# How much is left after its half-life?

Works out radioactive decay with the half-life formula N = N₀ × (1/2)^(t ÷ T½): the amount left, the initial amount, the time passed, or the half-life, with the decay constant and mean lifetime.

- Page: https://www.acalculator.org/chemistry/half-life-calculator
- JSON spec: https://www.acalculator.org/chemistry/half-life-calculator.json
- Version: 533b16d18d25

## Default answer

Example with the default inputs (Initial amount (N₀) 100, Time passed (t) 10,000 yr, Half-life (T½) 5,730 yr): After 10,000 yr, 29.8292 of 100 is left, with a half-life of 5,730 yr.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| n0 | Initial amount (N₀) | The amount at the start, in any unit: grams, atoms, counts per minute, or percent. |
| n | Amount left (N) | The amount left after the time t, in the same unit as the initial amount. |
| t | Time passed (t) | How long the substance has been decaying. |
| T | Half-life (T½) | The time it takes for half of the substance to decay. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| n0 | Initial amount (N₀) | The amount at the start, in any unit: grams, atoms, counts per minute, or percent. |
| n | Amount left (N) | N₀ × (1/2)^(t ÷ T½), in the unit of the initial amount. |
| t | Time passed (t) | How long the substance has been decaying. |
| half | Half-life (T½) | The time it takes for half of the substance to decay. |
| percent | Percent left | The share of the initial amount that is left: N ÷ N₀ × 100. |
| halfLives | Half-lives passed | How many half-lives the time passed is: t ÷ T½. |
| lambdaYear | Decay constant (λ, per year) | ln 2 ÷ T½, with the half-life in years of 365 days. |
| mean | Mean lifetime (τ) | The average time an atom lasts before it decays, T½ ÷ ln 2, in years of 365 days. |

## Method

N = N₀ × (1/2)^(t ÷ T½); N₀ = N × 2^(t ÷ T½); t = T½ × log₂(N₀ ÷ N); T½ = t ÷ log₂(N₀ ÷ N); λ = ln 2 ÷ T½; τ = T½ ÷ ln 2.

## Assumptions

- Decay is first order: the same fraction decays in each equal time, whatever the amount (OpenStax Chemistry 2e, 21.3).
- The amount left and the initial amount are in the same unit; any unit works.
- A year is 365 days.

## Worked examples

1. n0 = 1, t = 473,040,000, T = 166,194,720 gives n = 0.139052, percent = 13.905237%, lambdaYear = 0.131527. Source: OpenStax, Chemistry 2e, section 21.3 Radioactive Decay (λ = ln 2 ÷ t½, Nₜ = N₀e^(−λt)), https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay, retrieved 2026-10-02, example 21.6 (cobalt-60; the book rounds λ to 0.132 per year and gets 13.8%).
2. n0 = 13.6, n = 10.8, T = 180,701,280,000 gives t = 60,096,789,469.43156, halfLives = 0.332575. Source: OpenStax, Chemistry 2e, section 21.3 Radioactive Decay (λ = ln 2 ÷ t½, Nₜ = N₀e^(−λt)), https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay, retrieved 2026-10-02, example 21.7 (Dead Sea Scrolls, carbon-14 half-life 5,730 years, about 1,900 years old).
3. n0 = 100, n = 25, t = 864,000 gives T = 432,000.
4. n = 10, t = 94,608,000, T = 31,536,000 gives n0 = 80.
5. n0 = 100, t = 315,360,000,000, T = 180,701,280,000 gives n = 29.829244.

## FAQ

### What is a half-life?

The time it takes for half of a radioactive substance to decay. After one half-life 1/2 is left, after two 1/4, after three 1/8. Carbon-14 has a half-life of 5,730 years; cobalt-60, 5.27 years.

### How do I calculate the amount left?

Use N = N₀ × (1/2)^(t ÷ T½). For cobalt-60 (T½ = 5.27 years) after 15 years: (1/2)^(15 ÷ 5.27) = 0.139, so 13.9% is left.

### How do I find the half-life from two amounts?

Use T½ = t ÷ log₂(N₀ ÷ N). If 100 g drops to 25 g in 10 days, log₂(100 ÷ 25) = 2 half-lives passed, so T½ = 10 ÷ 2 = 5 days.

### How does carbon dating use the half-life?

Living things keep the same share of carbon-14; after death it decays. Comparing the activity now with that of living matter gives the age: t = T½ × log₂(N₀ ÷ N). For 10.8 counts per minute per gram against 13.6, t = 5,730 × log₂(13.6 ÷ 10.8) = about 1,906 years.

### What is the decay constant?

λ = ln 2 ÷ T½, the fraction of atoms that decays per unit of time, used in N = N₀e^(−λt). For cobalt-60, λ = 0.693 ÷ 5.27 = 0.1315 per year. The calculator gives λ per year.

### What is the mean lifetime?

The average time an atom lasts before it decays: τ = T½ ÷ ln 2, about 1.44 half-lives. After one mean lifetime, 1/e (about 36.8%) of the substance is left.

### Does the half-life depend on the amount?

No. Radioactive decay is first order, so the same fraction decays in each half-life whether you start with 1 g or 1 kg.

## Sources

- OpenStax, Chemistry 2e, section 21.3 Radioactive Decay (λ = ln 2 ÷ t½, Nₜ = N₀e^(−λt); cobalt-60 and carbon-14 examples), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay