Random decay, activity and half-life

Spontaneous and random

• Spontaneous: decay is not affected by external conditions such as temperature, pressure or chemical bonding.
• Random: you cannot predict which nucleus will decay next, or when. Each nucleus has a fixed probability of decaying per unit time. Count rates therefore fluctuate.

Decay constant and activity

The decay constant λ is the probability that a nucleus decays per unit time (s^-1). The activity A is the number of decays per second, measured in becquerels (Bq):

A = λN and dN/dt = -λN

The more nuclei there are, the higher the activity.

Half-life

The half-life t_1/2 is the average time for the number of undecayed nuclei (or the activity) to halve. It is linked to λ by

λ = ln 2 / t_1/2 ≈ 0.693 / t_1/2

To find the half-life from a graph of N or A against t, read the time to fall from any value to half that value; repeat for several starting points and average. After n half-lives the fraction remaining is (1/2)ⁿ.

Worked example

A sample has 3.0 x 10¹⁰ undecayed nuclei and a half-life of 2.0 hours. Find its activity.

1. λ = 0.693 / (2.0 x 3600) = 9.63 x 10^-5 s^-1
2. A = λN = 9.63 x 10^-5 x 3.0 x 10¹⁰

Answer: A = 2.9 x 10⁶ Bq

Check yourself

1. A radioactive isotope has a half-life of 8.0 days. Find its decay constant in day^-1, to 3 s.f.
   Show answer

   0.0866 day^-1. λ = ln 2 / t_1/2 = 0.693 / 8.0 = 0.0866 day^-1.

2. A sample contains 3.0 x 10¹⁰ nuclei with λ = 2.0 x 10^-6 s^-1. Find its activity, in units of 10⁴ Bq.
   Show answer

   6 x 10⁴ Bq. A = λN = 2.0 x 10^-6 x 3.0 x 10¹⁰ = 6.0 x 10⁴ Bq.

3. A sample has activity 500 Bq and half-life 30 s. Find the number of undecayed nuclei, in units of 10⁴, to 3 s.f.
   Show answer

   2.16 x 10⁴. λ = 0.693 / 30 = 0.0231 s^-1; N = A / λ = 500 / 0.0231 = 2.16 x 10⁴.

4. What fraction of the original nuclei remains undecayed after 3 half-lives? Give a decimal.
   Show answer

   0.125. (1/2)³ = 1/8 = 0.125.

5. A graph shows the activity of a sample falling from 800 Bq to 200 Bq in 12 hours. Find the half-life in hours.
   Show answer

   6 h. 800 to 400 to 200 is two half-lives, so t_1/2 = 12 / 2 = 6.0 h.

6. A source has a half-life of 2.5 h. How long does it take for its activity to fall from 6.4 x 10⁶ Bq to 4.0 x 10⁵ Bq, in hours?
   Show answer

   10 h. 6.4 x 10⁶ / 4.0 x 10⁵ = 16 = 2⁴, so 4 half-lives: 4 x 2.5 = 10 h.

7. What does it mean to say radioactive decay is random?
   • The half-life changes randomly
   • The type of radiation emitted is random
   • It is impossible to predict when a particular nucleus will decay
   • Heating a sample changes its decay rate randomly
   Show answer

   It is impossible to predict when a particular nucleus will decay. Only the probability of decay per unit time is fixed.

8. What does it mean to say radioactive decay is spontaneous?
   • It happens very quickly
   • It only happens when the nucleus is struck
   • It happens at a steady rate
   • It is not affected by external factors such as temperature or pressure
   Show answer

   It is not affected by external factors such as temperature or pressure. Nothing outside the nucleus triggers or changes the decay.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson Edexcel IAL Physics Unit 5 (WPH15) question paper, June 2023: list of data, formulae and relationships

Last updated 7 November 2026. Spotted a mistake? Tell us.