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Calculate remaining amount after radioactive decay, time for a substance to reach a target amount, or determine half-life from initial and final amounts. Covers nuclear isotopes, carbon dating, and drug pharmacokinetics.
Half-life = time for half a radioactive substance to decay. After 1 half-life: 50% remains. After 2: 25%. After 10: 0.098%. Formula: N(t) = N₀ × (½)^(t/t½).
| Isotope | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating to 50,000 yr |
| Uranium-238 | 4.47 billion years | Geological dating |
| Iodine-131 | 8.02 days | Thyroid cancer treatment |
| Technetium-99m | 6 hours | Medical imaging (most common) |
| Fluorine-18 | 110 minutes | PET scan imaging |
| Radon-222 | 3.82 days | Radioactive gas, lung cancer risk |
| Cobalt-60 | 5.27 years | Cancer radiation therapy |
Tc-99m (6-hour half-life): ideal for imaging — active long enough for scan, decays quickly after. After 10 half-lives (60 hr): only 0.098% remains. I-131 (8-day): treats thyroid cancer with targeted radiation.
¹⁴C/¹²C ratio constant in living organisms. After death, ¹⁴C decays. Measuring remaining ¹⁴C reveals time since death. Accurate to ~50,000 years.
Half-life is the defining characteristic of every radioactive isotope. It determines how long a radioactive material remains hazardous, when a medical imaging agent will clear from the body, how geologists date ancient rocks, and how archaeologists date ancient organic materials. The exponential nature of radioactive decay means the rate is always proportional to the amount remaining.
Nuclear medicine exploits short-lived isotopes for diagnosis and therapy. Diagnostic isotopes need half-lives long enough to perform the procedure but short enough to minimize patient radiation exposure. Technetium-99m (t½ = 6 hours) is the ideal diagnostic isotope — used in bone scans, cardiac perfusion studies, and kidney imaging. After 60 hours (10 half-lives), only 0.1% of the original dose remains. Therapeutic isotopes need longer half-lives to deliver sustained radiation: I-131 (8 days) for thyroid cancer, Lu-177 (6.6 days) for neuroendocrine tumors.
Nuclear power plants produce radioactive waste with enormously varying half-lives. Short-lived fission products (Kr-85: 10.7 years, Cs-137: 30 years) require storage for 300-600 years (10-20 half-lives). Long-lived actinides (Pu-239: 24,100 years, U-235: 704 million years) require geological isolation for hundreds of thousands of years. This is why nuclear waste disposal is a profound scientific and political challenge — the timescales exceed any human institution's lifespan.
Radioactive decay follows first-order kinetics: dN/dt = -λN, where λ is the decay constant. Solving this differential equation: N(t) = N₀e^(-λt). Since N = N₀/2 when t = t½: e^(-λt½) = 0.5, so λ = ln(2)/t½ = 0.6931/t½. Activity A = λN = λN₀e^(-λt). One Becquerel (Bq) = one decay per second. One Curie (Ci) = 3.7×10¹⁰ decays/second (original unit based on Ra-226 decay rate).
Radioactive decay is a spontaneous nuclear process in which an unstable atom emits radiation (alpha particles, beta particles, or gamma rays) and transforms into a different, more stable nucleus. The rate of decay is characterized by the half-life (t½) — the time required for exactly half of any initial quantity of radioactive nuclei to decay. Half-life is constant for any given isotope and cannot be altered by chemical reactions, temperature, pressure, or any external condition.
Half-life spans an enormous range: from nanoseconds for synthetic trans-uranic isotopes to billions of years for naturally occurring radioactive elements like uranium-238. This range makes radioactive decay useful for applications from medical imaging (minutes) to geological dating (billions of years). The exponential nature of decay ensures that while half is lost in each half-life, the sample never reaches exactly zero — only approaches it asymptotically.
Decay follows first-order kinetics: dN/dt = -λN, where λ is the decay constant (proportional to the probability of decay per unit time). Solving this differential equation: N(t) = N₀ × e^(-λt). Since N = N₀/2 when t = t½: e^(-λt½) = 0.5, therefore λ = ln(2)/t½ = 0.6931/t½. Alternative form: N(t) = N₀ × (1/2)^(t/t½). Activity A = λN (decays per second) — measured in Becquerel (Bq, 1 decay/second) or Curie (Ci, 3.7×10¹⁰ decays/second, based on Ra-226 decay rate). After n half-lives: fraction remaining = (1/2)ⁿ.
Diagnostic nuclear medicine requires isotopes with half-lives long enough to complete the imaging procedure but short enough to minimize patient radiation exposure. Technetium-99m (6-hour half-life): the most commonly used medical imaging isotope worldwide — used in bone scans, cardiac stress tests, kidney function studies, and lung perfusion scans. After 60 hours (10 half-lives): 0.098% remains. Fluorine-18 (110-minute half-life): used in PET (Positron Emission Tomography) scans for cancer staging. Iodine-123 (13.2-hour half-life): thyroid imaging. Gallium-68 (68-minute half-life): PET imaging for neuroendocrine tumors.
Carbon-14 (t½ = 5,730 years) is produced in the upper atmosphere by cosmic ray interaction with nitrogen. Living organisms maintain constant ¹⁴C/¹²C ratio through carbon exchange with the atmosphere. After death, ¹⁴C decays without replenishment. Measuring the remaining ¹⁴C ratio reveals time since death. Accurate range: 500-50,000 years (±30-80 years for recent samples). Calibrated using dendrochronology (tree rings) for greater accuracy. Uranium-Lead dating (U-238 t½ = 4.47 Gy, Pb-206 product): used for geological ages up to 4.5 billion years. Potassium-40/Argon-40 (K-40 t½ = 1.25 Gy): dates rocks and minerals 100,000 years to billions of years. These systems have revealed Earth's 4.54 billion year age.
Drug half-life is the time for plasma drug concentration to decrease by 50%. After 4-5 half-lives, 94-97% of the drug is eliminated — the body is considered "cleared." Ibuprofen: t½ = 2 hours, cleared in 10 hours. Diazepam (Valium): t½ = 20-100 hours, accumulates with repeated dosing. Amiodarone (heart medication): t½ = 40-55 days — takes months to reach steady state and months to clear. This extremely long half-life means dosing adjustments have very delayed effects. Loading doses are needed for drugs with long half-lives to reach therapeutic levels quickly without waiting 5-10 half-lives for steady state.
Nuclear power plants produce radioactive waste with enormously varying half-lives. Fission products from uranium splitting: Cesium-137 (t½ = 30 years, gamma emitter, bioaccumulates like potassium), Strontium-90 (t½ = 28.8 years, incorporates into bone like calcium). These require safe storage for 300-600 years (10-20 half-lives). Long-lived actinides: Plutonium-239 (t½ = 24,100 years), Am-241 (t½ = 432 years). These require geological isolation (deep repositories) for hundreds of thousands of years — far exceeding any human institutional lifespan, creating the fundamental challenge of nuclear waste management.