Seneca
Sign up

Radioactive Decay

Physics: IB Diploma Higher Level, topic 5.1.7

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.

Test yourself on Radioactive Decay

  • A radioactive sample initially weighs 4 g. After 2 half-lives, what is the mass of the radioactive sample?

Practise this topic for free