Relative dating puts events in order. To put numbers on them, geologists use radioactive atoms, which you met in Half-Life and Absolute Age and Radioactive Decay. A parent isotope decays into a daughter at a steady rate, and the decay can't be sped up or slowed down by heat, pressure or chemistry. That makes it a clock.
Why it works
In each half-life, half of the remaining parent atoms decay. After $n$ half-lives the parent left is $P = P_0\left(\tfrac12\right)^n$, and every decayed atom became a daughter: $D = P_0 - P$. Divide: $\dfrac{D}{P} = 2^n - 1$, so $n = \log_2\left(1 + \dfrac{D}{P}\right)$ and the age is $t = t_{1/2}\log_2\left(1 + \dfrac{D}{P}\right)$ (an exponential equation solved with a log). This assumes the mineral held no daughter atoms when it formed and lost none since; zircon, which rejects lead as it crystallizes, is ideal for uranium–lead dating.
| Half-lives passed | Parent left | Daughter ÷ parent |
|---|---|---|
| 1 | 1/2 | 1 |
| 2 | 1/4 | 3 |
| 3 | 1/8 | 7 |
| 4 | 1/16 | 15 |
Key idea
Count half-lives from the daughter-to-parent ratio: 1 means one half-life, 3 means two, 7 means three. In general $t = t_{1/2}\log_2(1 + D/P)$. The clock starts when a mineral crystallizes and traps the parent.
Worked examples
Worked example: Whole half-lives
A zircon has 7 atoms of lead-207 for every atom of uranium-235 (half-life 704 million years). How old is it?
- $D/P = 7$ means $\tfrac18$ of the parent is left: 3 half-lives.
- Age $= 3 \times 704 = 2,112$ million years, about 2.1 billion years.
Answer: About 2.1 billion years.
Worked example: Using the logarithm
A zircon has $D/P = 0.25$ for uranium-238 → lead-206 (half-life 4.47 billion years). How old is it?
- $n = \log_2(1.25) = \dfrac{\ln 1.25}{\ln 2} = \dfrac{0.2231}{0.6931} = 0.322$ half-lives.
- Age $= 0.322 \times 4,470 \approx 1,440$ million years.
Answer: About 1.44 billion years.
Common mistake
Many people think carbon-14 dates dinosaur bones and ancient rocks. With a half-life of 5,730 years, carbon-14 works only for once-living material up to about 50,000 years old; after that too little is left to measure. Dinosaurs died out 66 million years ago, more than 11,000 carbon-14 half-lives. Their ages come from uranium–lead and potassium–argon dates on volcanic ash and lava above and below their fossils. Check: match the half-life to the age you expect.
Watch atoms decay one at a time in the Build an Atom lab, then come back and date a rock.