The decay law turns a radioactive isotope into a clock. If you know how much of a parent isotope a sample started with, and how much is left, the exponential gives the time that has passed. Two clocks cover most of history: carbon-14 for things that were alive in the last few tens of thousands of years, and uranium–lead for rocks billions of years old. Try it in the Science Lab's Build an Atom, where half-lives date ancient wood and bone.
Key idea
From $N = N_0e^{-\lambda t}$, the age is $$t = \frac{T_{1/2}}{\ln 2}\,\ln\frac{N_0}{N}.$$ Carbon-14 ($T_{1/2} = 5730$ yr): living things keep the same $^{14}$C share as the air; after death it decays, giving ages up to about 50,000 years. Uranium–lead ($^{238}$U → $^{206}$Pb, $T_{1/2} = 4.47 \times 10^9$ yr): if a mineral started with no lead and kept every atom, $N_0 = N_{\text{U}} + N_{\text{Pb}}$, so $$t = \frac{T_{1/2}}{\ln 2}\,\ln\left(1 + \frac{N_{\text{Pb}}}{N_{\text{U}}}\right).$$
Why it works
Take logarithms of $\frac{N}{N_0} = e^{-\lambda t}$ (solving exponential equations): $\ln\frac{N_0}{N} = \lambda t$, and $\lambda = \frac{\ln 2}{T_{1/2}}$. The clock's useful range is set by the half-life. After about 9 half-lives (50,000 years for carbon-14), less than 0.2% of the original remains, too little to measure against background; for a clock to read billions of years, it needs a half-life in the billions. In the uranium–lead method, each decayed uranium atom becomes a lead atom (through a chain of short-lived steps), so the lead counts the decays. Zircon crystals are ideal: they take in uranium but reject lead when they form.
Worked examples
Worked example: Wood from a fire pit
Charcoal from an ancient fire pit has 35% of the carbon-14 found in living wood. How old is it?
- $t = \frac{5730}{\ln 2}\ln\frac{1}{0.35} = 8267 \times 1.050$ yr.
- $t = 8680$ yr. (Check: 35% is a little less than half of half, between one and two half-lives.)
Answer: About 8,700 years
Worked example: From activities
Carbon from living things decays at about 15.3 decays per minute per gram of carbon. A bone sample gives 4.1 decays per minute per gram. How old is it?
- Activity is proportional to the number of $^{14}$C nuclei, so $\frac{N_0}{N} = \frac{15.3}{4.1} = 3.73$.
- $t = 8267 \times \ln 3.73 = 8267 \times 1.317 = 10{,}900$ yr.
Answer: About 10,900 years
Worked example: An ancient zircon
A zircon crystal has 0.50 atoms of $^{206}$Pb for every atom of $^{238}$U. How old is it?
- $t = \frac{4.47 \times 10^9}{\ln 2}\ln(1 + 0.50) = 6.45 \times 10^9 \times 0.405$ yr.
Answer: $2.6 \times 10^9$ years. (The oldest Earth zircons are about 4.4 billion years old.)
Common mistake
Trying carbon dating on rocks millions of years old. Carbon-14's 5730-year half-life is far too short: after a million years, the fraction left is $0.5^{175}$, effectively zero, and the method also needs a sample that was once alive. For rocks, use a long-lived parent such as uranium-238. Check any method: the sample's age should be no more than about 10 half-lives of the clock you use.