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☢️ Half-Life Calculator

Calculate the remaining quantity of a radioactive substance using the half-life formula N(t) = N₀ × (1/2)^(t/t½). Enter the initial quantity, half-life and elapsed time to get the remaining amount, decayed amount, decay constant and more.

Initial quantity (N₀)

Half-life (t½)

Elapsed time (t)

What is this tool?

Half-life is the time required for a quantity to reduce to half of its initial value. The term is most commonly used in nuclear physics and chemistry to describe the radioactive decay of unstable isotopes. Every radioactive substance decays at its own characteristic rate, governed by the exponential decay law N(t) = N₀ × (1/2)^(t/t½), where N₀ is the initial quantity, t½ is the half-life, and t is the elapsed time. The concept of half-life applies to any process that follows exponential decay. This includes radioactive isotopes like Carbon-14 used in archaeological dating, Uranium-238 used in geochronology, and medical isotopes like Iodine-131 used in thyroid treatment. In pharmacology, half-life describes how quickly a drug is eliminated from the bloodstream. In each case, after one half-life, half of the substance remains; after two half-lives, a quarter remains; after three, an eighth, and so on. The decay never truly reaches zero, but after about ten half-lives the remaining quantity is less than 0.1% of the original, which is practically undetectable. This calculator handles any consistent time unit for the half-life and elapsed time — seconds, minutes, hours, days, or years — and also computes the decay constant λ = ln(2)/t½, which is the rate at which the substance decays per unit time.

How it works

The half-life calculator uses the exponential decay formula N(t) = N₀ × (1/2)^(t/t½). First, it divides the elapsed time t by the half-life t½ to determine how many half-lives have passed. This ratio is then used as the exponent of 1/2, giving the fraction of the substance remaining. Multiplying by the initial quantity N₀ yields the remaining amount. The decay constant λ = ln(2)/t½ ≈ 0.693/t½ is a closely related measure. It represents the probability per unit time that any single atom will decay. The two formulations are mathematically equivalent: N(t) = N₀ × e^(−λt). The calculator also derives the percentage remaining, the decayed quantity (N₀ − N(t)), and the number of completed half-lives.

Isotope	Half-Life	Use
Carbon-14 (C-14)	5,730 years	Radiocarbon dating
Uranium-238 (U-238)	4.47 billion years	Geological dating
Iodine-131 (I-131)	8.02 days	Thyroid therapy
Cobalt-60 (Co-60)	5.27 years	Radiotherapy
Radon-222 (Rn-222)	3.82 days	Environmental monitoring
Technetium-99m (Tc-99m)	6.01 hours	Medical imaging
Plutonium-239 (Pu-239)	24,100 years	Nuclear reactors
Tritium (H-3)	12.32 years	Self-luminous devices

How to use

Enter the initial quantity N₀ (must be greater than 0).
Enter the half-life t½ and select its time unit.
Enter the elapsed time t and select its time unit.
Click Calculate to see the remaining quantity and decay metrics.
Review the decay constant, percentage remaining, and number of half-lives.

Frequently Asked Questions

What is the half-life formula?

The half-life formula is N(t) = N₀ × (1/2)^(t/t½), where N₀ is the initial quantity, t½ is the half-life, and t is the elapsed time. It tells you how much of a substance remains after a given time period.

How is the decay constant λ related to half-life?

The decay constant λ = ln(2)/t½ ≈ 0.693/t½. It represents the probability of decay per unit time and is used in the alternative exponential form N(t) = N₀ × e^(−λt). Both formulations give identical results.

Can I use different time units for half-life and elapsed time?

The calculator assumes both values use the same time unit. If your half-life is in years and elapsed time is in days, convert one so they match (e.g., convert days to years by dividing by 365.25) before entering the values.

After how many half-lives is a substance considered gone?

Mathematically, the substance never truly reaches zero. However, after 7 half-lives less than 1% remains, and after 10 half-lives less than 0.1% remains, which is practically undetectable for most purposes.

What is carbon dating and how does it use half-life?

Carbon dating measures the remaining amount of the radioactive isotope Carbon-14 (half-life ≈ 5,730 years) in organic material. By comparing the C-14/C-12 ratio to the known atmospheric ratio, scientists can estimate how long ago the organism died.

Does this work for drug elimination in pharmacology?

Yes. In pharmacology, the half-life of a drug is the time it takes for the concentration in the bloodstream to reduce by half. The same formula applies to estimate how much drug remains after a given number of hours or days.

Tips & Advice

When working with half-life problems, always make sure the units of elapsed time match the units of the half-life before dividing. If the half-life is given in years but the elapsed time is in days, convert one of them first. Remember that after n half-lives, the fraction remaining is (1/2)^n — so after 5 half-lives only about 3.125% remains. The decay constant λ = ln(2)/t½ gives the decay rate per unit time and is useful when you need a continuous rate rather than a discrete half-life. For carbon dating, the ratio of C-14 to C-12 in the sample is compared to the atmospheric ratio, and the elapsed time is calculated from the known 5,730-year half-life. Note that half-life calculations assume a closed system with no external addition or removal of the substance.

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