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How does radioactive decay let geologists put an actual number of years on a rock?

Explain radioactive decay and half-life, and calculate the age of a sample or the fraction of parent remaining using the number of half-lives that have passed (Virginia 2018 Earth Science SOL ES.9).

A SOL-level answer on absolute dating for the Virginia Earth Science EOC: what radioactive decay and half-life mean, the parent-to-daughter ratio, how to count half-lives to find an age or the fraction remaining, why carbon-14 dates young organic material and uranium dates ancient rock, and how Earth's age (about 4.6 billion years) is known, with worked calculations.

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  1. What this topic is asking
  2. What radioactive decay is
  3. Half-life
  4. The two key calculations
  5. Choosing the right isotope
  6. Try this

What this topic is asking

Virginia Earth Science SOL standard ES.9 asks you to use radioactive decay to put a number of years on a rock or fossil, the absolute dating half of Earth's history. The EOC tests this with a half-life calculation: it gives a half-life and either a time or a parent-to-daughter ratio and asks for the age or the fraction remaining. This is the most quantitative Earth-history skill, so practice the counting until it is automatic.

What radioactive decay is

Because the rate is constant, the ratio of parent to daughter in a sample records how long decay has been going on. A sample that is mostly parent is young; one that is mostly daughter is old.

Half-life

The two key calculations

To find the number of half-lives that have passed:

number of half-lives=elapsed timehalf-life\text{number of half-lives} = \frac{\text{elapsed time}}{\text{half-life}}

To find the age of a sample once you know how many half-lives have passed (often from the parent-to-daughter ratio):

age=(number of half-lives)×(half-life)\text{age} = (\text{number of half-lives}) \times (\text{half-life})

A parent-to-daughter ratio converts to a fraction of parent remaining: 1:1 is half remaining (1 half-life), 1:3 is one-quarter remaining (2 half-lives), 1:7 is one-eighth remaining (3 half-lives).

Choosing the right isotope

Radiometric dating of the oldest meteorites and Earth materials gives Earth's age as about 4.6 billion years, a headline fact for the course.

Try this

Q1. An isotope has a half-life of 2000 years. What fraction of the parent remains after 6000 years? [2]

  • Cue. 60002000=3\frac{6000}{2000} = 3 half-lives, so 12→14→18\frac{1}{2} \to \frac{1}{4} \to \frac{1}{8}; one-eighth remains.

Q2. Explain why carbon-14 cannot be used to date a rock that is 500 million years old. [2]

  • Cue. Carbon-14's half-life is only about 5700 years, so after 500 million years essentially all of it has decayed away; there is no parent left to measure. A long half-life isotope like uranium-238 is used instead.

Exam-style practice questions

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

VA Earth Science SOL 2023 (style)1 marksA radioactive isotope has a half-life of 5000 years. After 15000 years, what fraction of the original parent isotope remains? (A) one-half. (B) one-quarter. (C) one-eighth. (D) one-sixteenth.
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A 1-point multiple-choice calculation item.

The correct answer is C. In 15000 years there are 150005000=3\frac{15000}{5000} = 3 half-lives. Each half-life halves the parent: after one, 12\frac{1}{2}; after two, 14\frac{1}{4}; after three, 18\frac{1}{8}. So one-eighth of the parent remains.

The test rewards counting half-lives (time divided by half-life) and halving the parent that many times.

VA Earth Science SOL 2024 (style)2 marksA rock sample contains a radioactive parent isotope and its stable daughter in a ratio of 1 part parent to 3 parts daughter. The half-life of the parent is 1.3 billion years. (a) How many half-lives have passed? (b) Calculate the age of the rock.
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A 2-point absolute-dating calculation.

(a) 1 point: a 1:3 parent-to-daughter ratio means one-quarter of the parent is left (14\frac{1}{4} parent, 34\frac{3}{4} decayed). Going from 1 to 12\frac{1}{2} to 14\frac{1}{4} is 2 half-lives.
(b) 1 point: age = number of half-lives times the half-life = 2×1.3=2.62 \times 1.3 = 2.6 billion years.

Markers reward converting the ratio to the fraction remaining and the number of half-lives in (a), and multiplying by the half-life in (b).

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