A half-life is the time it takes for half of something to disappear in a first-order process – one in which the rate is proportional to how much is left. Radioactive nuclei are the textbook case, but the same law describes the elimination of many drugs, first-order chemical reactions such as the decomposition of N₂O₅ or hydrogen peroxide, the discharge of a capacitor and the fading of fluorescence.
The key property of first-order decay is that the half-life does not depend on the amount: after one half-life 50% remains, after two 25%, after three 12.5%, after ten less than 0.1%. That makes the curve exponential, N(t) = N₀·(½)^(t/T½). The same process can be described by the decay constant λ (the fraction lost per unit time, λ = ln 2/T½) or by the mean lifetime τ = 1/λ, the average time a single particle survives; τ is about 1.443 times the half-life.
Knowing any three of initial amount, remaining amount, time and half-life fixes the fourth. Archaeologists use the carbon-14 half-life (5730 years) to date bones, pharmacists use drug half-lives to plan dosing intervals, and chemists find rate constants from the time a concentration takes to halve.
How to use the Half-Life Calculator
- Calculate
- Decay rate given as
- Initial amount N₀
- Amount remaining N(t)
- Time elapsed t
- Half-life T½
- Decay constant λ
- Mean lifetime τ
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Choose which value to calculate and how you know the decay rate (half-life, decay constant or mean lifetime), then fill in the remaining fields. Amounts can be in any unit – grams, moles, atoms, mg/L or becquerels – as long as N₀ and N(t) use the same one. The chart shows the decay curve with your point and each half-life marked.
Worked example
Carbon-14 (T½ = 5730 years), 100 g at the start, 10 000 years later: N = 100 × (½)^(10 000/5730) = 100 × 0.2983 = 29.83 g. That is 1.745 half-lives; λ = 0.6931/5730 = 1.2097 × 10⁻⁴ per year and τ = 8267 years.
Finding a half-life: 2.5 kg of a substance decays to 2.1 kg in 5 minutes. T½ = 5 × ln 2 / ln(2.5/2.1) = 5 × 0.69315 / 0.17435 = 19.88 minutes.
Half-life of different reaction orders
| Order | Rate law | Half-life | Depends on amount? |
|---|---|---|---|
| Zero | rate = k | [A]₀ / (2k) | Yes – gets shorter |
| First | rate = k[A] | ln 2 / k | No |
| Second | rate = k[A]² | 1 / (k[A]₀) | Yes – gets longer |
Tips
- Keep temperature in kelvin and pressure in consistent units.
- For gases, ideal-gas behaviour is a good approximation near room conditions.
- Half-life and rate constant are linked by t½ = ln 2 / k for first-order processes.
- Everything is calculated in your browser.
Common mistakes to avoid
- Using the mass number instead of the molar mass in Graham’s law.
- Mixing partial pressures and mole fractions.
- Forgetting that osmotic pressure uses the total particle concentration (van ‘t Hoff factor).
Formula
N(t) = N₀ · (½)^(t/T½) = N₀ · e^(−λt) = N₀ · e^(−t/τ) · T½ = ln 2 / λ = τ · ln 2 · T½ = t · ln 2 / ln(N₀/N(t))Worked example
Inputs — Calculate: Amount remaining N(t); Decay rate given as: Half-life T½; Initial amount N₀: 100; Time elapsed t: 10,000 years; Half-life T½: 5,730 years.
Results — Result = N(t) = 29.829244; Amount remaining N(t) = 29.829244; Initial amount N₀ = 100; Half-life T½ = 180,825,048,000; Time elapsed t = 315,576,000,000; Decay constant λ = 3.8332e-12.
Key terms
- Half-life T½
- Time for a quantity to fall to half its value in a first-order process.
- Decay constant λ
- Probability per unit time that a particle decays; λ = ln 2 / T½.
- Mean lifetime τ
- Average lifetime of a particle; τ = 1/λ.
- First-order process
- One whose rate is proportional to the remaining amount, giving exponential decay.
Frequently asked questions
How do you calculate half-life?
From two measurements: T½ = t·ln 2 / ln(N₀/N). If 2.5 kg falls to 2.1 kg in 5 minutes, T½ = 5 × 0.6931 / ln(1.1905) = 19.88 minutes.
How much is left after 3 half-lives?
(½)³ = 1/8 = 12.5% of the starting amount; after 5 half-lives 3.125%, after 10 about 0.098%.
How long until only 10% remains?
t = T½ × log₂(10) = 3.32 half-lives. For carbon-14 that is 3.32 × 5730 = 19 035 years.
What is the relationship between half-life and the decay constant?
T½ = ln 2 / λ ≈ 0.6931/λ. A decay constant of 0.1 per hour means a half-life of 6.93 hours.
What is the mean lifetime?
τ = 1/λ = T½/ln 2 ≈ 1.4427 × T½ – the average survival time of a single nucleus or molecule. After one τ, 1/e ≈ 36.8% remains.
Does the half-life depend on how much I start with?
Not for first-order processes – that is their defining feature. For second-order reactions T½ = 1/(k[A]₀) grows as the concentration falls, and zero-order T½ = [A]₀/(2k) shrinks.
Can the half-life formula be used for drugs?
Yes, for drugs with first-order elimination (most of them). It takes about 4–5 half-lives to reach steady state or to clear a drug almost completely.
What formula does the Half-Life Calculator use?
N(t) = N₀ · (½)^(t/T½) = N₀ · e^(−λt) = N₀ · e^(−t/τ) · T½ = ln 2 / λ = τ · ln 2 · T½ = t · ln 2 / ln(N₀/N(t)).