Radioactive Decay Kinetics
Radioactive decay is a first-order kinetic process, meaning the rate of decay at any instant is directly proportional to the number of radioactive atoms present. This simple mathematical relationship has profound consequences for nuclear science, archaeology, geology, and medicine.
The Decay Law
The fundamental equation of radioactive decay is:
N(t) = N_0 * e^(-lambda * t)
where N(t) is the number of radioactive atoms remaining at time t, N_0 is the initial number, and lambda is the decay constant -- a characteristic value for each radioactive isotope that represents the probability of decay per unit time. A large decay constant means rapid decay; a small one means the isotope is long-lived.
The activity (A) of a radioactive sample, measured in decays per second, follows the same exponential pattern:
A(t) = A_0 * e^(-lambda * t) = lambda * N(t)
Activity is measured in becquerels (Bq, 1 decay per second) in SI units, or curies (Ci) in older literature, where 1 Ci = 3.7 x 10^10 Bq, roughly the activity of 1 gram of radium-226.
Half-Life
The half-life (t_1/2) is the time required for half of a radioactive sample to decay. It is related to the decay constant by:
t_1/2 = ln(2) / lambda = 0.693 / lambda
Half-lives vary enormously across isotopes:
- Polonium-214: 164 microseconds
- Radon-222: 3.82 days
- Cobalt-60: 5.27 years
- Carbon-14: 5,730 years
- Uranium-238: 4.47 billion years
- Tellurium-128: 2.2 x 10^24 years (the longest measured half-life)
After one half-life, 50% of the original atoms remain. After two half-lives, 25% remain. After ten half-lives, only about 0.1% remains. A useful rule of thumb is that after approximately 10 half-lives, a radioactive sample has decayed to negligible levels.
Radiocarbon Dating
Carbon-14 dating, developed by Willard Libby in 1949 (earning the 1960 Nobel Prize in Chemistry), exploits the 5,730-year half-life of carbon-14. Cosmic rays continuously produce carbon-14 in the upper atmosphere, where it oxidizes to CO2 and enters the biosphere. Living organisms maintain a constant ratio of carbon-14 to carbon-12 by continuously exchanging carbon with the environment. When an organism dies, this exchange stops, and the carbon-14 begins to decay.
By measuring the remaining carbon-14 activity in an artifact and comparing it to the expected activity of living material, scientists can calculate the time since death. The technique is reliable for ages up to about 50,000 years (roughly 9 half-lives). Beyond that, too little carbon-14 remains for accurate measurement.
Radiometric Dating of Rocks
For geological timescales, scientists use isotopes with much longer half-lives:
- Potassium-40 / Argon-40 (t_1/2 = 1.25 billion years): Used for dating volcanic rocks. Potassium-40 decays to argon-40, which is trapped in mineral crystals.
- Uranium-238 / Lead-206 (t_1/2 = 4.47 billion years): Used for zircon crystals in igneous rocks. Concordia diagrams comparing U-238/Pb-206 and U-235/Pb-207 ratios provide highly precise ages.
- Rubidium-87 / Strontium-87 (t_1/2 = 49.6 billion years): Used for dating ancient metamorphic and igneous rocks.
The age of the Earth (approximately 4.54 billion years) and the oldest known minerals (4.4 billion-year-old zircons from Western Australia) were determined using uranium-lead dating.
Branching Decay and Secular Equilibrium
Some nuclei can decay by more than one pathway. Potassium-40, for example, undergoes beta-minus decay to calcium-40 (89.3%) and electron capture to argon-40 (10.7%). The total decay constant is the sum of the partial decay constants for each branch.
In long decay chains, a condition called secular equilibrium can develop when the parent has a much longer half-life than any of its daughters. At secular equilibrium, the activity of every daughter in the chain equals the activity of the parent. This principle allows geologists to determine uranium content by measuring the gamma rays from daughter products like bismuth-214.
Practical Calculations
A common problem type: A hospital receives a 10.0 mCi shipment of iodine-131 (t_1/2 = 8.02 days) for thyroid therapy. What is the activity after 24 days?
Number of half-lives: 24 / 8.02 = 2.99, approximately 3 half-lives. Remaining activity: 10.0 * (1/2)^3 = 10.0 * 0.125 = 1.25 mCi.
More precisely using the exponential formula: lambda = 0.693 / 8.02 = 0.0864 per day. A = 10.0 * e^(-0.0864 * 24) = 10.0 * e^(-2.074) = 10.0 * 0.1257 = 1.26 mCi.