14.1 Atomic Nucleus and Radioactive Decay

Nuclear physics unravels the mysteries of atomic nuclei and radioactivity. We'll explore the structure of nuclei, including protons and neutrons, and how they're held together by the strong nuclear force. Understanding nuclear stability and binding energy sets the stage for grasping radioactive decay.

Radioactive decay comes in different flavors: alpha, beta, and gamma. We'll learn how these processes change atomic nuclei and emit particles or energy. Half-life and decay rates help us predict radioactive behavior, while decay series show us how unstable nuclei transform over time.

Atomic Structure

Components of the Atomic Nucleus

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• Atomic nucleus forms the dense central core of an atom containing most of its mass
• Protons carry positive electric charge and contribute to the atomic number of an element
• Neutrons possess no electric charge but add to the mass number of an atom
• Isotopes represent atoms of the same element with different numbers of neutrons
  • Affect the atomic mass while maintaining the same chemical properties
  • Can be stable or unstable (radioactive)
• Strong nuclear force binds protons and neutrons together within the nucleus
  • Overcomes the electrostatic repulsion between positively charged protons
• Nucleus size ranges from ~1-10 femtometers (1 fm = 10^-15 m)
  • Significantly smaller than the overall atom (10^-10 m)

Nuclear Stability and Binding Energy

• Nuclear stability depends on the ratio of protons to neutrons
  • Stable nuclei generally have roughly equal numbers of protons and neutrons
• Binding energy represents the energy required to break apart a nucleus into its constituent nucleons
  • Calculated using Einstein's mass-energy equivalence formula: $E = mc^2$
• Nuclear binding energy per nucleon peaks around iron (Fe-56)
  • Explains why fusion and fission reactions release energy
• Mass defect refers to the difference between the mass of a nucleus and the sum of its constituent nucleon masses
  • Directly related to binding energy through $E = mc^2$

Radioactive Decay

Types of Radioactive Decay

• Radioactivity involves the spontaneous emission of particles or energy from unstable atomic nuclei
• Alpha decay occurs when a nucleus emits an alpha particle (two protons and two neutrons)
  • Reduces the atomic number by 2 and the mass number by 4
  • Common in heavy nuclei (uranium, thorium)
• Beta decay happens when a neutron converts to a proton or vice versa
  • Beta minus (β^-) decay emits an electron and an antineutrino
  • Beta plus (β^+) decay emits a positron and a neutrino
  • Changes the atomic number by 1 while keeping the mass number constant
• Gamma radiation involves the emission of high-energy photons
  • Often accompanies alpha or beta decay as nuclei transition to lower energy states
  • Does not change the atomic number or mass number

Nuclear Equations and Conservation Laws

• Nuclear equations represent radioactive decay processes using atomic symbols and particle notation
  • $_{92}^{238}U \rightarrow _{90}^{234}Th + _{2}^{4}He$ (alpha decay)
  • $_{6}^{14}C \rightarrow _{7}^{14}N + e^- + \bar{\nu}_e$ (beta minus decay)
• Conservation laws apply to nuclear reactions
  • Conservation of electric charge
  • Conservation of nucleon number (mass number)
  • Conservation of energy (including rest mass energy)
• Decay energy (Q-value) calculated from mass difference between parent and daughter nuclei
  • Determines the kinetic energy of emitted particles and radiation

Half-Life and Decay Series

Half-Life and Radioactive Decay Rates

• Half-life represents the time required for half of a radioactive sample to decay
  • Characteristic property of each radioisotope
  • Ranges from fractions of a second to billions of years
• Decay constant (λ) relates to half-life through the equation: $t_{1/2} = \frac{\ln(2)}{\lambda}$
• Exponential decay law describes the number of undecayed nuclei over time
  • $N(t) = N_0e^{-\lambda t}$, where N_0 is the initial number of nuclei
• Activity (A) measures the rate of radioactive decays per unit time
  • $A = \lambda N$, often expressed in becquerels (Bq) or curies (Ci)
• Carbon-14 dating uses the known half-life of ^14C (5,730 years) to determine the age of organic materials
  • Effective for dating objects up to about 50,000 years old

Decay Series and Radioactive Equilibrium

• Decay series describe the sequential decay of radioactive nuclei through multiple steps
  • End with a stable nuclide (often an isotope of lead)
• Three naturally occurring decay series
  • Uranium series (starts with ^238U)
  • Thorium series (starts with ^232Th)
  • Actinium series (starts with ^235U)
• Branching occurs when a nucleus can decay through multiple pathways
  • Probabilities of different decay modes sum to 100%
• Secular equilibrium established in decay chains when parent half-life greatly exceeds that of daughters
  • Activity of each member in the chain becomes equal
  • Important in geological dating and understanding natural radioactivity