Radiometric Dating: Age Equation for U-Pb and K-Ar
How do geologists know that a rock is hundreds of millions or billions of years old? The answer is radioactive decay. A parent isotope decays at a fixed rate into a daughter isotope, so the ratio of daughter to parent atoms in a mineral acts as a clock. The same first-order kinetics that you use for half-life problems is the entire mathematics behind it. This topic is a standard application of first-order decay in physical and nuclear chemistry for IIT-JAM, GATE, CSIR-NET and CUET-PG.
The age equation
t = (1 / λ) ln (1 + D / P) λ = ln 2 / t½
P is the number of parent atoms now present (N) and D is the number of daughter atoms that have formed from it. λ is the decay constant and t½ the half-life. Note that the equation uses atoms (or moles) of daughter and parent, not their masses, unless you convert. This is a different case from radiocarbon dating, where the parent is measured against a living standard; here the daughter product accumulates inside the mineral.
The assumptions behind the method
- The mineral formed with no daughter atoms, or the initial daughter amount can be corrected for.
- The mineral has been a closed system: neither parent nor daughter has entered or left since it formed.
- The decay constant is unchanged by temperature, pressure or chemical state.
- For a decay chain such as U-238 → Pb-206, all the short-lived intermediates are small, so the end product Pb-206 stands for the number of parent atoms decayed.
Worked example 1: uranium-238 to lead-206
A zircon crystal contains Pb-206 and U-238 atoms in the ratio D/P = 0.30. Take t½ of U-238 = 4.468 × 109 years. Find the age.
λ = 0.6931 / 4.468 × 109 = 1.551 × 10−10 yr−1.
t = (1 / 1.551 × 10−10) × ln (1 + 0.30) = 6.447 × 109 × 0.2624 = 1.69 × 109 years.
Check: the fraction of parent left is 1 / 1.30 = 0.769. After 1.69 × 109 years, the number of half-lives is 1.69 / 4.468 = 0.378, and (1/2)0.378 = 0.769. It matches.
Worked example 2: a young sample and a sanity check
Another mineral has D/P = 0.05 for the same pair. Find its age.
t = 6.447 × 109 × ln (1.05) = 6.447 × 109 × 0.04879 = 3.14 × 108 years, about 314 million years.
For small D/P, ln (1 + x) is close to x, so t ≈ (1 / λ) (D/P) = 6.447 × 109 × 0.05 = 3.22 × 108 years. This quick estimate is within 3% of the exact answer and is useful for checking.
Worked example 3: the half-life checkpoint
What D/P ratio corresponds to exactly one half-life?
After one half-life, half the parent has decayed, so P = N0/2 and D = N0/2. D/P = 1. Substituting in the age equation: t = (1/λ) ln 2 = t½. Good. So D/P = 1 means one half-life, D/P = 3 means two half-lives (since 1 + D/P = 4 = 22), and D/P = 7 means three.
Worked example 4: potassium-40 to argon-40
K-40 decays by two routes. Most of it forms Ca-40, which cannot be used because ordinary calcium is already present in rocks. About 10.7% decays by electron capture to Ar-40. This daughter is a gas that is trapped in the mineral when it cools, so its amount can be measured. Use the commonly quoted values: total λ = 5.543 × 10−10 yr−1 and λEC = 0.581 × 10−10 yr−1 (these appear in tables; use the numbers given in your paper).
Only a fraction of decays gives Ar, so the equation becomes t = (1 / λ) ln [1 + (λ / λEC)(Ar / K)].
Suppose a volcanic rock has Ar-40/K-40 = 0.0010 (atom ratio). λ / λEC = 5.543 / 0.581 = 9.540.
t = (1 / 5.543 × 10−10) × ln (1 + 9.540 × 0.0010) = 1.804 × 109 × 0.009495 = 1.71 × 107 years, about 17 million years.
Check: the total half-life is 0.6931 / 5.543 × 10−10 = 1.25 × 109 years, and 17 million years is about 1.4% of one half-life, so only a very small part of the K-40 has decayed. That is consistent with a tiny Ar/K ratio.
Choosing the isotope pair
| Parent → Daughter | Half-life (given as data) | Suited for |
|---|---|---|
| U-238 → Pb-206 | 4.468 × 109 years | Very old minerals |
| K-40 → Ar-40 | 1.25 × 109 years (total) | Rocks that cooled from a melt, as argon escapes while hot |
| C-14 → N-14 | about 5730 years | Organic remains (the parent itself is measured; see the radiocarbon article) |
The rule: the half-life of the clock should be comparable to the age you want to measure. A short half-life gives almost no parent left in an old rock, and a very long half-life gives almost no daughter in a young one.
Common mistakes
- Using ln (D/P) instead of ln (1 + D/P). The parent present now is part of the original total.
- Using masses directly for D/P without converting to moles or atoms. Pb-206 and U-238 have different masses.
- Ignoring the branching in K-40, which gives a result about 9 to 10 times too low.
- Mixing time units. If t½ is in years, the age comes out in years.
- Assuming a closed system when the data say otherwise, for example if the question says that some argon was lost.
Questions in this area are mostly calculation-based on first-order kinetics. For the current exam syllabus and pattern, check the official notification of the exam you are preparing for.
The half-life calculator helps you check decay constants and half-life steps.
Open the Half-Life Calculator →Previous-year CSIR-NET questions and timed quizzes to check what you just read.
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