Radioactive Half-Life and Decay Kinetics

The half-life T_½ of a radioactive nuclide is the time for half of the nuclei in a sample to decay (or for the activity to fall to half), and the number of undecayed nuclei follows the exponential decay law N = N₀e^−λt = N₀(½)^t/T½, where the decay constant λ = ln 2 ÷ T_½. In brief: the activity A = λN (in becquerels, 1 Bq = 1 decay per second) decreases with the same law. After n half-lives, the fraction remaining is (½)ⁿ: ½, ¼, ⅛, 1/16, …. The half-life is a constant for each nuclide, ranging from fractions of a second to billions of years, and is not changed by temperature, pressure or chemical form.

Radiocarbon dating, medical tracers and the disposal of nuclear waste all depend on how quickly radioactive nuclei decay. This guide explains the random nature of decay, the exponential law, activity and the decay constant, compares half-lives in a table, works through numerical examples and corrects common mistakes.

Why is decay random?

Each unstable nucleus has a fixed probability per unit time λ of decaying, independent of its history and of the other nuclei. It is impossible to predict when a particular nucleus will decay, but for a large sample, the statistics are very reliable. This is like throwing dice: if each of a large number of dice is rolled every second and removed when it shows a six, the number left falls with a regular exponential pattern. In a given short time interval, the number of decays is, on average, proportional to the number of undecayed nuclei: dN/dt = −λN. The fact that λ is constant, independent of the age of the nucleus, shows that nuclei do not “age”: a nucleus that has survived for a billion years is as likely to decay in the next second as a newly made one. The randomness gives statistical fluctuations of about √N in a count of N decays.

What is the exponential decay law?

Solving dN/dt = −λN gives N = N₀ e^−λt. The half-life is found by setting N = N₀/2: e^−λT½ = ½, so T_½ = ln 2 ÷ λ = 0.693 ÷ λ. Therefore N = N₀(½)^t/T½. The mean lifetime is τ = 1 ÷ λ = 1.44 T_½. The graph of N against t is a decreasing curve, which halves in each interval of one half-life; the graph of ln N against t is a straight line with gradient −λ. The same law applies to the mass of the radioactive isotope remaining, the activity, and the count rate (corrected for background). After 10 half-lives, the fraction is about 0.1 percent (1/1,024); after 20, about one in a million.

What are activity and the decay constant?

The activity A is the number of decays per second: A = |dN/dt| = λN = A₀e^−λt. The SI unit is the becquerel (Bq); the older unit is the curie (1 Ci = 3.7 × 10¹⁰ Bq, roughly the activity of 1 g of radium). The number of nuclei can be found from the mass: N = (m ÷ M) N_A, where M is the molar mass. A short half-life means a large λ and a high activity for the same number of atoms: such sources are intense but short-lived. A long half-life gives a low activity but persists for a long time (this is the problem of nuclear waste). The specific activity is the activity per unit mass.

Half-lives compared

Worked example: decay calculations

1. Number of half-lives. Iodine-131 (T_½ = 8.0 days): after 24 days, there have been 3 half-lives, so the fraction remaining is (½)³ = 1/8 = 12.5 percent. A sample with an initial activity of 400 MBq has 50 MBq.

2. Decay constant. For carbon-14: λ = 0.693 ÷ (5,730 × 3.156 × 10⁷ s) = 3.83 × 10⁻¹² s⁻¹.
3. Activity of radium. 1.00 g of radium-226 (T_½ = 1,600 y): N = (1.00 ÷ 226) × 6.022 × 10²³ = 2.66 × 10²¹ nuclei. λ = 0.693 ÷ (1,600 × 3.156 × 10⁷) = 1.37 × 10⁻¹¹ s⁻¹. A = λN = 3.66 × 10¹⁰ Bq = 0.99 Ci, the origin of the curie.
4. Time from a fraction. A sample has 25 percent of its original carbon-14: t = 2 half-lives = 11,460 years. For 60 percent remaining: t = T_½ × log₂(1 ÷ 0.60) = 5,730 × 0.737 = 4,220 years.
5. Technetium-99m. A dose of 400 MBq with T_½ = 6.0 h is given. After 12 h (2 half-lives), the activity is 100 MBq; after 24 h (4 half-lives), 25 MBq. The short half-life limits the radiation dose to the patient.
6. Using the exponential form. For a sample with N₀ = 1.0 × 10⁶ nuclei and λ = 0.10 per hour, after 5.0 hours: N = 10⁶ × e^−0.50 = 6.1 × 10⁵. The half-life T_½ = 0.693 ÷ 0.10 = 6.9 hours.

How was the law found?

Ernest Rutherford and Frederick Soddy showed in 1902 to 1903 that the activity of a radioactive substance falls exponentially with time. Rutherford suggested in 1904 that radioactive decay could be used to measure the age of rocks, and in 1907 Bertram Boltwood used the ratio of lead to uranium to date minerals to hundreds of millions of years and more, opening the field of radiometric dating. Willard Libby developed radiocarbon dating in the late 1940s (Nobel Prize in Chemistry 1960). Modern radiometric dating of meteorites gives an age of the solar system of about 4.57 billion years.

Which mistakes are common?

Mistake 1: Thinking that the sample is gone after two half-lives

One quarter remains. The decay never reaches zero in the exponential law.

Mistake 2: Using the half-life as the time for each nucleus to decay

It is a statistical property of large numbers. A single nucleus may decay at any time.

Mistake 3: Confusing the decay constant and the half-life

T_½ = 0.693 ÷ λ. A large λ means a short half-life.

Mistake 4: Forgetting to correct for background radiation

Subtract the background count rate before analyzing the decay.