Calculate radioactive decay, half-life, remaining quantity, and time elapsed. Perfect for nuclear physics, carbon dating, medical isotopes, and scientific analysis with detailed step-by-step solutions.
Half-life is a fundamental concept in nuclear physics and chemistry that describes the time required for half of a given quantity of a radioactive substance to decay. This concept is crucial for understanding radioactive decay, carbon dating, nuclear medicine, and many scientific applications.
The half-life of a substance is constant and characteristic of each radioactive isotope. It provides a way to predict how much of a radioactive material will remain after any given time period, making it invaluable for scientific research, medical applications, and dating ancient artifacts.
Where: N(t) = remaining quantity, N₀ = initial quantity, λ = decay constant, t = time, t₁/₂ = half-life
Where: λ = decay constant, τ = mean lifetime, t₁/₂ = half-life
| Isotope | Half-Life | Decay Type | Primary Application |
|---|---|---|---|
| Carbon-14 | 5,730 years | Beta decay | Archaeological dating |
| Uranium-238 | 4.468 billion years | Alpha decay | Nuclear fuel, dating |
| Technetium-99m | 6.01 hours | Gamma decay | Medical imaging |
| Iodine-131 | 8.02 days | Beta decay | Thyroid treatment |
| Plutonium-239 | 24,100 years | Alpha decay | Nuclear weapons/power |
| Cobalt-60 | 5.27 years | Beta/Gamma | Cancer treatment |
| Radon-222 | 3.82 days | Alpha decay | Environmental monitoring |
Alpha Decay: Emission of an alpha particle (helium nucleus). Reduces atomic number by 2 and mass number by 4. Common in heavy elements like uranium and radium.
Beta Decay: Conversion of a neutron to a proton (β⁻) or proton to neutron (β⁺). Changes atomic number by ±1 while maintaining mass number. Common in carbon-14 and other isotopes.
Gamma Decay: Emission of high-energy electromagnetic radiation. No change in atomic or mass number, but releases excess nuclear energy. Often accompanies other decay types.
Electron Capture: Absorption of an inner orbital electron by the nucleus. Reduces atomic number by 1. Common in artificially produced isotopes.
Archaeological Dating: Carbon-14 dating allows scientists to determine the age of organic materials by measuring the remaining carbon-14 content and calculating how many half-lives have passed.
Nuclear Medicine: Precise half-life calculations ensure proper dosing of radioactive isotopes for medical imaging and treatment, balancing effectiveness with patient safety.
Nuclear Waste Management: Understanding half-lives is crucial for safely storing radioactive waste, determining how long materials remain hazardous, and planning storage facilities.
Nuclear Power: Half-life calculations help determine fuel efficiency, waste production, and safety protocols in nuclear reactors.
Independence from External Conditions: Radioactive decay rates are unaffected by temperature, pressure, chemical environment, or electromagnetic fields. The half-life remains constant under all normal conditions.
Random Nature: While the overall decay rate is predictable for large numbers of atoms, individual decay events are random and cannot be predicted with certainty.
Statistical Nature: Half-life calculations become more accurate with larger sample sizes. Small samples may show apparent variations due to statistical fluctuations.
Mean Lifetime vs Half-Life: Mean lifetime (τ) represents the average time an atom exists before decaying, while half-life represents the time for half the sample to decay. They are related by τ = t₁/₂ / ln(2).
Decay Chains: Many radioactive isotopes decay into other radioactive isotopes, creating decay chains. Each step has its own half-life, complicating calculations for complex systems.
Secular Equilibrium: In long decay chains where the parent half-life is much longer than daughter half-lives, a steady state is reached where production equals decay for intermediate products.