Science

Half-Life Calculator

Calculate remaining amount after radioactive or exponential decay. Solve for remaining quantity or time to reach a target amount.

Key Formula

N(t) = N₀ × (1/2)^(t / t½)  |  t = t½ × ln(N₀/N) / ln(2)

Calculate

Enter values below and results update instantly.

Remaining
250
Decayed
750
% Remaining
25%
Half-Lives Passed
2
Decay Constant (λ)
0.00012097

Decay Curve

Time (years)Amount017,19001,000

About This Tool

The Half-Life Calculator computes radioactive and exponential decay using the standard half-life formula N(t) = N₀ × (1/2)^(t/t½). Enter an initial amount and half-life period to find remaining quantity after a given time, or enter a target amount to find how long until that level is reached. This tool is essential for nuclear physics, pharmacology, chemistry, archaeology (radiocarbon dating), and any field dealing with exponential decay processes.

The Mathematics of Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. The differential equation dN/dt = -λN has the solution N(t) = N₀e^(-λt), where λ is the decay constant. The half-life relates to the decay constant by t½ = ln(2)/λ ≈ 0.693/λ. The mean lifetime (average time before a single atom decays) is τ = 1/λ = t½/ln(2). After one half-life, 50% remains; after two, 25%; after three, 12.5%; after ten half-lives, only about 0.1% of the original amount remains.

Radioactive Decay

Radioactive decay is a quantum mechanical process where unstable atomic nuclei release energy by emitting radiation (alpha particles, beta particles, or gamma rays). Each radioactive isotope has a characteristic, immutable half-life ranging from fractions of a second to billions of years. The decay of individual atoms is random and unpredictable, but for large numbers of atoms, the statistical behavior follows the exponential decay law precisely. This is because each atom has an independent, constant probability of decaying per unit time.

Radiocarbon Dating

Carbon-14 dating, developed by Willard Libby in 1949 (for which he received the Nobel Prize), uses the 5,730-year half-life of Carbon-14 to date organic materials. Living organisms maintain a constant ratio of C-14 to C-12 through metabolic exchange with the atmosphere. After death, C-14 decays without replenishment. By measuring the remaining fraction, the time since death can be calculated. The practical limit is about 50,000 years (roughly 9 half-lives), beyond which the remaining C-14 is too small to measure reliably.

Pharmacological Half-Life

In medicine, the half-life of a drug determines its dosing schedule and duration of effect. After administration, the body eliminates the drug through metabolism and excretion at a rate that (for many drugs) follows first-order kinetics, making the half-life concept directly applicable. A drug is considered effectively eliminated after 4-5 half-lives (6.25% to 3.125% remaining). Drugs with short half-lives (2-4 hours) like ibuprofen need frequent dosing, while drugs with long half-lives (days) like fluoxetine can be taken once daily. Drug interactions can alter half-life by affecting liver enzymes.

Geological and Cosmological Applications

Long-lived radioactive isotopes serve as geological clocks. Uranium-238 (half-life 4.47 billion years) decays to Lead-206, providing a reliable method for dating rocks billions of years old. Potassium-40 (1.25 billion years) decays to Argon-40, useful for dating volcanic rocks. These radiometric dating methods have established the age of the Earth at approximately 4.54 billion years and the age of the oldest known minerals at 4.4 billion years. The consistency across different isotope systems provides strong confidence in these dates.

Frequently Asked Questions

What is half-life?
Half-life is the time required for a quantity to reduce to half of its initial value. It is most commonly associated with radioactive decay, where unstable atomic nuclei transform into more stable forms by emitting radiation. However, the concept applies to any exponential decay process, including drug elimination from the body (pharmacological half-life), chemical reaction rates, and even the decay of internet memes in public attention. The half-life is a constant for any given radioactive isotope regardless of the initial amount, temperature, or pressure.
What is the half-life formula?
The fundamental decay formula is N(t) = N_0 x (1/2)^(t/t_half), where N(t) is the remaining amount at time t, N_0 is the initial amount, and t_half is the half-life. This can also be written as N(t) = N_0 x e^(-lambda x t), where lambda = ln(2)/t_half is the decay constant. To solve for time: t = t_half x ln(N_0/N) / ln(2). The mean lifetime (average time before decay) is tau = t_half / ln(2), which is approximately 1.443 times the half-life.
What is the half-life of Carbon-14?
Carbon-14 has a half-life of 5,730 years, which makes it ideal for radiocarbon dating of organic materials up to about 50,000 years old. Living organisms constantly replenish their Carbon-14 through metabolism, maintaining a constant ratio with Carbon-12. After death, the Carbon-14 decays without replacement. By measuring the remaining fraction of Carbon-14 in a sample, scientists can determine when the organism died. After about 10 half-lives (57,300 years), so little Carbon-14 remains that measurements become unreliable.
How is half-life used in medicine?
In pharmacology, half-life determines how long a drug remains active in the body and how frequently it must be administered. A drug with a 4-hour half-life drops to 25% after 8 hours and 6.25% after 12 hours. Drugs typically need 4-5 half-lives to be considered eliminated. Doctors use this to set dosing schedules: short half-life drugs need more frequent doses. In nuclear medicine, radioactive tracers with short half-lives (hours to days) are used for imaging, as they provide diagnostic information while minimizing radiation exposure.
Can half-life be changed?
For radioactive decay, the half-life is an intrinsic property of the nucleus and cannot be changed by chemical reactions, temperature, pressure, or any ordinary physical condition. However, in extreme conditions like the interior of stars or in particle accelerators, certain decay rates can be slightly altered. For chemical and biological half-lives, the rate can be influenced by temperature (Arrhenius equation), pH, catalysts, and other environmental factors. Drug half-life in the body varies with liver function, kidney function, age, and drug interactions.
What are some common radioactive half-lives?
Half-lives span an enormous range. Polonium-214 has a half-life of 164 microseconds, while Bismuth-209 has the longest measured half-life at 1.9 x 10^19 years (over a billion times the age of the universe). Common isotopes include: Iodine-131 (8 days, used in thyroid treatment), Cobalt-60 (5.27 years, used in radiation therapy), Strontium-90 (28.8 years, nuclear fallout concern), Cesium-137 (30.17 years, Chernobyl/Fukushima contaminant), Carbon-14 (5,730 years, radiocarbon dating), and Uranium-238 (4.47 billion years, geological dating).

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