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): $4\,^{1}<em>{1}H \;\rightarrow\;^{4}</em>{2}He + 2e^{+}+2\nu_{e}+\text{energy}$
  • 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 $^{2}<em>{1}H+^{3}</em>{1}H$ or $^{2}<em>{1}H+^{2}</em>{1}H$.
  • 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: $^{235}<em>{92}U + ^{1}</em>{0}n \;\rightarrow\;^{236}_{92}U^{*}$ (excited).
  2. Fission products: $^{236}<em>{92}U^{*}\;\rightarrow\;^{140}</em>{54}Xe + ^{94}<em>{38}Sr + x\,^{1}</em>{0}n$.
  • 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 $^{4}<em>{2}\alpha \,(^{4}</em>{2}He^{2+})$.
• Daughter: $Z<em>{\text{daughter}} = Z</em>{\text{parent}}-2$; $A<em>{\text{daughter}} = A</em>{\text{parent}}-4$.
• Low penetration (stopped by few cm air / paper / skin).
• Example balance:
  $^{238}<em>{92}U \;\rightarrow\;^{234}</em>{90}Th + ^{4}_{2}\alpha$.

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

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

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

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

Gamma ( $\gamma$ ) Emission

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

Electron Capture (EC)

• Inner orbital electron captured by nucleus: $p+e^{-}\rightarrow n+\nu_{e}$.
• Equation: $^{A}<em>{Z}X + e^{-} \;\rightarrow\;^{A}</em>{Z-1}Y$.
• 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): $^{245}<em>{97}Bk \;\rightarrow\;^{241}</em>{95}Am + ^{4}_{2}\alpha$.
  2. Prior (\beta^{+}): $^{245}<em>{98}Cf \;\rightarrow\;^{245}</em>{97}Bk + ^{0}_{+1}\beta^{+}$.
  3. First (\gamma): $^{245}<em>{98}Cf^{*} \;\rightarrow\;^{245}</em>{98}Cf + \gamma$.
• 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: $\lambda = \frac{\ln 2}{t<em>{1/2}} = \frac{0.693}{t</em>{1/2}}$.
• 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: $^{1}<em>{0}n,\ ^{0}</em>{-1}e,\ ^{0}<em>{+1}e,\ ^{4}</em>{2}\alpha,\ \gamma$.
• 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.