Lesson 3 · 25 min

Half-life and radiometric dating

Radioactive decay is the most reliable clock there is: carbon-14 for tens of thousands of years, potassium–argon and uranium–lead for billions. Compute ages from isotope ratios and meet the hidden assumptions — calibration and closed systems.

In this lesson

Use the first-order decay law to calculate amounts left, activities and elapsed times
Explain how radiocarbon dating works, the conventions behind a conventional age, and why calibration is needed
Compute rock ages from ⁴⁰Ar/⁴⁰K or ²⁰⁶Pb/²³⁸U ratios and state the assumptions they need
Choose a suitable dating method for samples of different ages

Sandstone in the Jack Hills of Western Australia hides zircon crystals smaller than sand grains. The lead inside them comes almost entirely from uranium decay, and the uranium–lead method dates them at 4.4 billion years — the oldest minerals known on Earth, only about 140 million years younger than the planet itself. Such ages rest on a law you met in kinetics: the half-life of a first-order process does not depend on how much you start with. Radioactive decay is the perfect first-order process, untouched by temperature, pressure or chemistry, which makes it nature’s most reliable clock.

Predict first

A piece of charcoal has only 1/8 of the ¹⁴C/¹²C ratio of living wood. Roughly how old is it?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

About 2100 years About 5700 years About 17 000 years About 46 000 years

The decay law and activity

N = N_0\,e^{-\lambda t} = N_0\left(\tfrac12\right)^{t/t_{1/2}},\qquad t_{1/2} = \frac{\ln 2}{\lambda},\qquad A = \lambda N

Activity A is proportional to the number of atoms left, so it falls by the same exponential law.

$N_0,\ N$: number of atoms at the start and at time t
$\lambda$: decay constant (s⁻¹ or a⁻¹)
$t_{1/2}$: half-life
$A$: activity: decays per second, in becquerels (Bq); 1 Ci = 3.7 × 10¹⁰ Bq

This is exactly the first-order law of the integrated rate laws lesson, with atoms in place of concentration. Activity links the half-life to the amount of sample: 1 g of radium-226 holds 2.66 × 10²¹ atoms and, with a half-life of 1600 years, has an activity of 3.66 × 10¹⁰ Bq — about one curie, which is where that unit came from. The shorter the half-life, the higher the activity of the same number of atoms.

Radiocarbon dating

Cosmic rays make neutrons high in the atmosphere, and the neutrons turn nitrogen into carbon-14: $\ce{^{1}_{0}n} + \ce{^{14}_{7}N} \mathrel{\ce{->}} \ce{^{14}_{6}C} + \ce{^{1}_{1}H}$ . The ¹⁴C is oxidised to CO₂ and enters every living thing through photosynthesis. While alive, organisms keep exchanging carbon with the air, holding about 0.226 Bq of ¹⁴C per gram of carbon; at death the exchange stops and ¹⁴C decays with a half-life of 5700 years. Measure the sample’s “fraction modern” $F^{14}\mathrm{C}$ (its ¹⁴C/¹²C over that of the AD 1950 atmosphere) and you have its age. The concentration countdown tells the basic idea of this clock; real archaeological dating must also reckon with three things:

Conventions: conventional radiocarbon ages use the Libby half-life, 5568 years (2.3 % shorter than the best value, 5700), and count back from AD 1950, written “years BP”.
Calibration: atmospheric ¹⁴C has varied with cosmic-ray intensity, the geomagnetic field and the carbon cycle, so a conventional age must be converted to a calendar age with the IntCal20 calibration curve (built from dated tree rings, corals and cave deposits); the two can differ by centuries.
Contamination and the limit: burning fossil fuels has diluted atmospheric ¹⁴C, and 1960s nuclear tests nearly doubled it in the northern hemisphere; after about 50 000 years only about 0.2 % of the ¹⁴C remains, so the slightest trace of modern carbon badly distorts the result.

t = \frac{t_{1/2}}{\ln 2}\,\ln\frac{N_0}{N},\qquad t_{^{14}\mathrm{C}} = -8033\ \mathrm{a}\times\ln F^{14}\mathrm{C}

8033 years is the mean life for the Libby half-life, 5568/ln 2; conventional radiocarbon ages are computed this way.

$N_0/N$: ratio of parent atoms at the start and now
$F^{14}\mathrm{C}$: fraction modern: today’s ¹⁴C/¹²C over that of the AD 1950 atmosphere

Dating rocks

Rocks are far too old for ¹⁴C, which decayed away long ago. They need nuclides with half-lives of billions of years, and the measurement is the ratio of daughter to parent:

Potassium–argon: ⁴⁰K has a half-life of about 1.25 billion years, and only 10.5 % of its decays give ⁴⁰Ar (by electron capture); the rest give ⁴⁰Ca. Argon escapes from molten rock, so a lava’s clock is “reset” as it cools, and all the ⁴⁰Ar* that accumulates afterwards comes from decay.
Uranium–lead: zircon (ZrSiO₄) takes up uranium but rejects lead when it crystallises, so its lead is almost all decay product. Two clocks, ²³⁸U → ²⁰⁶Pb and ²³⁵U → ²⁰⁷Pb, check each other; when they agree (“concordant”) the crystal has not lost lead. In 1956 Clair Patterson used the lead isotopes of meteorites to date the Earth at 4.55 billion years; the best value today is 4.54 ± 0.05 billion years.

t = \frac{1}{\lambda}\ln\!\left(1 + \frac{\lambda}{\lambda_\mathrm{e}}\,\frac{^{40}\mathrm{Ar}^*}{^{40}\mathrm{K}}\right),\qquad t = \frac{1}{\lambda_{238}}\ln\!\left(1 + \frac{^{206}\mathrm{Pb}^*}{^{238}\mathrm{U}}\right)

The asterisk marks radiogenic daughter (made by decay); λ = 5.543 × 10⁻¹⁰ a⁻¹, λe = 0.581 × 10⁻¹⁰ a⁻¹, λ₂₃₈ = 1.551 25 × 10⁻¹⁰ a⁻¹.

$\lambda$: total decay constant of ⁴⁰K
$\lambda_\mathrm{e}$: decay constant of the branch that makes ⁴⁰Ar (electron capture)
$\lambda_{238}$: decay constant of ²³⁸U

Ages from measured isotope ratios

Radiometric dating calculator

Radiogenic ²⁰⁶Pb* to ²³⁸U atom ratio · 0.980 Relative standard uncertainty of the ratio · 1.0 %

How the ratio grows (or falls) in a closed system

Age: 4.4 billion years ± 32 million years

That is 0.99 half-lives

If no lead was lost, ²⁰⁷Pb*/²⁰⁶Pb* should be 0.5587

Zircon takes up uranium but rejects lead when it crystallises, so its lead is almost all decay product. The second clock, ²³⁵U → ²⁰⁷Pb, gives an independent age; when the two agree (“concordant”) the crystal has not lost lead.

A ²⁰⁶Pb/²³⁸U of 0.98 is what the Jack Hills zircons hold — read off the age, then the ²⁰⁷Pb/²⁰⁶Pb* expected if no lead was lost: the second clock’s prediction. Switch to potassium–argon and compare the age error for the same 1 % error in the ratio.

Worked example

The potassium–argon age of a basalt

A basalt contains radiogenic ⁴⁰Ar* and ⁴⁰K in the atom ratio 0.0100. How old is it?

Concept check

Which method would you use to date a wooden spear shaft about 40 000 years old?

Carbon-14 Potassium–argon Uranium–lead None of them

Try the calculation

Wood from an ancient tomb has $F^{14}\mathrm{C} = 0.62$ . What is its conventional radiocarbon age in years BP?

Model limits Every clock has assumptions

Every radiometric age assumes that the decay constant is constant; that the sample is a closed system, with no parent or daughter gained or lost before measurement; and that the initial daughter content is known (or can be subtracted). A potassium–argon sample that lost argon looks too young; weathering or heating can make zircon lose lead, and then the two uranium–lead clocks disagree. ¹⁴C also assumes the atmosphere’s ¹⁴C/¹²C is known — exactly what the calibration curve supplies. Finally, the constants geochronologists use differ slightly from NUBASE2020 (a ²³⁸U half-life of 4.468 against 4.463 billion years, and revised ⁴⁰K branching); the differences are about 0.1–1 %, and high-precision work must state its constants.

Key takeaways

N = N₀(1/2)^{t/t½} and activity A = λN; decay ignores temperature and chemistry, which makes it a reliable clock.
Conventional radiocarbon ages use the Libby half-life of 5568 years counted from AD 1950 and must be calibrated with IntCal20 to give calendar ages; the limit is about 50 000 years.
Rocks are dated from daughter-to-parent ratios: potassium–argon t = (1/λ)ln(1 + (λ/λe)Ar/K), uranium–lead t = ln(1 + Pb/U)/λ₂₃₈.
A method’s half-life must match the sample’s age, and the sample must have been a closed system.

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Data and references

F. G. Kondev et al., “The NUBASE2020 evaluation of nuclear physics properties”, Chinese Physics C 45, 030001 (2021).
M. Stuiver, H. A. Polach, “Discussion: reporting of ¹⁴C data”, Radiocarbon 19, 355–363 (1977).
P. J. Reimer et al., “The IntCal20 Northern Hemisphere radiocarbon age calibration curve (0–55 cal kBP)”, Radiocarbon 62, 725–757 (2020).
R. H. Steiger, E. Jäger, “Subcommission on geochronology: convention on the use of decay constants in geo- and cosmochronology”, Earth and Planetary Science Letters 36, 359–362 (1977).
A. H. Jaffey et al., “Precision measurement of half-lives and specific activities of ²³⁵U and ²³⁸U”, Physical Review C 4, 1889–1906 (1971).
S. A. Wilde, J. W. Valley, W. H. Peck, C. M. Graham, “Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago”, Nature 409, 175–178 (2001).
G. B. Dalrymple, “The age of the Earth in the twentieth century: a problem (mostly) solved”, Geological Society, London, Special Publications 190, 205–221 (2001).