GCSE — physicsHard5 min read

Half-Life and Radioactive Dating

Half-life calculations; radioactive dating; decay curves

Tutor AI Team
Tutor AI Team

# Half-Life and Radioactive Dating — GCSE Physics

Radioactive decay is random — we can't predict when any single nucleus will decay. But with large numbers of nuclei, we can predict how the activity of a sample changes over time using the concept of half-life.

1. What Is Half-Life?

Half-life (t1/2t_{1/2}) is the time it takes for:

  • The number of radioactive nuclei in a sample to halve, OR
  • The activity (count rate) of a sample to halve

Half-life is constant for a given isotope — it doesn't matter how much of the substance you start with.

Examples of Half-Lives

Isotope Half-Life Use
Uranium-238 4.5 billion years Dating rocks
Carbon-14 5730 years Dating organic material
Iodine-131 8 days Medical thyroid treatment
Technetium-99m 6 hours Medical imaging
Polonium-218 3 minutes Research

2. Decay Curves

A decay curve is a graph of activity (or number of nuclei) against time. It shows exponential decay.

Reading Half-Life from a Graph

  1. Find the initial activity (count rate) on the y-axis
  2. Find the value at half the initial activity
  3. Read across to the curve, then down to the x-axis
  4. This time = one half-life

Check: the second half-life should end at ¼ of the original value.

3. Half-Life Calculations

Method: Repeated Halving

After nn half-lives: Remaining=Initial×(12)n\text{Remaining} = \text{Initial} \times \left(\frac{1}{2}\right)^n

Half-lives (nn) Fraction remaining Percentage
0 1 100%
1 1/2 50%
2 1/4 25%
3 1/8 12.5%
4 1/16 6.25%
5 1/32 3.125%

Finding the Number of Half-Lives

n=total timehalf-lifen = \frac{\text{total time}}{\text{half-life}}

4. Radioactive Dating

Carbon Dating (Archaeological)

  • Living organisms absorb carbon-14 from the atmosphere
  • When they die, no more C-14 is absorbed
  • C-14 decays with a half-life of 5730 years
  • By measuring how much C-14 remains, we can calculate when the organism died
  • Works for objects up to ~50,000 years old

Uranium-Lead Dating (Geological)

  • Uranium-238 decays to lead-206 with a half-life of 4.5 billion years
  • By measuring the ratio of U-238 to Pb-206 in a rock, its age can be estimated
  • Used to date very old rocks (billions of years)
  • This is how we know the Earth is ~4.5 billion years old

5. Uses Based on Half-Life

Short half-life (hours/days) — medical tracers:

  • Active enough to be detected
  • Decays quickly so patient isn't exposed for long

Medium half-life — industrial/home use:

  • Smoke detectors use americium-241 (half-life 432 years)

Long half-life — dating:

  • Carbon-14 (5730 years) for archaeology
  • Uranium-238 (4.5 billion years) for geology

Worked Example: Example 1

Problem

Question: A sample has an activity of 800 Bq. The half-life is 3 hours. What is the activity after 9 hours?

Number of half-lives: n=9/3=3n = 9/3 = 3 800→400→200→100 Bq800 \rightarrow 400 \rightarrow 200 \rightarrow 100 \text{ Bq}

Solution

Worked Example: Example 2

Problem

Question: The activity of a source drops from 6400 Bq to 200 Bq in 10 hours. Calculate the half-life.

6400→3200→1600→800→400→2006400 \rightarrow 3200 \rightarrow 1600 \rightarrow 800 \rightarrow 400 \rightarrow 200 = 5 half-lives

Half-life = 10/5=210/5 = 2 hours.

Solution

Worked Example: Example 3

Problem

Question: A sample of carbon-14 has 25% of the original amount remaining. How old is the sample? (t1/2=5730t_{1/2} = 5730 years)

100% → 50% (1 half-life) → 25% (2 half-lives)

Age = 2×5730=11,4602 \times 5730 = 11{,}460 years.

Solution

Worked Example: Example 4

Problem

Question: A radioactive source has a half-life of 6 hours. It currently has an activity of 4000 Bq. What was its activity 18 hours ago?

18 hours = 3 half-lives. Working backwards (activity was higher in the past):

4000×23=4000×8=32,0004000 \times 2^3 = 4000 \times 8 = 32{,}000 Bq

Solution

7. Radioactive Waste

Nuclear power produces radioactive waste with varying half-lives:

Category Half-life Disposal
Low-level Short Sealed in containers, buried in shallow sites
Intermediate Medium Encased in concrete, stored underground
High-level Very long (thousands of years) Vitrified (turned to glass), stored deep underground

High-level waste remains dangerous for thousands of years — disposal is a major challenge.

8. Practice Questions

    1. Define half-life. (2 marks)
    1. A source has an activity of 1600 Bq and a half-life of 4 days. Calculate the activity after 16 days. (3 marks)
    1. The activity of a source decreases from 12,000 Bq to 750 Bq in 8 hours. Calculate the half-life. (3 marks)
    1. A wooden artefact contains 12.5% of the original carbon-14. Estimate its age. (t1/2t_{1/2} of C-14 = 5730 years) (3 marks)
    1. Explain why medical tracers need a short half-life. (2 marks)

    Answers

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Summary

  • Half-life = time for activity (or number of nuclei) to halve
  • After nn half-lives: remaining = initial × (1/2)n(1/2)^n
  • Decay curves show exponential decay
  • Carbon dating: C-14 (t1/2t_{1/2} = 5730 years) for archaeology
  • Uranium dating: U-238 (t1/2t_{1/2} = 4.5 billion years) for geology
  • Half-life determines the isotope's suitability for different uses
Tags:
#gcse#physics#atomic-structure#half-life#radioactive-dating#decay

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