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How do we measure the absolute age of a rock using radioactive decay?

Radiometric dating: radioactive decay of unstable parent isotopes to stable daughter isotopes; the concept of half-life as a constant; the use of parent-to-daughter ratios to calculate absolute ages; the main isotopic systems (uranium-lead, potassium-argon and carbon-14) and their suitable age ranges; the assumptions and limitations of radiometric dating; the combination of absolute and relative dating.

A focused answer to the OCR H414 dot point on radiometric dating. Covers radioactive decay of parent to daughter isotopes, half-life as a constant, calculating absolute ages from parent-to-daughter ratios, the uranium-lead, potassium-argon and carbon-14 systems and their ranges, the assumptions and limitations, and combining absolute with relative dating.

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  1. What this dot point is asking
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What this dot point is asking

OCR wants you to explain radioactive decay of parent to daughter isotopes, define half-life as a constant, calculate absolute ages from parent-to-daughter ratios, name the main isotopic systems (uranium-lead, potassium-argon, carbon-14) and their suitable ranges, state the assumptions and limitations, and combine absolute with relative dating.

The answer

Radioactive decay and half-life

Some isotopes are unstable (radioactive) and decay spontaneously and randomly into a stable daughter isotope, releasing radiation. The rate is constant for a given isotope and is described by the half-life.

Because the half-life is constant, the proportion of parent left falls predictably: after 11 half-life 12\frac{1}{2} remains, after 22 half-lives 14\frac{1}{4}, after 33 half-lives 18\frac{1}{8}, and so on. This is captured by:

N=N0(12)nN = N_0 \left(\tfrac{1}{2}\right)^{n}

where N0N_0 is the original number of parent atoms, NN the number remaining, and nn the number of half-lives elapsed.

Calculating an age from the parent-to-daughter ratio

When a mineral crystallises it traps parent atoms but (ideally) no daughter. Over time, parent decays to daughter, so the parent-to-daughter ratio records the age. The method:

  1. Find the fraction of parent remaining from the ratio.
  2. Work out how many half-lives that fraction corresponds to.
  3. Multiply the number of half-lives by the half-life to get the age.

The main isotopic systems

The half-life must suit the age being measured:

  • Uranium-lead (U-Pb). Very long half-lives (hundreds of millions to billions of years); used to date the oldest rocks and the age of the Earth.
  • Potassium-argon (K-Ar). Intermediate half-life; widely used to date igneous rocks (for example lavas and ash bands).
  • Carbon-14 (C-14). Short half-life (about 57305730 years); used only for recent organic material (up to roughly 5000050\,000 to 6000060\,000 years).

Assumptions and limitations

Radiometric ages are only valid if certain assumptions hold:

  • The system has remained closed (no parent or daughter added or lost since formation).
  • The initial amount of daughter is known (ideally zero, or correctable).
  • A suitable material and isotope are used for the age range.

If the rock has been heated, weathered or altered, daughter (for example argon gas) can escape, giving a wrong age.

Combining absolute and relative dating

Radiometric dating gives absolute ages (in years) but only for suitable minerals (mostly igneous). Relative dating orders all the rocks (including sediments). Combining them, an absolute date from a lava or ash band can be tied into a relative sequence, calibrating the geological time scale in years.

Examples in context

Example 1. Dating the Earth with uranium-lead. The very long half-lives of the uranium-lead system allow it to date the oldest meteorites and minerals, giving the age of the Earth at about 4.64.6 billion years.

Example 2. Calibrating an ash band. Potassium-argon dating of a volcanic ash band within a sedimentary sequence ties an absolute age into the relative succession, anchoring the surrounding biozones in years.

Try this

Q1. Define the half-life of a radioactive isotope. [1 mark]

  • Cue. The time taken for half of the parent atoms in a sample to decay to the daughter isotope.

Q2. A mineral has a parent-to-daughter ratio of 1:11{:}1. State how many half-lives have elapsed. [1 mark]

  • Cue. One half-life (half the parent remains, half has become daughter).

Q3. Explain why uranium-lead, not carbon-14, is used to date a 33 billion year old rock. [2 marks]

  • Cue. Carbon-14 has a short half-life and is useful only up to about 6000060\,000 years; uranium-lead has very long half-lives suited to dating ancient rocks billions of years old.

Exam-style practice questions

Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

OCR H414/01 20204 marksA mineral contains a parent isotope and its stable daughter in the ratio 1 part parent to 3 parts daughter. The half-life of the parent is 700 million years. Calculate the age of the mineral, showing your working.
Show worked answer →

Convert the ratio to a fraction remaining, count half-lives, then multiply.

Fraction of parent remaining. The ratio is 11 parent to 33 daughter, so out of every 44 original parent atoms, 11 remains and 33 have decayed. The fraction of parent remaining is 14\frac{1}{4}.

Number of half-lives. After each half-life the parent halves: after 11 half-life 12\frac{1}{2} remains, after 22 half-lives 14\frac{1}{4} remains. So 14\frac{1}{4} corresponds to 22 half-lives.

Age.

age=2×700 Ma=1400 million years\text{age} = 2 \times 700\ \mathrm{Ma} = 1400\ \mathrm{million\ years}

The mineral is 1.41.4 billion years old. Markers reward converting the ratio to 14\frac{1}{4} remaining, recognising this as 22 half-lives, and multiplying by the half-life.

OCR H414/01 20184 marksExplain why carbon-14 dating cannot be used to date a granite that is 400 million years old, and name a more suitable isotopic system.
Show worked answer →

Tie the half-life to the age range, then name an alternative.

Carbon-14 has a short half-life (about 5730 years5730\ \mathrm{years}). After about 1010 half-lives (roughly 5000050\,000 to 60000 years60\,000\ \mathrm{years}) so little carbon-14 remains that it cannot be measured accurately. A granite 400 million400\ \mathrm{million} years old is far beyond this range, so essentially all the carbon-14 would have decayed and no measurable age could be obtained.

Carbon-14 also needs organic carbon, which a granite does not contain.

A more suitable system. Uranium-lead (or potassium-argon) dating, which uses isotopes with very long half-lives (hundreds of millions to billions of years), is suitable for dating ancient igneous rocks such as granite.

Markers reward the short half-life and limited range of carbon-14 (and its need for organic material) and naming a long-half-life system (uranium-lead or potassium-argon).

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