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

Half-life 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 decay constant and activity are key concepts that help quantify radioactive decay processes.

Radioactive Decay Fundamentals

Understanding Radioactive Decay and Exponential Decay

• 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: $N(t) = N_0 e^{-λt}$
  • 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: $A = λN$
  • 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: $-\frac{dN}{dt} = λN$
  • 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: $t_{1/2} = \frac{\ln(2)}{λ}$
  • 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: $τ = \frac{1}{λ}$
  • τ (tau) denotes mean lifetime
  • λ represents decay constant
• Relates to half-life through equation: $τ = \frac{t_{1/2}}{\ln(2)}$
• Provides alternative measure of radioactive decay rate
• Used in theoretical calculations and modeling of radioactive processes

Radioactive Decay Products

Parent Nuclide Characteristics

• Parent nuclide 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 (alpha decay, beta decay, gamma emission)
• Examples include uranium-238 (parent nuclide in uranium decay series) and carbon-14 (used in radiocarbon dating)

Daughter Nuclide Formation and Properties

• Daughter nuclide 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

Becquerel and Related Units

• Becquerel (Bq) represents SI unit of radioactivity
• Defined as one decay per second
• Named after physicist Henri Becquerel, discoverer of radioactivity
• Replaced older unit curie (Ci) in scientific applications
• Conversion: 1 Ci = 3.7 × 10¹⁰ Bq
• Related units include:
  • Gray (Gy) measures absorbed radiation dose (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 nuclear medicine