Superposition and cross-cutting give the order of events. To find ages in years, geologists use radioactive atoms. Some atoms, like uranium-238 and potassium-40, are unstable: one by one they change (decay) into other atoms. Each kind decays at its own steady rate, set by its half-life, the time for half of the atoms to decay.
| Parent atom | Daughter atom | Half-life | Used to date |
|---|---|---|---|
| Carbon-14 | Nitrogen-14 | 5,730 years | Once-living things, up to about 50,000 years old |
| Potassium-40 | Argon-40 | 1.25 billion years | Volcanic rock (igneous) |
| Uranium-238 | Lead-206 | 4.47 billion years | Crystals in igneous rock, up to the age of Earth |
Key idea
After each half-life, half of the remaining parent atoms are left: $\tfrac{1}{2}$, then $\tfrac{1}{4}$, then $\tfrac{1}{8}$. After $n$ half-lives the fraction left is $\left(\tfrac{1}{2}\right)^n$, and the age is $n$ × the half-life.
Why it works
Each unstable atom has the same chance of decaying in a given time, no matter how old it is or how many others are around. So in each half-life, half of whatever is left decays. The rate does not change with heat, pressure or chemistry, which makes it a dependable clock. That is multiplying by one half again and again, an exponent: $\tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} = \left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8}$.
Worked example: After three half-lives
A sample starts with 800 carbon-14 atoms. How many are left after 3 half-lives, and how many years is that?
- Halve three times: $800 \to 400 \to 200 \to 100$.
- Or: $800 \times \left(\tfrac{1}{2}\right)^3 = 800 \times \tfrac{1}{8} = 100$.
- Time: $3 \times 5{,}730 = 17{,}190$ years.
Answer: 100 atoms, after 17,190 years
Worked example: Dating a volcanic rock
A volcanic rock has only $\tfrac{1}{4}$ of its original potassium-40 left. How old is it?
- $\tfrac{1}{4} = \left(\tfrac{1}{2}\right)^2$, so 2 half-lives have passed.
- Age: $2 \times 1.25 = 2.5$ billion years.
Answer: About 2.5 billion years
Common mistake
Thinking all the atoms are gone after two half-lives (half, then the other half). Each half-life halves what is left, not what you started with: after two half-lives, one quarter remains; after three, one eighth. Check: the amount left never reaches exactly zero in a few steps; it keeps halving.
Watch atoms decay one at a time in the Atom Lab.