Half-Life Calculator

Background.

A half-life calculator computes the remaining quantity of a substance after a specified elapsed time, given its half-life—the interval required for half of the material to decay or be eliminated. The concept is fundamental to nuclear chemistry, where it describes radioactive decay; to pharmacology, where it determines dosing intervals; and to archaeology, where carbon-14 half-life measurements date organic artifacts. The underlying mathematics is first-order exponential decay, a process in which the rate of loss is proportional to the amount present.

Students in chemistry and physics courses search for half-life tools most intensively during units on nuclear reactions and kinetics. Medical students and pharmacists use the same calculations to determine how long a drug remains at therapeutic concentration in a patient's bloodstream. Radiocarbon dating laboratories at universities and museums apply half-life calculations to samples ranging from ancient Egyptian papyri to Pleistocene mammoth bones. The 1949 development of radiocarbon dating by Willard Libby at the University of Chicago revolutionized archaeology by providing absolute dates for organic materials up to approximately 50,000 years old, and it earned Libby the 1960 Nobel Prize in Chemistry.

The mathematical model assumes that decay is a random process at the atomic level but deterministic at the macroscopic level. Each radioactive atom has a constant probability of decaying in a given time interval, independent of the presence of other atoms. This independence means that the overall decay rate is proportional to the number of atoms remaining, producing the characteristic exponential curve. The half-life is the median lifetime of a population of atoms: after one half-life, half remain; after two, one-quarter remain; after three, one-eighth remain. The formula N = N₀(1/2)^(t/t₁/₂) captures this relationship exactly.

In pharmacokinetics, half-life determines dosing frequency and accumulation. A drug with a 4-hour half-life administered every 4 hours will approach steady-state concentration after approximately five half-lives (20 hours), at which point the amount eliminated between doses equals the amount administered. The U.S. Food and Drug Administration requires half-life data for every new drug application, and physicians use it to adjust doses for patients with renal or hepatic impairment.

What is half-life calculator?

Half-life is the time required for half of a quantity of substance to decay, be eliminated, or otherwise transform through a first-order process. It is denoted t₁/₂ and is independent of the initial amount: one gram and one kilogram of the same isotope both decay to half their mass in the same interval. The half-life is related to the decay constant λ by the equation t₁/₂ = ln(2) / λ. The concept applies to radioactive isotopes, drug concentrations in blood plasma, chemical reaction intermediates, and any system where the rate of change is proportional to the current state. Different substances have vastly different half-lives: oxygen-15 decays in 122 seconds, carbon-14 in 5,730 years, and uranium-238 in 4.468 billion years. In pharmacology, biological half-life refers to the time for plasma concentration to fall by half, which may differ from the chemical half-life if metabolism or excretion accelerates elimination. Half-life is inversely related to the decay constant and directly related to the mean lifetime τ by t₁/₂ = τ × ln(2). All three quantities describe the same exponential decay process from different perspectives. Because half-life is a statistical property of large ensembles, individual atoms may decay immediately or survive many half-lives, but the median time for half the population to decay remains constant.

How to use this calculator.

1. 01Enter the half-life of the substance in your chosen time unit (seconds, minutes, hours, days, or years).
2. 02Enter the elapsed time since the initial measurement, using the same unit.
3. 03Enter the initial amount of the substance. This can be mass, volume, concentration, or any unit.
4. 04Select the time unit from the dropdown to ensure consistency.
5. 05Review the primary output: the remaining amount after the elapsed time.
6. 06Check secondary outputs: decayed amount, percent remaining, and number of half-lives elapsed.
7. 07Use the results for lab reports, dosing schedules, or radiometric dating calculations.

The formula.

The half-life formula derives from the first-order differential equation that governs exponential decay. The rate of change of the quantity N with respect to time is proportional to N itself: dN/dt = −λN, where λ is the decay constant with units of inverse time. The negative sign indicates that N decreases over time. Separating variables and integrating from N₀ at t = 0 to N at time t gives ln(N/N₀) = −λt, which rearranges to N = N₀e^(−λt). The half-life is defined as the time at which N = N₀/2. Substituting into the integrated equation: N₀/2 = N₀e^(−λt₁/₂). Dividing by N₀ and taking the natural logarithm yields ln(1/2) = −λt₁/₂. Since ln(1/2) = −ln(2), the equation simplifies to t₁/₂ = ln(2)/λ. This shows that half-life and decay constant are inversely proportional; a large λ means rapid decay and a short half-life. The alternative form N = N₀(1/2)^(t/t₁/₂) is algebraically equivalent to the exponential form but more intuitive for counting half-lives. If t = t₁/₂, the exponent is 1 and N = N₀/2. If t = 2t₁/₂, the exponent is 2 and N = N₀/4. This discrete stepping is easier to explain to students than the continuous exponential, though both describe the same physical process. The percent remaining is simply (N/N₀) × 100 = 100 × (1/2)^(t/t₁/₂). Dimensional analysis confirms consistency: λ has units of per time (e.g., yr⁻¹), so ln(2)/λ has units of time. The exponent t/t₁/₂ is dimensionless, as required for any exponent. The initial amount N₀ can be in grams, moles, becquerels, or any other unit; it cancels in the ratio N/N₀, so the percent remaining is independent of the unit chosen. From a probabilistic perspective, λ is the probability per unit time that a given atom will decay. For a large population, the law of large numbers ensures that the fraction decaying in any interval is predictable, even though individual decay events are random.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Godwin, H. (1962). "Half-life of radiocarbon." Nature 195(4845):984.
2. [2]Libby, W.F. (1952). Radiocarbon Dating. Chicago: University of Chicago Press.
3. [3]Reimer, P.J. et al. (2020). "The IntCal20 Northern Hemisphere Radiocarbon Age Calibration Curve." Radiocarbon 62(4):725–757.
4. [4]Abramowitz, M. & Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: National Bureau of Standards.
5. [5]Rowland, M. & Tozer, T.N. (2011). Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications, 4th ed. Philadelphia: Lippincott Williams & Wilkins.
6. [6]Atkins, P. & de Paula, J. (2014). Atkins' Physical Chemistry, 10th ed. Oxford: Oxford University Press.