# Carbon Dating Calculator

> Computes the age of a sample from the percentage of carbon-14 remaining by first-order decay with the Cambridge half-life of 5730 years, or the remaining percentage from a known age, and also reports the conventional radiocarbon age based on the Libby half-life of 5568 years.

- Calculator id: `carbon-dating` · Category: Biology, Earth & Space (`science`) · Tool name: `calculate_carbon_dating_age`
- Canonical page: https://tttkmbb.com/science/carbon-dating · This document: https://tttkmbb.com/science/carbon-dating.md · JSON definition: https://tttkmbb.com/science/carbon-dating.json

## Purpose

Computes the age of a sample from the percentage of carbon-14 remaining by first-order decay with the Cambridge half-life of 5730 years, or the remaining percentage from a known age, and also reports the conventional radiocarbon age based on the Libby half-life of 5568 years.

**Use when:** You know the fraction of ¹⁴C left relative to modern carbon and want an uncalibrated age, or want the ¹⁴C remaining after a given number of years.

**Do not use when:** You need calibrated calendar dates (that requires a calibration curve such as IntCal20), the sample is older than about 50,000 years, or you are working with another isotope or drug (use half-life).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `percent_c14_remaining` | number | % | optional | Carbon-14 activity of the sample as a percentage of the modern reference (100 = modern). Give this or age_years. (> 0, max 100) |
| `age_years` | number | years | optional | Known age; when given, the remaining percentage is computed instead. Give this or percent_c14_remaining. (min 0, max 1000000) |
| `half_life_years` | number | years | optional, default 5730 | Half-life used for the age: default is the Cambridge value 5730 ± 40 years for ¹⁴C; the Libby value 5568 is reported separately. (> 0, max 1000000000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `age_years` | number | years | t = −(t½ / ln 2) × ln(fraction remaining), uncalibrated. |
| `radiocarbon_age_libby_years` | number | years BP | Same fraction with the Libby half-life 5568 years, as used for conventional radiocarbon ages (before calibration). |
| `percent_remaining` | number | % | 100 × 2^(−t / t½). |
| `fraction_remaining` | number |  | N / N0. |
| `half_lives_elapsed` | number |  | t / t½ = log2(N0 / N). |
| `decay_constant_per_year` | number | 1/year | ln 2 / t½. |
| `mean_lifetime_years` | number | years | t½ / ln 2 (8267 years for 5730). |
| `solved_for` | string |  | age_years or percent_remaining. |

## Formula

`t = −(t½ / ln 2) × ln(fraction_remaining); fraction_remaining = 2^(−t / t½); λ = ln 2 / t½; τ = t½ / ln 2; Libby age = −(5568 / ln 2) × ln(fraction_remaining)`

First-order decay with the Cambridge half-life 5730 ± 40 years. Laboratories report conventional radiocarbon ages with the Libby half-life (5568 years, mean life 8033 years) per Stuiver & Polach (1977); both are uncalibrated and must be converted to calendar years with a calibration curve. Practical limit about 50,000 years (≈ 0.2 % remaining).

## Data Sources

- Wikipedia – Radiocarbon dating — https://en.wikipedia.org/wiki/Radiocarbon_dating (reference, retrieved 2026-09-24)
- Stuiver M & Polach HA (1977) Discussion: Reporting of 14C data, Radiocarbon 19(3):355-363 — https://doi.org/10.1017/S0033822200003672 (peer_reviewed, retrieved 2026-09-24)

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/carbon-dating?`
- `POST https://tttkmbb.com/api/v1/calculate/carbon-dating` 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/carbon-dating · OpenAPI operationId `calculate_carbon_dating_age` 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": "carbon-dating", "inputs": {…}}`

## Example

- 10 % of ¹⁴C remaining: inputs `{"percent_c14_remaining":10}` → `{"age_years":19035,"radiocarbon_age_libby_years":18496,"half_lives_elapsed":3.3219,"decay_constant_per_year":0.00012097,"mean_lifetime_years":8266.6,"solved_for":"age_years"}`
- Sample 2,000 years old: inputs `{"age_years":2000}` → `{"percent_remaining":78.5106,"fraction_remaining":0.785106,"half_lives_elapsed":0.349,"radiocarbon_age_libby_years":1943,"solved_for":"percent_remaining"}`

```
GET https://tttkmbb.com/api/v1/calculate/carbon-dating?percent_c14_remaining=10
```

## Limitations

You need calibrated calendar dates (that requires a calibration curve such as IntCal20), the sample is older than about 50,000 years, or you are working with another isotope or drug (use half-life). First-order decay with the Cambridge half-life 5730 ± 40 years. Laboratories report conventional radiocarbon ages with the Libby half-life (5568 years, mean life 8033 years) per Stuiver & Polach (1977); both are uncalibrated and must be converted to calendar years with a calibration curve. Practical limit about 50,000 years (≈ 0.2 % remaining). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Why two half-lives?**

Libby's original 5568 years is kept by convention for reported radiocarbon ages (years BP) so that all dates stay comparable; the more accurate Cambridge value 5730 gives the physically better age. Calibration curves correct the Libby-based age anyway.

**Is the age a calendar date?**

No. Atmospheric ¹⁴C varied over time, so an uncalibrated age must be calibrated (IntCal20, SHCal20, Marine20) to get a calendar range; differences reach several thousand years for old samples.

**How is percent_c14_remaining measured?**

As the sample's ¹⁴C/¹²C ratio (or activity) divided by that of the 1950 modern standard, after correcting for isotopic fractionation (δ¹³C); 100 % means modern carbon.

## Related

- [Half-Life Calculator](https://tttkmbb.com/chemistry/half-life.md) — General first-order decay for any isotope or elimination half-life.
- [Logarithm Calculator](https://tttkmbb.com/math/logarithm.md) — Natural logarithm arithmetic used to solve for the age.