Radioactive decay is a fundamental concept in nuclear physics, describing the spontaneous transformation of unstable atomic nuclei into more stable forms. This process is pivotal in understanding nuclear reactions, energy production, and various applications in medicine, industry, and environmental science. For students preparing for the College Board AP Physics 2: Algebra-Based exam, mastering radioactive decay is essential for both theoretical comprehension and practical problem-solving.

Radioactive decay refers to the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay leads to the transformation of the parent nucleus into a daughter nucleus, which may be a different element or isotope. The stability of a nucleus depends on the balance between protons and neutrons, and when this balance is disrupted, the nucleus becomes radioactive.

There are several types of radioactive decay, each characterized by the emission of different particles or radiation:

• Alpha Decay: Involves the emission of an alpha particle, which consists of 2 protons and 2 neutrons. This process decreases the atomic number by 2 and the mass number by 4, resulting in a new element.
• Beta Decay: Involves the transformation of a neutron into a proton with the emission of a beta particle (electron) and an antineutrino, or the transformation of a proton into a neutron with the emission of a positron and a neutrino. This changes the atomic number by ±1.
• Gamma Decay: Entails the emission of gamma rays, which are high-energy photons. Gamma decay usually accompanies other types of decay, leading to the release of excess energy without changing the atomic number or mass number.
• Positron Emission: A form of beta decay where a proton is converted into a neutron, emitting a positron and a neutrino. This reduces the atomic number by one.

The half-life of a radioactive isotope is the time required for half of the nuclei in a sample to undergo decay. It is a characteristic property of each radioactive isotope and is independent of initial quantity, temperature, pressure, or chemical state. The concept of half-life is crucial for calculating the age of objects, managing nuclear waste, and determining dosages in medical treatments. The mathematical representation of half-life (\( t_{1/2} \)) can be derived from the decay constant (\( \lambda \)): $$ t_{1/2} = \frac{\ln(2)}{\lambda} $$ Where \( \ln(2) \) is the natural logarithm of 2.

The decay law describes the number of undecayed nuclei as a function of time. It is given by the exponential decay formula: $$ N(t) = N_0 e^{-\lambda t} $$ Where:

• N(t):b
Number of undecayed nuclei at time \( t \)
• N₀: Initial number of nuclei at time \( t = 0 \)
• \( \lambda \): Decay constant, representing the probability of decay per unit time

This equation highlights that the decay process is random and memoryless, meaning the probability of decay in the next instant is independent of how much time has already elapsed.

A radioactive series, or decay chain, occurs when a radioactive isotope undergoes a series of decays to reach a stable end product. Each step in the series involves different types of decay and intermediate isotopes. An example is the decay series of Uranium-238: $$ \text{U-238} \rightarrow \text{Th-234} \rightarrow \text{Pa-234} \rightarrow \text{U-234} \rightarrow \ldots \rightarrow \text{Pb-206} $$ Understanding decay series is important in fields like radiometric dating, where the age of rocks and minerals is determined based on the ratios of parent and daughter isotopes.

Radioactive decay has numerous applications across various fields:

• Medicine: Radioisotopes are used in diagnostic imaging (e.g., PET scans) and in cancer treatment through targeted radiation therapy.
• Archaeology: Carbon-14 dating utilizes the half-life of C-14 to determine the age of organic materials.
• Nuclear Energy: Understanding radioactive decay is essential for the management and utilization of nuclear fuels in reactors.
• Environmental Science: Radioisotopes help trace pollutants and study environmental processes.
• Industrial Applications: Radioactive sources are used in non-destructive testing and material analysis.

Radioactive decay is inherently probabilistic. The likelihood of a nucleus decaying in a given time interval is described by the decay constant (\( \lambda \)). The expected number of decays over time can be predicted, but the exact moment of decay for any individual nucleus cannot be determined. Key statistical concepts include:

• Exponential Distribution: The time between decays follows an exponential distribution, characterized by the decay constant.
• Poisson Distribution: The number of decay events in a fixed interval of time can be modeled by a Poisson distribution.

Various conservation laws govern radioactive decay processes:

• Conservation of Mass-Energy: Total mass-energy remains constant. Energy is released or absorbed during decay.
• Conservation of Charge: Total electrical charge is conserved. The emitted particles carry charge that balances the change in the nucleus.
• Conservation of Momentum: Momentum is conserved in decay events, affecting the recoil of the nucleus.
• Conservation of Lepton Number: Leptons, such as electrons and neutrinos, are conserved in beta decays.

Detecting radioactive decay involves measuring emitted particles or radiation. Common detection methods include:

• Geiger-Müller Counters: Detect ionizing particles by measuring electrical pulses produced during their passage through the gas-filled tube.
• Scintillation Detectors: Use materials that emit light when struck by radiation, with the light being converted into electrical signals.
• Semiconductor Detectors: Utilize semiconductor materials to detect energy deposited by radiation, providing precise energy measurements.
• Cloud Chambers and Bubble Chambers: Visualize the paths of charged particles as trails of condensed vapor or bubbles.

The energy released during radioactive decay can be calculated using the mass-energy equivalence principle. The mass difference (\( \Delta m \)) between the parent and daughter nuclei is converted into energy (\( E \)): $$ E = \Delta m \cdot c^2 $$ Where:

• \( \Delta m \): Mass defect, the difference in mass between reactants and products
• \( c \):b> Speed of light in vacuum (\( 3 \times 10^8 \) m/s)

This energy is carried away by the emitted particles and radiation, contributing to their kinetic energy.

In decay chains, secular equilibrium occurs when the half-life of the parent isotope is much longer than that of its decay products. Under these conditions, the activity (decay rate) of the daughter isotopes becomes equal to that of the parent. This equilibrium simplifies calculations in systems where multiple radioactive species are present, such as in natural radioactive materials.

Radioactive decay is the foundation of various dating methods used to determine the age of materials:

• Carbon-14 Dating: Measures the decay of C-14 in organic materials to estimate ages up to ~50,000 years.
• Uranium-Lead Dating: Utilizes the decay of U-238 to Pb-206 and U-235 to Pb-207 for dating geological formations billions of years old.
• Potassium-Argon Dating: Based on the decay of K-40 to Ar-40, used in volcanic rocks and archaeological samples.

Exposure to radioactive materials poses significant health risks due to ionizing radiation, which can damage biological tissues and DNA. Understanding radioactive decay is crucial for:

• Radiation Protection: Implementing measures to minimize exposure in medical, industrial, and nuclear settings.
• Nuclear Waste Management: Safely storing and disposing of radioactive waste to prevent environmental contamination.
• Medical Treatments: Balancing the benefits of radiation therapy against potential side effects.

• Radioactive decay is the spontaneous transformation of unstable nuclei into more stable forms.
• Types of decay include alpha, beta, and gamma, each with distinct particles emitted and effects.
• Half-life is a crucial concept for predicting decay rates and applications like dating and medicine.
• Understanding decay laws and conservation principles is essential for analyzing nuclear processes.
• Radioactive decay has widespread applications in various scientific and industrial fields.