, the exponent adds another factor of one-half.

Use this calculator any time you are working with a process that follows first-order exponential decay: radioactive isotopes, drug pharmacokinetics, biological elimination, fluorescence decay, or capacitor discharge in RC circuits. The formula applies whenever the rate of decrease is proportional to the current amount rather than a fixed absolute rate.

This calculator is appropriate for single-compartment pharmacokinetic models — where a drug distributes rapidly and uniformly and clears by a single elimination pathway. Many common drugs approximate this well enough for rough planning. It is not appropriate for two-compartment models where the drug redistributes between blood and tissue over time, producing a biphasic decay curve.

Do not use this calculator for radioactive equilibria, where a parent isotope decays into a radioactive daughter isotope that in turn decays further. Uranium-238 decaying through a long chain of intermediaries requires a system of coupled differential equations, not a single half-life calculation. Similarly, do not use it to model processes with a production term — situations where new material is being continuously generated while simultaneously decaying.

The most common mistake is mixing time units between the half-life and elapsed time. Using a half-life in years alongside elapsed time in days gives a wildly wrong answer. The formula only works when both values use identical units. If the half-life of Iodine-131 is 8.02 days, elapsed time must also be in days — not hours, not weeks.

A second frequent error is confusing half-life with mean lifetime (also called the average lifetime or tau). The mean lifetime is approximately 1.443 times the half-life. After one mean lifetime, about 36.8% of the substance remains — not 50%. The two quantities measure different things. Half-life is always shorter than mean lifetime for the same substance.

A third mistake is applying the half-life formula to processes that are not truly exponential. Biological clearance of a drug follows first-order kinetics only when the drug concentration stays below the saturation threshold of metabolic enzymes. At high concentrations some drugs switch to zero-order kinetics — constant elimination rate regardless of concentration. Ethanol is the classic example: the body eliminates it at a roughly fixed rate per hour, not a fixed fraction per hour.

The core equation is N(t) = N0 x (1/2)^(t/t½). Breaking it down: N0 is the starting quantity, t is elapsed time, t½ is the half-life, and N(t) is the remaining quantity at time t. The exponent t/t½ tells you how many complete half-life periods have elapsed — it does not need to be a whole number.

To solve for elapsed time given a remaining quantity, rearrange the equation: t = t½ x log(N/N0) / log(0.5). The ratio N/N0 must be between 0 and 1 (exclusive) for the result to be a positive time. Using natural log (ln) gives the same answer: t = -t½ x ln(N/N0) / ln(2). Both forms are algebraically identical.

The percent remaining at any point is simply (N/N0) x 100. After one half-life: 50%. After two: 25%. After three: 12.5%. After ten half-lives: 0.098% — less than one-tenth of a percent. After 20 half-lives: roughly 0.0001%. For practical purposes most materials are considered fully decayed after 5 to 10 half-lives, depending on the sensitivity of the measurement being made.

The half-life formula assumes the decay constant is truly constant — which holds for nuclear decay to extraordinary precision but breaks down for biological and chemical processes under changing conditions. Temperature shifts enzymatic activity, pH changes reaction rates, and genetic variation alters metabolic enzyme expression. A drug half-life measured in healthy 25-year-olds may be two to three times longer in elderly patients with reduced renal clearance. The formula gives a mathematically exact answer from your inputs, but the inputs themselves carry the uncertainty.