Radioactivity, Uses & Dangers

Definition

• Because individual decays are random, you cannot say "this particular nucleus will decay at 3:42 pm"
• What you can measure precisely is how long it takes for a large sample to halve. This is the half-life:

the half-life of a radioactive isotope is the time taken for the number of unstable nuclei in a sample to fall to half of its original value

• Equivalently, since activity is proportional to the number of unstable nuclei, half-life is the time for the activity of the sample to drop to half its starting value
• Half-life is a property of the isotope itself. Every nucleus of a given isotope has the same probability of decaying per second, so the half-life is constant, and temperature, pressure and chemistry have no effect

Half-lives vary enormously

• Different isotopes have wildly different half-lives:

• Short half-lives mean a very high activity for a short time; long half-lives mean a much lower activity but lasting far longer than any human timescale

Halving step by step

• After each half-life, the number of unstable nuclei (and the activity) is halved:

• The pattern: after n half-lives, the fraction remaining is (1/2)ⁿ

Reading a half-life from a graph

• To find the half-life from an activity-time graph:
  1. Read off the initial activity A₀ from the y-axis at t = 0
  2. Halve it to get A₀/2
  3. Draw a horizontal line from A₀/2 across to the curve, then drop straight down to the time axis
  4. The reading on the time axis is the half-life
• It is a good idea to do this a second time (read at A₀/4) and divide the resulting time by 2 to check; you should get the same answer

Example — A radioactive sample has an initial activity of 6400 Bq. After 9 hours the activity has fallen to 800 Bq. Find the half-life.

• Step 1 — Work out the number of halvings between 6400 Bq and 800 Bq
  • 6400 → 3200 (one halving)
  • 3200 → 1600 (two halvings)
  • 1600 → 800 (three halvings)
  • So three half-lives have passed in 9 hours
• Step 2 — Divide the elapsed time by the number of half-lives
  • half-life = 9 ÷ 3 = 3 hours

Example — A sample of iodine-131 has an initial mass of 80 mg. The half-life of iodine-131 is 8 days. What mass remains after 32 days?

• Step 1 — Work out how many half-lives have passed: 32 ÷ 8 = 4 half-lives
• Step 2 — Halve the mass four times: 80 → 40 → 20 → 10 → 5 mg
• (Or use the fraction directly: (1/2)⁴ = 1/16; remaining mass = 80 × 1/16 = 5 mg)
• After 32 days, 5 mg of iodine-131 remains; the other 75 mg has decayed into other isotopes