# Half-Life Calculator

URL: https://speedycalculator.com/calculators/chemistry/half-life/
Last updated: 2026-10-30

> The half-life calculator solves the first-order decay law for whichever value you need – the amount left, the half-life, the elapsed time or the starting amount – and also gives the decay constant, the mean lifetime and a decay curve with every half-life marked.

## Inputs

- Calculate
- Decay rate given as
- Initial amount N₀
- Amount remaining N(t)
- Time elapsed t (years, s, min, h, days, ms)
- Half-life T½ (years, s, min, h, days, ms)
- Decay constant λ (1/year, 1/s, 1/min, 1/h, 1/day)
- Mean lifetime τ (years, s, min, h, days, ms)

## Results

- Result
- Decay curve
- Amount remaining N(t)
- Initial amount N₀
- Half-life T½
- Time elapsed t
- Decay constant λ
- Mean lifetime τ
- Fraction remaining
- Fraction decayed
- Number of half-lives elapsed
- Amount after each half-life

## 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))
```

## Explanation

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

OrderRate lawHalf-lifeDepends on amount?Zerorate = k[A]₀ / (2k)Yes – gets shorterFirstrate = k[A]ln 2 / kNoSecondrate = k[A]²1 / (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).

## 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.

## FAQ

**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)).


## Sources

- [IUPAC Compendium of Chemical Terminology (Gold Book)](https://goldbook.iupac.org/)
- [NIST – Radionuclide half-life measurements](https://www.nist.gov/pml/radionuclide-half-life-measurements)
- [OpenStax – Chemistry 2e](https://openstax.org/details/books/chemistry-2e)