# Half-Life Calculator

> Computes first-order exponential decay: given two of half-life, elapsed time and remaining amount (with an initial amount) it solves the third and reports the remaining fraction, number of half-lives, decay constant and mean lifetime.

- Calculator id: `half-life` · Category: Chemistry (`chemistry`) · Tool name: `calculate_half_life_decay`
- Canonical page: https://tttkmbb.com/chemistry/half-life · This document: https://tttkmbb.com/chemistry/half-life.md · JSON definition: https://tttkmbb.com/chemistry/half-life.json

## Purpose

Computes first-order exponential decay: given two of half-life, elapsed time and remaining amount (with an initial amount) it solves the third and reports the remaining fraction, number of half-lives, decay constant and mean lifetime.

**Use when:** You need how much of a radioactive isotope, drug or other first-order decaying quantity remains after a time, or the half-life or elapsed time implied by a measured remaining amount.

**Do not use when:** The process is not first-order (e.g. zero-order elimination, growth, or compound interest – use compound-interest for money) or several isotopes decay in a chain.

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `initial_amount` | number |  | optional, default 100 | Starting quantity in any unit (g, Bq, atoms, mg/L). Default 100 makes remaining_amount a percentage. (> 0) |
| `half_life` | number |  | optional | Half-life in any time unit (same unit as elapsed_time). Leave empty to solve for it. (> 0) |
| `elapsed_time` | number |  | optional | Time elapsed, same unit as half_life. Leave empty to solve for it. (min 0) |
| `remaining_amount` | number |  | optional | Quantity left after elapsed_time, same unit as initial_amount. Leave empty to solve for it. (> 0) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `remaining_amount` | number |  | N(t) = N0 × (1/2)^(t / half_life). |
| `remaining_fraction` | number |  | N(t) / N0. |
| `remaining_percent` | number | % | 100 × N(t) / N0. |
| `decayed_amount` | number |  | N0 − N(t). |
| `number_of_half_lives` | number |  | elapsed_time / half_life. |
| `half_life` | number |  | Half-life, given or solved (time unit of the inputs). |
| `elapsed_time` | number |  | Elapsed time, given or solved (time unit of the inputs). |
| `decay_constant` | number |  | ln 2 / half_life, per time unit of the inputs. |
| `mean_lifetime` | number |  | 1 / λ = half_life / ln 2. |
| `solved_for` | string |  | remaining_amount, half_life or elapsed_time. |

## Formula

`N(t) = N0 × (1/2)^(t / t½) = N0 × e^(−λt); λ = ln 2 / t½; τ = 1 / λ; t½ = t / log2(N0 / N); t = t½ × log2(N0 / N)`

First-order kinetics: the decay rate is proportional to the amount present, so the same fraction decays in every half-life regardless of the starting amount.

## Data Sources

- OpenStax Chemistry 2e – 21.3 Radioactive Decay — https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay (textbook, retrieved 2026-09-23)
- Wikipedia – Half-life — https://en.wikipedia.org/wiki/Half-life (reference, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/half-life?`
- `POST https://tttkmbb.com/api/v1/calculate/half-life` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/half-life · OpenAPI operationId `calculate_half_life_decay` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "half-life", "inputs": {…}}`

## Example

- Carbon-14 after 11,460 years: inputs `{"initial_amount":100,"half_life":5730,"elapsed_time":11460}` → `{"remaining_amount":25,"remaining_fraction":0.25,"number_of_half_lives":2,"decay_constant":0.00012097,"mean_lifetime":8266.6,"solved_for":"remaining_amount"}`
- 80 mg of iodine-131 falls to 10 mg in 24.06 days: inputs `{"initial_amount":80,"remaining_amount":10,"elapsed_time":24.06}` → `{"half_life":8.02,"number_of_half_lives":3,"remaining_percent":12.5,"decay_constant":0.086427,"solved_for":"half_life"}`

```
GET https://tttkmbb.com/api/v1/calculate/half-life?initial_amount=100&half_life=5730&elapsed_time=11460
```

## Limitations

The process is not first-order (e.g. zero-order elimination, growth, or compound interest – use compound-interest for money) or several isotopes decay in a chain. First-order kinetics: the decay rate is proportional to the amount present, so the same fraction decays in every half-life regardless of the starting amount. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Which time unit should I use?**

Any, as long as half_life and elapsed_time share it; the decay constant and mean lifetime are then per that unit (e.g. per year for a half-life in years).

**Does this work for drug elimination?**

Yes for drugs with first-order (linear) elimination, using the plasma half-life; about 97% is eliminated after 5 half-lives. Zero-order elimination (e.g. ethanol at high doses) is not modelled.

## Related

- [Moles to Grams Calculator](https://tttkmbb.com/chemistry/moles-mass.md) — Convert the remaining mass of an isotope to a number of atoms.
- [Logarithm Calculator](https://tttkmbb.com/math/logarithm.md) — Log2 and natural log arithmetic used in solving for time.