Glossary

4.1 Radioactive decay law and half-life

4 min read•august 9, 2024

is a natural process where unstable atomic nuclei emit particles or energy. This phenomenon follows an pattern, described by a mathematical function. The decay rate is proportional to the number of radioactive nuclei present.

is the time it takes for half of a radioactive sample to decay. It's crucial for understanding decay rates and is used in radiometric dating. The and are key concepts that help quantify radioactive decay processes.

Radioactive Decay Fundamentals

Understanding Radioactive Decay and Exponential Decay

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  • Radioactive decay involves spontaneous emission of particles or energy from unstable atomic nuclei
  • Process transforms unstable isotopes into more stable daughter nuclei
  • Follows exponential decay pattern described by mathematical function
  • Decay rate proportional to the number of radioactive nuclei present
  • Exponential decay formula: N(t)=N0e−λtN(t) = N_0 e^{-λt}
    • N(t) represents number of radioactive nuclei at time t
    • N₀ denotes initial number of radioactive nuclei
    • λ (lambda) signifies decay constant
    • t indicates elapsed time
  • Graphical representation shows rapid initial decrease followed by gradual decline

Decay Constant and Activity

  • Decay constant (λ) measures probability of a single atom decaying per unit time
  • Unique value for each radioactive isotope
  • Expressed in units of inverse time (s⁻¹, min⁻¹, or yr⁻¹)
  • Activity quantifies rate of radioactive decay in a sample
  • Defined as number of decays per unit time
  • Calculated using formula: A=λNA = λN
    • A represents activity
    • λ denotes decay constant
    • N indicates number of radioactive nuclei
  • Activity units include becquerels (Bq) or curies (Ci)

Decay Rate and Its Significance

  • Decay rate refers to number of radioactive nuclei decaying per unit time
  • Equivalent to activity of a radioactive sample
  • Varies among different isotopes (uranium-238 decays slowly, while radon-222 decays rapidly)
  • Influences radiation exposure and safety considerations in nuclear applications
  • Decay rate formula: −dNdt=λN-\frac{dN}{dt} = λN
    • dN/dt represents rate of change in number of nuclei
    • Negative sign indicates decrease in number of nuclei over time

Half-Life and Mean Lifetime

Concept and Calculation of Half-Life

  • Half-life defines time required for half of radioactive sample to decay
  • Remains constant for a given isotope regardless of initial quantity
  • Calculated using formula: t1/2=ln⁡(2)λt_{1/2} = \frac{\ln(2)}{λ}
    • t₁/₂ represents half-life
    • ln(2) denotes natural logarithm of 2
    • λ signifies decay constant
  • Varies widely among isotopes (carbon-14 half-life: 5,730 years, uranium-238 half-life: 4.5 billion years)
  • Used in radiometric dating techniques to determine age of materials (geological samples, archaeological artifacts)
  • Multiple half-lives reduce radioactive material to negligible levels (after 10 half-lives, less than 0.1% of original material remains)

Mean Lifetime and Its Relationship to Half-Life

  • Mean lifetime represents average time a radioactive nucleus exists before decaying
  • Calculated as reciprocal of decay constant: τ=1λτ = \frac{1}{λ}
    • τ (tau) denotes mean lifetime
    • λ represents decay constant
  • Relates to half-life through equation: τ=t1/2ln⁡(2)τ = \frac{t_{1/2}}{\ln(2)}
  • Provides alternative measure of radioactive decay rate
  • Used in theoretical calculations and modeling of radioactive processes

Radioactive Decay Products

Parent Nuclide Characteristics

  • refers to original unstable radioactive isotope
  • Undergoes decay process to form more stable configuration
  • Characterized by specific atomic number and mass number
  • Decay mode depends on neutron-to-proton ratio (, , gamma emission)
  • Examples include uranium-238 (parent nuclide in uranium ) and carbon-14 (used in )

Daughter Nuclide Formation and Properties

  • results from decay of parent nuclide
  • May be stable or undergo further decay (decay chains)
  • Often has different chemical properties than parent nuclide
  • Decay series continue until reaching stable end product (lead-206 in uranium-238 decay series)
  • Ratio of parent to daughter nuclides used in radiometric dating techniques
  • Daughter nuclide accumulation rate depends on parent nuclide half-life and initial concentration

Units of Radioactivity

  • (Bq) represents SI unit of radioactivity
  • Defined as one decay per second
  • Named after physicist , discoverer of radioactivity
  • Replaced older unit (Ci) in scientific applications
  • Conversion: 1 Ci = 3.7 × 10¹⁰ Bq
  • Related units include:
    • Gray (Gy) measures absorbed (1 Gy = 1 J/kg)
    • Sievert (Sv) quantifies equivalent biological effect of radiation
  • Activity often expressed in multiples of becquerel (kBq, MBq, GBq) for practical applications
  • Used in radiation protection, environmental monitoring, and

Key Terms to Review (20)

Activity: Activity is the measure of the rate at which a radioactive substance decays, defined as the number of decays per unit time, typically expressed in units such as becquerels (Bq) or curies (Ci). Understanding activity is crucial for determining how quickly a radioactive material will release energy and decay into other elements or isotopes, and it ties into concepts like selection rules, radioactive equilibrium, and decay laws, including half-life.
ALARA Principle: The ALARA principle stands for 'As Low As Reasonably Achievable' and is a key concept in radiation protection that aims to minimize exposure to ionizing radiation. This principle underscores the importance of keeping radiation doses to workers, patients, and the public as low as possible while still achieving the desired results in various applications, including medical imaging and radiation therapy. The ALARA principle connects to the measurement of radiation dose and its biological effects, emphasizing that even small doses can have cumulative effects on health over time.
Alpha decay: Alpha decay is a type of radioactive decay in which an unstable atomic nucleus emits an alpha particle, consisting of two protons and two neutrons, resulting in a new element with a lower atomic number. This process is significant in understanding the stability of nuclei, the historical development of nuclear physics, and the broader implications for nuclear reactions and safety.
Becquerel: The becquerel (Bq) is the SI unit of radioactivity that quantifies the decay of radioactive substances. One becquerel is defined as one disintegration or decay event per second, indicating the activity of a radioactive source. Understanding the becquerel is crucial for measuring radioactivity in nuclear physics, relating to atomic structure and decay processes, and analyzing radioactive series and branching paths of decay.
Beta Decay: Beta decay is a type of radioactive decay in which an unstable atomic nucleus transforms into a more stable one by emitting a beta particle, which can be an electron or a positron. This process involves the conversion of a neutron into a proton or vice versa, resulting in a change in the atomic number and potentially the element itself.
Curie: A curie (Ci) is a unit of radioactivity defined as the amount of radioactive material that undergoes 3.7 x 10^{10} disintegrations per second. This unit connects to various concepts in nuclear physics, including measuring the intensity of radioactive sources, understanding the stability of nuclei in the chart of nuclides, analyzing the pathways in radioactive decay series, and evaluating the decay law and half-life of isotopes.
Daughter nuclide: A daughter nuclide is a product that results from the decay of a parent nuclide during radioactive decay processes. This term connects closely with concepts of radioactive series, where a sequence of decays leads to the formation of various daughter nuclides, and also relates to the understanding of decay laws and half-lives, which describe the probabilities of these transformations occurring over time.
Decay Constant: The decay constant is a parameter that quantifies the rate at which a radioactive isotope decays over time. It is defined as the probability per unit time that a nucleus will decay and is crucial for understanding various processes related to radioactivity, such as reaction rates, decay rates, and the energetics of decay mechanisms.
Decay Series: A decay series is a sequence of radioactive decays that occur when a radioactive isotope transforms into a series of different isotopes until a stable isotope is formed. This process involves multiple steps, often resulting in the emission of various particles and radiation types. Understanding decay series is essential to grasp the overall behavior of radioactive materials and their half-lives.
Exponential Decay: Exponential decay is a process where a quantity decreases at a rate proportional to its current value, leading to a rapid decline over time. This concept is fundamental in understanding various phenomena, including the behavior of radioactive substances as they undergo transformation and the interactions of photons with matter. It helps describe how unstable nuclei lose energy and particles, leading to half-lives that characterize decay rates.
Gamma decay: Gamma decay is a type of radioactive decay in which an unstable atomic nucleus releases energy in the form of gamma radiation, a highly penetrating electromagnetic radiation. This process usually occurs after other forms of decay, such as alpha or beta decay, and helps the nucleus reach a more stable energy state without changing its number of protons or neutrons.
Half-life: Half-life is the time required for half of the radioactive atoms in a sample to decay into a different element or isotope. This concept is crucial for understanding the stability and behavior of isotopes, and it connects to various aspects such as safety, monitoring, and the applications of nuclear science.
Henri Becquerel: Henri Becquerel was a French physicist who is best known for discovering radioactivity, a groundbreaking phenomenon that laid the foundation for modern nuclear physics. His work in the late 19th century revealed the spontaneous emission of radiation from certain materials, which significantly advanced the understanding of atomic structure and radioactive decay processes. This discovery directly connects to the study of radioactive decay laws and half-life, as it provided crucial insights into how unstable isotopes transform over time.
Marie Curie: Marie Curie was a pioneering physicist and chemist best known for her groundbreaking research on radioactivity, a term she coined. Her work laid the foundation for modern nuclear physics and significantly advanced the understanding of natural and artificial radioactivity, radioactive decay, and radioactive equilibrium, which are essential concepts in the study of nuclear science.
N(t) = n0 e^(-λt): The equation n(t) = n0 e^(-λt) describes the exponential decay of a radioactive substance over time, where n(t) represents the remaining quantity of the substance at time t, n0 is the initial quantity, λ (lambda) is the decay constant, and e is the base of natural logarithms. This formula illustrates how the amount of radioactive material decreases as time progresses, showcasing the fundamental principles behind radioactive decay and half-life.
Nuclear medicine: Nuclear medicine is a medical specialty that uses radioactive materials for diagnosis and treatment of diseases, particularly cancers and various other disorders. By utilizing specific radioactive isotopes, it enables imaging of the body's organs and tissues, providing vital information about their function and condition. This field connects closely with concepts like reaction rates and cross sections in nuclear physics, as well as the underlying principles of radioactive decay.
Parent nuclide: A parent nuclide is the original radioactive isotope that undergoes decay to transform into a different, more stable nuclide, known as the daughter nuclide. This concept is essential for understanding the processes of radioactive decay and how isotopes change over time, which directly ties into the principles of decay law and half-life, as well as how these transformations occur within radioactive series that may branch out into various decay paths.
Radiation dose: Radiation dose is the amount of radiation energy absorbed by a substance or tissue, measured to assess the potential biological effects. Understanding radiation dose is crucial in evaluating risks associated with exposure, particularly in medical treatments and natural occurrences, as it helps to quantify how much radiation has been delivered, its biological implications, and how different types of radiation can interact with matter.
Radioactive decay: Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This process transforms the original atom into a different element or isotope, leading to changes in the atomic structure and releasing energy, which is crucial in various applications, including energy generation and medical treatments.
Radiocarbon dating: Radiocarbon dating is a scientific method used to determine the age of an object containing organic material by measuring the amount of carbon-14 it contains. This technique relies on the principle of radioactive decay, specifically the predictable rate at which carbon-14 decays into nitrogen-14 over time, allowing researchers to estimate the time since the death of the organism. By using half-life calculations, radiocarbon dating can provide precise dates for archaeological findings and ancient biological samples.
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