Half Life

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Although each nuclear decay is random, with a large collection of nuclei, we can statistically predict how many will decay in a certain time.

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Time taken to halve

  • Half-life, T1/2, is the time taken:
    • For the number of radioactive nuclei in a sample to halve.
    • For the activity (the number of decays per second) to halve.
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  • The activity of a sample is directly proportional to the number of nuclei remaining:
    • Activity = decay constant, λ x number of nuclei remaining.
  • Activity is related to half-life:
    • λ=ln(2)T12\lambda =\frac{ln (2)}{T_\frac{1}{2}} = 0.69 ÷ T1/2
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Activity 2

  • By knowing the activity, we can infer how many nuclei are remaining.
  • Knowing the atomic mass of an isotope and the mass of a sample of isotope enables the number of nuclei and so the activity to be found.
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  • Find the half-life of a sample of plutonium-239, which has a mass of 1200 g and an activity of 2.8 × 1012 Bq:
    • Number of moles of Pu-239 = 1200 ÷ 239 = 5.021.
    • Number of nuclei = number of moles × Avogadro’s constant = 3.02 × 1024.
    • λ = activity ÷ number of nuclei = 9.26 × 10-13.
    • So, half-life T1/2 = ln(2) ÷ λ = 7.5 × 1011 s.


The half-life of a sample is the time taken for the number of radioactive counts to fall to half the initial value.

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  • Half-life is the time taken for half of the sample to decay.
  • This is the same as saying the time taken for the count rate to halve.
  • Knowing the half-life of a substance and the number of nuclei it has means we can find its age.
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Uses of half-life

  • We can find the age of a material by considering the count rate of the material.
  • By knowing the original count rate, the current count rate, and the half-life of the material - we can use the following equation to calculate the time passed:
    • A=A0eλtA={A_0}e^{-{\lambda}t}

Graphical Method

Half life can be found in many ways, the easiest is to use a graph.

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Graphical method

  • Firstly plot a graph of count rate or number of atoms against time.
  • The plot should show exponential decay.
  • Look for the initial count rate / number of atoms (the y-intercept).
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Graphical method 2

  • Divide the initial count rate or number of atoms by two, this is the half count rate, or half the initial number of atoms.
  • Draw a line across from the half count rate to the decay curve.
  • Draw a line down to the time axis (x axis) and read off the time. This is the half-life.
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Graphical method 3

  • Each half-life will be the same length.
  • So it will take two half-lives for the sample to decay to a quarter of the initial count, or a quarter of the initial atoms.

Jump to other topics

1Measurements & Errors

2Particles & Radiation


4Mechanics & Materials


6Further Mechanics & Thermal Physics (A2 only)

7Fields & Their Consequences (A2 only)

8Nuclear Physics (A2 only)

9Option: Astrophysics (A2 only)

10Option: Medical Physics (A2 only)

11Option: Engineering Physics (A2 only)

12Option: Turning Points in Physics (A2 only)

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