Half-Life Calculator
Give any three of the four decay quantities — starting amount N₀, remaining amount Nₜ, elapsed time t, and half-life t½ — and this solves the fourth, with steps. A second mode converts between half-life, mean lifetime τ, and decay constant λ.
Example: with Tool Decay solver (N₀, Nₜ, t, t½) · Solve for Elapsed time t · Initial quantity N₀ 100 · Remaining quantity Nₜ 25 · Elapsed time t 11460 · Half-life t½ 5730 · … → Elapsed time t: 11,460.
Computed by the calculator below using its default values. Change any input to see your own numbers.
Half-life in the real world
Half-life is the time for an exponentially decaying quantity to fall by half — after one half-life 50% remains, after two 25%, after three 12.5%. Standard nuclear data (IAEA/NNDC) lists carbon-14 at about 5,730 years (the basis of radiocarbon dating), technetium-99m at ~6 hours (medical imaging), iodine-131 at ~8 days (thyroid treatment), cobalt-60 at ~5.3 years, cesium-137 at ~30 years, and uranium-238 at ~4.5 billion years. The same equation governs drug elimination in pharmacology and capacitor discharge in electronics.
How it’s calculated
Nₜ = N₀(½)^(t/t½). Rearranged: t = t½·ln(N₀/Nₜ)/ln 2; t½ = t·ln 2/ln(N₀/Nₜ); N₀ = Nₜ·2^(t/t½). Constants: t½ = ln(2)·τ = ln(2)/λ, so λ = ln 2 ÷ t½ and τ = 1/λ. Use any consistent time unit.
Assumes pure exponential (first-order) decay with a constant rate — real mixtures of isotopes or multi-compartment drugs need more advanced models.
Worked example
A fossil sample retains 25 units of carbon-14 per 100 original units (25%). Since N₀/Nₜ = 4, ln(4)/ln(2) = 2 half-lives have passed, so t = 2 × 5,730 = 11,460 years. From the same half-life, λ = ln 2 ÷ 5,730 = 0.000121 per year and τ = 5,730 ÷ ln 2 = 8,266.6 years.
Common mistakes
- Treating decay as linear — two half-lives leave 25%, not 0%.
- Mixing time units (half-life in years, elapsed time in days) without converting.
- Swapping N₀ and Nₜ, which flips the sign of the logarithm and gives negative time.
- Confusing mean lifetime τ with half-life — τ is about 44% longer (1/ln 2).
Frequently asked questions
What is the half-life formula?
Nₜ = N₀ × (1/2)^(t / t½): the remaining quantity equals the starting quantity times one-half raised to the number of half-lives elapsed. Rearranged, t = t½ × ln(N₀/Nₜ) ÷ ln 2, which is how the calculator solves for time.
How are half-life, mean lifetime, and decay constant related?
t½ = ln(2) × τ = ln(2) ÷ λ. The decay constant λ is the fraction decaying per unit time, the mean lifetime τ = 1/λ is the average survival time of an atom, and the half-life is ln 2 ≈ 0.693 of the mean lifetime. Knowing any one gives the other two.
Does half-life only apply to radioactivity?
No — any exponential decay has a half-life: drug concentration in the bloodstream, charge on a capacitor, foam settling in a glass, or the value of a depreciating asset modeled exponentially. The math on this page applies to all of them.
How does carbon-14 dating use this equation?
Living things hold a steady share of carbon-14 that starts decaying at death, with a half-life of about 5,730 years. Measuring the fraction remaining and solving for t gives the age: a sample with 25% remaining is two half-lives old, about 11,460 years. The method works to roughly 50,000 years.
What units should I use?
Any, as long as they are consistent: if the half-life is in years, the elapsed time answer is in years and λ is per year. Quantities N₀ and Nₜ can be grams, atoms, becquerels, or percentages — only their ratio matters.