5.1.7

Radioactive Decay

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Radioactive Decay

When an object is radioactive it releases radioactive particles but as time passes the rate of particles decreases. This is called decay.

Random decay

Random decay

  • The decay of a radioactive substance is random and unpredictable.
  • To measure decay, we must look at the count rate over a long time to see if it decreases.
Probability

Probability

  • The probability that a given nucleus will decay in a given time is proportional to the number of nuclei. The equation for calculating the rate of decay is:
    • The rate of decay of nuclei = decay constant x the number of nuclei
    • ΔNΔt=−λN\frac{{\Delta}N}{{\Delta}t}=-{\lambda}N
      • λ{\lambda} is the decay constant.
      • NN is the number of nuclei.

The Exponential Law

The reduction in the rate of decay decreases according to an exponential pattern.

Exponential decay

Exponential decay

  • From the equation for the rate of decay, we can find an exponential relationship between the number of nuclei and time.
  • The equation for the rate of decay of nuclei is:
    • ΔNΔt=−λN\frac{{\Delta}N}{{\Delta}t}=-{\lambda}N
  • The exponential relationship corresponding to this is:
    • N=N0e−λtN={N_0}e^{-{\lambda}t}
      • N0{N_0} is the initial number of nuclei.
Activity

Activity

  • The activity of a sample can be found by using the decay constant and the number of nuclei. The equation for activity is:
    • Activity = decay constant x number of nuclei
    • A=λNA={\lambda}N
  • The exponential relationship corresponding to the activity is:
    • A=A0e−λtA={A_0}e^{-{\lambda}t}

Half-Life

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

Time taken to halve

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.
Activity

Activity

  • 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 the half life:
    • λ=ln⁡(2)T1/2=0.693T1/2λ =\frac{ \ln (2)}{T_{1/2}} = \frac{0.693}{T_{1/2}}
Activity 2

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.
Example

Example

  • 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 × 1010 s.
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Practice questions on Radioactive Decay

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  1. 1
    Decay of a radioactive substance is:Fill in the list
  2. 2
    The probability that a given nucleus will decay is proportional to the:Multiple choice
  3. 3
    $$N_0$$ represents:Multiple choice
  4. 4
    The graphs for activity and number of radioactive nuclei are identical apart from:Multiple choice
  5. 5
    A radioactive sample initially weighs 4 g. After 2 half-lives, what is the mass of the radioactive sample?Multiple choice
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Scientific Practices - Unstable Nuclei

IB Multiple Choice - Atomic Physics