Exponential decay and radiometric ages
Radioactive decay is exponential, and the decay equation can be turned into an age equation. This lesson derives it and uses uranium–lead to date a meteorite.
- Use N = N₀e^(−λt) and λ = ln 2 / t½.
- Derive the age equation t = (1/λ) ln(1 + D/P).
- Explain the role of isochrons in removing the effect of initial daughter isotopes.
Links to: A-level physics, radioactivity: the decay constant, activity, half-life and exponential decay; maths: natural logarithms.
The decay law
The rate of decay of a sample is proportional to the number of undecayed nuclei: dN/dt = −λN, where λ is the decay constant. Its solution is N = N₀e^(−λt). Setting N = N₀/2 gives the half-life: t½ = ln 2 / λ ≈ 0.693/λ.
From decay to age
If a mineral formed with P₀ parent atoms and no daughter atoms, then after time t the parent remaining is P = P₀e^(−λt) and the radiogenic daughter is D = P₀ − P. Dividing, D/P = e^(λt) − 1, so
t = (1/λ) ln(1 + D/P).
The age therefore depends only on the measured ratio D/P and the decay constant, not on how much material was present.
Isochrons
Most minerals contained some daughter isotope when they formed. The isochron method solves this. Several minerals from one rock, formed at the same time with the same initial isotope ratio, but with different amounts of parent, are measured. Plotted against each other, ratios such as ⁸⁷Sr/⁸⁶Sr and ⁸⁷Rb/⁸⁶Sr fall on a straight line whose gradient is e^(λt) − 1 and whose intercept is the initial ratio. Meteorites dated this way by several independent methods agree, giving confidence in an age of about 4.56 billion years for most chondrites and many achondrites.
A uranium–lead age
A mineral in a meteorite has a radiogenic ²⁰⁶Pb/²³⁸U ratio of 1.01. The half-life of ²³⁸U is 4.468 × 10⁹ years. Calculate the age.
- λ = ln 2 / t½ = 0.6931 ÷ 4.468 × 10⁹ = 1.551 × 10⁻¹⁰ yr⁻¹.
- t = (1/λ) ln(1 + D/P) = (1 ÷ 1.551 × 10⁻¹⁰) × ln(2.01).
- ln(2.01) = 0.698, and 1 ÷ 1.551 × 10⁻¹⁰ = 6.447 × 10⁹ yr.
- t = 6.447 × 10⁹ × 0.698 ≈ 4.50 × 10⁹ years.
Answer: About 4.50 billion years. In practice scientists combine both uranium decay chains in the lead–lead method, which reaches precisions of a few hundred thousand years for the oldest solids.
In a spreadsheet, plot N/N₀ against t for ²³⁸U (t½ = 4.468 Gyr), ⁴⁰K (t½ = 1.25 Gyr) and ⁸⁷Rb (t½ ≈ 49 Gyr) over 0 to 5 Gyr. Then plot ln(N/N₀) against t and explain why the lines are straight. Which isotope would give the most precise age for a 4.5-billion-year-old rock, and why?
Quiz
5 questions. Choose an answer to see whether you are right and why. Finish the quiz to complete the lesson.
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Question 1Which is the relationship between decay constant and half-life?
Setting N = N₀/2 in N = N₀e^(−λt) gives e^(−λt½) = 1/2, so t½ = ln 2 / λ.
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Question 2In the age equation t = (1/λ) ln(1 + D/P), what is D?
D is the daughter produced by decay; P is the parent remaining.
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Question 3A sample has D/P = 3. How many half-lives have passed?
1 + D/P = 4 = 2², so the parent has halved twice.
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Question 4What does the intercept of an isochron give?
At zero parent, the line gives the daughter ratio the rock started with; the gradient gives the age.
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Question 5Why is agreement between different decay systems important?
Different parent–daughter pairs respond differently to disturbance, so agreement shows the age is robust.