Nuclear Reactions & Radioactive Decay – Comprehensive Study Notes

Binding Energy, Stability & General Features of Nuclear Reactions

  • Binding-energy curve
    • Binding energy per nucleon peaks for intermediate-mass nuclei (around Fe, Ni).
    • ⟹ Fusing very light nuclei or fissioning very heavy nuclei moves products toward this maximum and releases large energy (mass defect → E=mc^{2}).
  • Isotopic (nuclide) notation: ^{A}_{Z}X
    • Z = atomic number = # protons.
    • A = mass number = # protons + neutrons (nucleons).
  • Conservation rules when writing/ balancing nuclear equations
    • Total A on reactant side = total A on product side.
    • Total Z on reactant side = total Z on product side.
    • Energy, momentum & lepton number also conserved (latter omitted on MCAT except for noting antineutrino/ neutrino).

Fusion

  • Definition: Combination of two (or more) light nuclei → heavier nucleus.
  • Stellar example (proton–proton chain simplified): 4\,^{1}{1}H \;\rightarrow\;^{4}{2}He + 2e^{+}+2\nu_{e}+\text{energy}
    • Sun outputs 3.85\times10^{26}\ \text{J·s}^{-1} (≈ 385 YW).
    • Energy corresponds to small mass defect: missing mass converted to radiant energy.
  • Terrestrial research
    • Magnetic/ inertial confinement reactors (ITER, NIF) attempt ^{2}{1}H+^{3}{1}H or ^{2}{1}H+^{2}{1}H.
    • Text’s given prototype: deuterium + lithium reactions inside experimental fusion power plants.

Fission

  • Definition: Splitting of a heavy nucleus → two (or more) lighter nuclei + free neutrons + energy.
  • Spontaneous fission rare (large Coulomb barrier); induced fission via absorption of low-energy (thermal) neutron practical.
  • Chain reaction concept
    • If each fission releases ≥ 1 extra neutron capable of causing another fission, a self-sustaining chain can occur.
    • Controlled in reactors (moderator, control rods); uncontrolled in weapons.
  • Classical MCAT example
    1. Capture: ^{235}{92}U + ^{1}{0}n \;\rightarrow\;^{236}_{92}U^{*} (excited).
    2. Fission products: ^{236}{92}U^{*}\;\rightarrow\;^{140}{54}Xe + ^{94}{38}Sr + x\,^{1}{0}n.
    • Balancing A: 236-140-94 = 2 ⇒ x=2 neutrons.
    • Balancing Z: 92-54-38=0 ✔️
      • Released neutrons propagate the chain.

Radioactive Decay Overview

  • Natural, spontaneous transmutation accompanied by emission of characteristic particles/ photons.
  • MCAT problem types:
    1. Arithmetic of balancing nuclear symbols.
    2. Half-life calculations.
    3. Exponential-decay constant usage.

Alpha ( \alpha ) Decay

  • Emission of an alpha particle ^{4}{2}\alpha \,(^{4}{2}He^{2+}).
  • Daughter: Z{\text{daughter}} = Z{\text{parent}}-2; A{\text{daughter}} = A{\text{parent}}-4.
  • Low penetration (stopped by few cm air / paper / skin).
  • Example balance:
    ^{238}{92}U \;\rightarrow\;^{234}{90}Th + ^{4}_{2}\alpha.

Beta-Minus ( \beta^{-} ) Decay

  • A neutron → proton + e^{-} + \bar{\nu}_{e}.
  • Emitted particle: ^{0}_{-1}\beta^{-} (electron).
  • Daughter: Z+1, A unchanged.
  • Example: ^{146}{61}Pm \;\rightarrow\;^{146}{62}Sm + ^{0}_{-1}\beta^{-}.

Beta-Plus / Positron ( \beta^{+} ) Decay

  • A proton → neutron + e^{+} + \nu_{e}.
  • Emitted particle: ^{0}_{+1}\beta^{+}.
  • Daughter: Z-1, A unchanged.
  • Sample equation form: ^{A}{Z}X \;\rightarrow\;^{A}{Z-1}Y + ^{0}_{+1}\beta^{+}.

Gamma ( \gamma ) Emission

  • High-energy photon (\gamma) emitted when nucleus transitions from excited ( * ) to lower energy.
  • No change in A or Z.
  • Notation: ^{A}{Z}X^{*} \;\rightarrow\;^{A}{Z}X + \gamma.
  • Highly penetrating; requires dense shielding (lead, concrete).

Electron Capture (EC)

  • Inner orbital electron captured by nucleus: p+e^{-}\rightarrow n+\nu_{e}.
  • Equation: ^{A}{Z}X + e^{-} \;\rightarrow\;^{A}{Z-1}Y.
  • Mass number same; atomic number decreases by 1.
  • Often competes with \beta^{+} decay; thought of as its reverse.

Worked Decay-Chain Puzzle (combined modes)

  • Given final ^{241}_{95}Am after: \alpha decay ← \beta^{+} decay ← \gamma emission.
    1. Last (\alpha): ^{245}{97}Bk \;\rightarrow\;^{241}{95}Am + ^{4}_{2}\alpha.
    2. Prior (\beta^{+}): ^{245}{98}Cf \;\rightarrow\;^{245}{97}Bk + ^{0}_{+1}\beta^{+}.
    3. First (\gamma): ^{245}{98}Cf^{*} \;\rightarrow\;^{245}{98}Cf + \gamma.
  • Starting excited nucleus: ^{245}_{98}Cf^{*}.

Half-Life \bigl(t_{1/2}\bigr)

  • Definition: Time required for ½ of initial radioactive nuclei to decay.
  • After n half-lives, fraction remaining = (\tfrac{1}{2})^{n}.
  • Example: t_{1/2}=4\,\text{yr}. After 12\,\text{yr}=3 half-lives ⇒ remaining fraction (\tfrac{1}{2})^{3}=\tfrac{1}{8}.

Exponential Decay Mathematics

  • Differential form: \frac{dN}{dt} = -\lambda N.
  • Solution: N(t)=N_{0}\,e^{-\lambda t}.
  • Relation to half-life: \lambda = \frac{\ln 2}{t{1/2}} = \frac{0.693}{t{1/2}}.
  • Sample calculation (transcript example):
    • N_{0}=2\,\text{mol}, \lambda=2\,\text{h}^{-1}, t=0.75\,\text{h}.
    • N = 2\,\text{mol}\,e^{-2(0.75)} = 2\,\text{mol}\,e^{-1.5} \approx 2\,\text{mol}\times0.22 = 0.44\,\text{mol}.
    • Nuclei count: 0.44\,\text{mol}\times6.02\times10^{23}\,\frac{\text{nuclei}}{\text{mol}} \approx 2.64\times10^{23}\, nuclei remain.

Practical, Ethical & Real-World Connections

  • Energy production
    • Controlled fission supplies ≈ 10 % of global electricity; challenges: waste management, meltdown risk.
    • Fusion promises abundant, cleaner energy; still net-negative today (2024).
  • Medical imaging/ therapy
    • \gamma emitters in PET/ SPECT; \beta^{+} tracers; radiotherapy uses \beta^{-} and \gamma.
  • Astrophysics & cosmology
    • Stellar nucleosynthesis, supernovae explain elemental abundances.
  • Ethical considerations
    • Nuclear weapons proliferation, accidents (Chernobyl, Fukushima), long-lived waste necessitate stringent regulation & international policy.

Quick MCAT Strategy Reminders

  • Translate words ⟶ nuclide symbols; immediately check A & Z balance.
  • Recognize particle symbols: ^{1}{0}n,\ ^{0}{-1}e,\ ^{0}{+1}e,\ ^{4}{2}\alpha,\ \gamma.
  • For half-life mental math: set up powers of \tfrac{1}{2}; memorize \ln 2\approx0.693.
  • Chain reactions: watch for extra neutrons (fission) or missing ones (fusion) to infer net effect.
  • Keep units straight: J, eV (1 eV = 1.6\times10^{-19}\,\text{J}), W (J·s⁻¹), mol, nuclei.