Half Life Calculator

May 11, 2026

Whether you are dating an archeological find with carbon‑14, calculating how long a radioactive tracer stays in a patient’s body, or determining medication dosage intervals – the concept of half‑life lies at the core. Without it, predictions of residual activity or required decay time become guesswork.

Our half‑life calculator instantly solves the exponential decay equation for any unknown variable. No manual logarithms, no unit conversions – just enter the values you have and get the result.

Half‑Life Calculator Overview

The calculator operates with four quantities: initial amount, remaining amount, elapsed time, and half‑life. Provide any three and it computes the fourth. This covers all practical scenarios:

You know the isotope’s half‑life and how long a sample has been decaying. Find the fraction left.
You have measurements of initial and current amounts plus the time between them. Find the half‑life.
You need to know how long it takes for a substance to drop below a safety threshold. Enter the half‑life and the allowed residue.

Try an Example

Formula Reference

N(t) = N₀ × (½)^(t / T½): Remaining amount after elapsed time
t = T½ × ln(N₀ / N(t)) / ln(2): Elapsed time from initial and remaining amounts
T½ = t × ln(2) / ln(N₀ / N(t)): Half-life from experimental data
N₀ = N(t) / (½)^(t / T½): Initial amount from remaining amount
λ = ln(2) / T½ ≈ 0.6931 / T½: Decay constant
τ = T½ / ln(2) ≈ 1.443 × T½: Mean lifetime

Behind the tool is the standard exponential decay law, which applies equally to radioactive nuclei, chemical kinetics, and pharmacokinetics. The calculations ignore second‑order effects like chain decays or saturation – so results are exact for isolated first‑order processes.

How to Use the Calculator

If you want the remaining quantity after a known time: input the initial mass or activity, the half‑life, and the elapsed time. The tool applies the formula N = N₀ × (1/2)^(t/T₁/₂) and returns the remaining amount in the same units.

If you want to determine half‑life from experimental data: supply the starting amount, the amount measured after a certain time, and that time interval. The calculator solves for T₁/₂ using logarithms, eliminating error‑prone manual work.

If you want to find the time needed to reach a target level: enter the initial amount, the desired residual, and the half‑life. It outputs the time, which is especially useful for safety planning – for instance, how long before I‑131 thyroid treatment waste can be handled without extra shielding.

All fields accept decimal and scientific notation (e.g., 2.5e‑3 for 0.0025), and the time units are fully flexible: seconds, minutes, days, or years.

Formula for Half‑Life Calculation

The underlying equation is:

N(t) = N₀ × (½)⁽ᵗ⁄T½⁾

Where:

N(t) – quantity remaining after time t
N₀ – initial quantity
T½ – half‑life (in the same time units as t)
t – elapsed time

Often the decay is expressed through the decay constant λ (lambda). The relationship is:

T½ = ln(2) / λ ≈ 0.693147 / λ

And the exponential decay form becomes:

N(t) = N₀ × e⁻λᵗ

Both representations are equivalent; the calculator uses the first form internally because it aligns more directly with the half‑life concept. When you provide the decay constant instead of half‑life, simply divide 0.6931 by your λ value and use the result as T½.

Example Calculations

Example 1: Carbon‑14 dating.
A wooden artifact originally contained 1,000 mg of carbon‑14. Today it contains 250 mg. The half‑life of ¹⁴C is 5,730 years.
How old is the artifact?
Two half‑lives (250 mg = 1,000 mg × ¼) must have passed, so elapsed time = 2 × 5,730 = 11,460 years. The calculator confirms this instantly.

Example 2: Medical isotope decay.
Technetium‑99m (T½ = 6.01 hours) is prepared from a molybdenum‑99 generator. A 50‑mg sample is eluted at 08:00.
How much remains at 14:00 the same day?
t = 6 hours. N(t) = 50 × (½)^(6/6.01) ≈ 50 × 0.501 ≈ 25.0 mg. The result approximates 25 mg, but the calculator gives the exact 25.05 mg – showing that even a small deviation from exactly one half‑life is handled precisely.

Example 3: Determining half‑life from lab data.
A researcher measures 100 g of a substance at time zero and 35 g after 12 days.
Entering N₀ = 100, N(t) = 35, t = 12 days, the calculator solves for T½. Using the formula: T½ = t × (ln(2) / ln(N₀/N(t))) = 12 × 0.6931 / ln(100/35) ≈ 12 × 0.6931 / 1.0498 ≈ 7.92 days. The half‑life is about 7.9 days.

Applications Beyond Radioactivity

Half‑life mathematics is not exclusive to nuclear physics. Use the same calculator for:

Pharmacokinetics: how long a drug stays in the bloodstream. The biological half‑life of ibuprofen is roughly 2 hours; the calculator tells you that after 6 hours, only 12.5% of the peak concentration remains.
Chemical kinetics: first‑order reaction half‑life. Hydrogen peroxide decomposition, for example, follows this pattern.
Environmental science: pollutant degradation in soil or water, if it follows first‑order kinetics.
Finance: the time required for an investment to halve in real value under constant inflation (a negative growth‑rate scenario).

In each case, replace “amount” with the relevant quantity – concentration, activity, pressure – and the same exponential logic applies.

This calculator is provided for educational and estimation purposes; for safety‑critical or medical decisions, always validate with professional instruments and official guidelines.

Frequently Asked Questions

What is half-life?

Half-life is the time required for a quantity to reduce to half its initial value. It applies to radioactive decay, chemical reactions, drug clearance, and any process exhibiting exponential decay. The term is most commonly associated with unstable atomic nuclei, but the underlying mathematics works identically for all first-order processes.

How do you calculate half-life from the decay constant?

Half-life (T₁/₂) and the decay constant (λ) are inversely related: T₁/₂ = ln(2) / λ ≈ 0.6931 / λ. If you know λ, simply divide 0.6931 by it. Conversely, λ = 0.6931 / T₁/₂. This formula derives directly from the exponential decay equation N(t) = N₀e⁻λᵗ.

Can half-life change over time?

For a true first‑order process, the half‑life is constant and does not depend on the initial amount. In nuclear physics, each radioactive isotope has a fixed, unchanging half‑life. However, in some biological or complex chemical systems, clearance may deviate from simple exponential behavior, making apparent half‑life vary.

What is the difference between half-life and mean lifetime?

Mean lifetime (τ) is the average time a particle exists before decaying. It equals 1/λ, whereas half‑life equals ln(2)/λ. Numerically, τ ≈ 1.44 × half‑life. Both describe the same decay process, but half‑life is more intuitive for the public, while mean lifetime is preferred in particle physics.

What are some common half-life values?

Examples: Carbon‑14 – 5,730 years, Uranium‑238 – 4.468 billion years, Iodine‑131 – 8.02 days, Radon‑222 – 3.82 days, Technetium‑99m – 6.01 hours. In medicine, caffeine has a biological half‑life of about 3–5 hours in adults. Values vary widely across isotopes and substances.

Is the calculator accurate for very large or very small numbers?

Yes, the calculator handles values from picoseconds to billions of years and masses from micrograms to tons. It uses double‑precision arithmetic and the same exponential formulas that govern actual nuclear decay. As of 2026, results are reliable for educational and professional estimation purposes.