Half-Life Calculator

About the Half-Life Calculator

A half-life calculator computes the remaining quantity of any substance or entity after radioactive decay, drug metabolism, or any first-order decay process, given the initial amount, the half-life period, and the elapsed time. The concept of half-life — the time required for exactly half of a substance to disappear — appears across nuclear physics (radioactive isotopes), pharmacology (drug elimination from the body), environmental science (pollutant degradation), and engineering (reliability and aging). The mathematical law is universal: N(t) = N₀ × (½)^(t/t½). Our calculator accepts any unit of time (seconds, minutes, hours, days, years), making it applicable from short-lived medical isotopes (Technetium-99m: 6 hours) to geological dating (Carbon-14: 5,730 years; Uranium-238: 4.47 billion years). Results include the remaining amount, percentage remaining, number of half-lives elapsed, the decay constant (λ = ln2/t½), and the mean lifetime (τ = 1/λ) — all the quantities needed for complete first-order kinetics analysis. Used by students in chemistry, physics, and pharmacology; nuclear engineers; environmental scientists; and medical professionals working with radiopharmaceuticals.

How It Works

N(t) = N₀ × (1/2)^(t/t½). Equivalently: N(t) = N₀ × e^(-λt), where λ = ln(2) / t½ is the decay constant. Number of half-lives elapsed = t / t½. Percentage remaining = (N(t) / N₀) × 100 = (1/2)^(t/t½) × 100. Mean lifetime τ = 1/λ = t½ / ln(2) ≈ 1.4427 × t½. Example 1 (Carbon-14 dating): N₀ = 100%, t½ = 5,730 years, t = 11,460 years. t/t½ = 2 half-lives. N(t) = 100 × (1/2)² = 25% remaining. Example 2 (Ibuprofen): N₀ = 400mg, t½ = 2 hours, t = 8 hours. t/t½ = 4. N(t) = 400 × (1/2)^4 = 400/16 = 25mg. 93.75% eliminated. Example 3 (Technetium-99m medical imaging): t½ = 6 hours, 12-hour scan-to-discharge delay. t/t½ = 2. N = (1/4) of original dose remains (25%).

Tips & Best Practices

• ✓Radiocarbon dating practical limits: below ~1% remaining (>6.6 half-lives, ~37,800 years), measurement uncertainty in C-14 content dominates and dating reliability decreases. Accelerator mass spectrometry (AMS) can date material with as little as 0.001% C-14 (up to ~55,000 years). Older samples require different isotopes (potassium-argon, uranium-lead).
• ✓Drug accumulation at steady state: when a drug is taken at regular intervals, it accumulates until a steady state is reached at approximately 4-5 half-lives. At steady state, the amount eliminated per dose interval equals the dose taken. Warfarin (t½ = 36-42 hours) reaches steady state in 7-10 days — explaining why INR monitoring is important during the first 2 weeks.
• ✓Uranium-238 decay chain: U-238 decays through 14 intermediate isotopes before reaching stable Lead-206, with a series of different half-lives. The longest step (U-238 → Th-234) has t½ = 4.47 billion years — roughly the age of the solar system. This extreme half-life makes uranium useful for dating ancient geological formations.

Who Uses This Calculator

Chemistry and physics students working radioactive decay problems for coursework. Pharmacology students and healthcare professionals calculating drug clearance times and dosing intervals. Nuclear medicine professionals calculating residual activity of radiopharmaceuticals in patients. Environmental scientists calculating the time for radioactive contamination or chemical pollutant concentrations to fall to safe levels.