13.1 Nuclear reactions and Q-values

Nuclear reactions are the powerhouses of the universe. From the stars to nuclear power plants, these processes shape our world. Understanding how they work and calculating their energy output is crucial for harnessing their potential and predicting their behavior.

Q-values are the key to unlocking nuclear reactions' secrets. By measuring the mass difference between reactants and products, we can determine if a reaction releases or absorbs energy. This knowledge helps us predict which reactions are possible and how much energy they'll produce.

Nuclear reaction types and characteristics

Types of nuclear decay

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• Alpha decay involves atomic nucleus emitting an alpha particle (two protons and two neutrons)
  • Reduces mass number by 4 and atomic number by 2
  • Example: Uranium-238 decaying to Thorium-234
• Beta decay occurs in three forms
  • Beta-minus (electron emission)
  • Beta-plus (positron emission)
  • Electron capture
  • Each changes atomic number by 1 while maintaining mass number
  • Example: Carbon-14 undergoing beta-minus decay to Nitrogen-14
• Gamma decay involves emission of high-energy photons from excited nucleus
  • Does not change number of protons or neutrons
  • Example: Cobalt-60 emitting gamma rays after beta decay

Nuclear fission and fusion

• Nuclear fission splits heavy nucleus into lighter nuclei
  • Often releases neutrons and large amount of energy
  • Example: Uranium-235 splitting into Barium-141 and Krypton-92
• Nuclear fusion combines light nuclei to form heavier nuclei
  • Releases energy in the process
  • Serves as primary energy source in stars
  • Example: Hydrogen nuclei fusing to form helium in the Sun's core

Q-value calculation for nuclear reactions

Mass-energy equivalence principle

• Einstein's mass-energy equivalence principle $E = mc^2$ fundamental for Q-value calculations
  • c represents speed of light in vacuum
• Q-value represents energy released or absorbed during reaction
  • Based on mass difference between reactants and products
• Atomic mass units (amu) typically used in calculations
  • Conversion factor: 1 amu = 931.5 MeV/c²

Q-value calculation process

• Calculate Q-value by subtracting sum of product masses from sum of reactant masses
  • Multiply result by c²
• Positive Q-value indicates exothermic reaction (energy released)
• Negative Q-value indicates endothermic reaction (energy absorbed)
• Account for binding energy of nucleons in calculations
  • Mass of nucleus less than sum of constituent nucleon masses
• Example: Calculate Q-value for fusion of deuterium and tritium to form helium-4 and a neutron

Energy release vs absorption in nuclear reactions

Exothermic reactions

• Positive Q-value corresponds to energy released in reaction
• Released energy distributed among reaction products as kinetic energy
• Example: Alpha decay of Polonium-210 releasing 5.3 MeV of energy

Endothermic reactions

• Negative Q-value indicates energy must be supplied for reaction to occur
• Energy often supplied as kinetic energy of incident particle
• Example: Photodisintegration of deuterium requiring 2.2 MeV of energy

Energy distribution and considerations

• Energy distribution follows principles of conservation of energy and momentum
• Consider recoil energy of residual nucleus for reactions involving particle or photon emission
• In decay processes, Q-value represents total decay energy
  • Shared between daughter nucleus and emitted particle(s) or radiation
• Example: Beta decay of Tritium distributing energy between electron, antineutrino, and daughter nucleus

Conservation laws in nuclear reactions

Mass-energy and charge conservation

• Law of conservation of mass-energy states total mass-energy of isolated system remains constant
  • Crucial in nuclear reactions where mass converts to energy
• Conservation of charge requires total electric charge remains constant before and after reaction
• Example: Beta decay conserving total charge as proton converts to neutron and electron emitted

Nucleon and lepton number conservation

• Conservation of nucleon number (mass number) requires total number of nucleons remains constant
• In beta decay, conservation of lepton number must be considered
  • Accounts for emission or absorption of neutrinos or antineutrinos
• Example: Neutron decay conserving both nucleon and lepton numbers

Applications and implications

• Conservation laws provide framework for balancing nuclear equations
• Allow prediction of possible reaction products
• Concept of baryon number conservation extends nucleon number conservation
  • Includes all particles composed of quarks
• Violations of conservation laws in theoretical reactions provide insights into fundamental physics
  • Can suggest existence of new particles or forces
• Example: Proton decay theories exploring possible violation of baryon number conservation