Formula & Calculator
Radiometric Age Dating (Half-Life)
Calculates the age of a rock or mineral sample from the ratio of remaining radioactive parent isotope to its original amount, using the isotope's half-life.
Interpretation
t = (t₁/₂ / ln(2)) × ln(N₀/N). Age of a sample from the ratio of parent to daughter isotopes. Used in geochronology and archaeology.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| t | Age of sample | years |
| t1/2 | Isotope half-life | years |
| N0 | Original amount of parent isotope | |
| N | Remaining amount of parent isotope |
What it means
This formula calculates the age of a sample from radiometric dating, based on the decay of a radioactive isotope. It uses the half‑life t₁/₂ and the ratio of initial to remaining parent atoms (N₀/N). This is applied in methods like carbon‑14 dating, uranium‑lead dating, and potassium‑argon dating. It is fundamental in geology, archaeology, and paleontology for determining the age of rocks, fossils, and artifacts. Understanding this is essential for interpreting the timing of geological events and for constructing the history of Earth and life.
Worked example
Radiometric Age Dating – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| Half‑life (years) | 4.47e9 |
| N₀/N ratio | 1.5 |
| Parameter | Value |
|---|---|
| Half‑life | 5730 |
| N₀/N | 8 |
Common mistakes
- Radiometric age dating: t = (t₁/₂ / ln(2)) × ln(N₀/N) – based on radioactive decay.
- t₁/₂: Half‑life of the isotope – in years (or other time units).
- N₀: Initial number of parent atoms – often estimated from present‑day daughter ratio.
- N: Current number of parent atoms.
- Assumptions: Closed system, no initial daughter, constant decay rate – check for validity.
Applications
Radiometric age dating using half‑life, t = (t₁/₂ / ln2)·ln(N₀/N), calculates the age of a rock or fossil based on the measured ratio of parent to daughter isotopes (N₀/N). This is the foundation of geochronology, allowing the determination of absolute ages of geological events. Geologists use it to date rocks, to constrain the timing of tectonic events, and to calibrate the geological timescale. By applying this formula, scientists can understand the age of Earth, the timing of extinctions, and the rates of geological processes. Radiometric dating is essential for archaeological dating (e.g., carbon‑14) and for understanding planetary histories.
- Absolute dating of rocks and minerals in geological studies
- Calibration of the geological timescale
- Archaeological dating using carbon‑14
- Understanding of Earth's formation and evolution
- Correlation of sedimentary sequences and tectonic events
Frequently Asked Questions
t = (t₁/₂ / ln(2)) × ln(N₀/N), where t₁/₂ is the half‑life, N₀ is the initial amount of parent isotope, and N is the remaining amount. This gives the age of a sample.
It comes from the decay law: N = N₀ e^(−λt), where λ = ln(2)/t₁/₂. Solving for t gives the formula with the natural log.
- U‑Pb – for zircon, meteorites, very old rocks.
- K‑Ar and Ar‑Ar – for volcanic rocks and minerals.
- Rb‑Sr – for igneous and metamorphic rocks.
- Sm‑Nd – for ancient crustal rocks.
- C‑14 – for organic materials up to ~50,000 years.
The half‑life of ¹⁴C is 5,730 years. It is used for dating archaeological and organic materials up to about 50,000 years.
N₀ is estimated from the current amount of parent and daughter isotopes, assuming that the system was closed and that the initial daughter amount is known (or can be subtracted).
No loss or gain of parent or daughter isotopes has occurred since the rock formed. If the system has been open, the age estimate is invalid.
- Requires a closed system.
- Assumes initial daughter concentration is zero or known.
- May be affected by metamorphism or alteration.
- Only applicable to certain minerals.
From the oldest meteorites and lunar rocks, using Pb‑Pb isochron dating, giving an age of about 4.54 billion years.
It plots the ratio of daughter/parent vs. daughter/other isotope for several minerals from the same rock. The slope of the line gives the age, without needing to know initial daughter concentration.
For a given counting error, a shorter half‑life allows more precise dating for younger samples. A longer half‑life is better for very old samples (since more parent remains).