Relative dating and absolute dating
Relative dating tells which rock or fossil is older or younger, but not by how many years. Simple rules help:
- Superposition: in undisturbed layers, the lower layer formed first.
- Cross-cutting: a crack or magma body that cuts layers is younger than those layers.
- Index fossils: fossils of species that lived for a short time and spread widely link layers in different places.
Absolute dating gives an age in years. The main method is radiometric dating, which uses radioactive decay as a clock.
How radioactive decay works as a clock
Some atoms are unstable. A parent isotope decays into a stable daughter isotope. We cannot say when one atom will decay, but for many atoms the rate is fixed. Heat, pressure and chemistry do not change it.
The half-life (T½) is the time for half of the parent atoms to decay. After n half-lives, the fraction of parent left is (½)ⁿ.
| Half-lives | Parent left | Daughter |
|---|---|---|
| 0 | 100% | 0% |
| 1 | 50% | 50% |
| 2 | 25% | 75% |
| 3 | 12.5% | 87.5% |
Calculating an age
The clock starts when the mineral forms (or when a living thing dies, for carbon-14). Measure parent (P) and daughter (D):
Number of half-lives n = log₂(1 + D/P), and age t = n × T½.
Easy case: if P = 25% and D = 75%, then 1 + D/P = 4 = 2², so n = 2.
The formula assumes the rock started with no daughter atoms and has not lost or gained any. Geologists check this by testing several minerals.
Common dating methods and Earth's history
- Carbon-14 → nitrogen-14, T½ ≈ 5 730 years. Living things take in carbon; after death, C-14 decays. Useful up to about 50 000 years: bones, wood, charcoal.
- Potassium-40 → argon-40, T½ ≈ 1.25 billion years. Used for volcanic rocks.
- Uranium-238 → lead-206, T½ ≈ 4.47 billion years. Used for very old rocks and zircon crystals.
Choose an isotope whose half-life suits the age: too short and nothing is left; too long and almost nothing has decayed.
Radiometric dating sets the geological time scale. Meteorites give Earth an age of about 4.6 billion years. Fossils in dated layers show when life forms appeared and when mass extinctions happened.
Key formulas and definitions
- Parent left after n half-lives = N₀ × (½)ⁿ
- n = log₂(1 + D/P)
- Age t = n × T½
- N = N₀ × (½)^(t / T½)
Worked examples
1. A sample has 1/8 of its original parent atoms. How many half-lives have passed?
1/8 = (½)³, so 3 half-lives.
2. Charcoal has 25% of its original C-14 (T½ = 5 730 y). Find its age.
25% = (½)², so 2 half-lives. Age = 2 × 5 730 = 11 460 years.
3. A rock has D/P = 7 for U-238 → Pb-206. Find its age.
1 + 7 = 8 = 2³ → 3 half-lives. 3 × 4.47 billion ≈ 13.4 billion years. That is older than the universe, so the rock must have started with some lead or been disturbed: the assumption failed.
4. A volcanic rock has 50% of its K-40 left. Find its age.
1 half-life = 1.25 billion years.
5. Bone has 12.5% of its C-14 left. Age?
12.5% = 1/8 → 3 half-lives → 3 × 5 730 = 17 190 years.
6. A mineral has D/P = 1.5 for a parent with T½ = 1.0 × 10⁹ years. Find the age.
n = log₂(2.5) ≈ 1.32. Age ≈ 1.32 × 10⁹ years.
Common mistakes
- Thinking all the parent is gone after two half-lives: 25% is still left.
- Using carbon-14 on rocks millions of years old: almost no C-14 would remain.
- Mixing up the ratio: D/P is daughter divided by parent, not the other way.
- Believing heat or pressure can speed up radioactive decay.