Nuclear Reactions
High-Yield Summary
- Nuclear notation: superscript A (mass number = protons + neutrons), subscript Z (atomic number = protons), e.g. ¹⁶₈O.
- Fusion: small nuclei combine into a larger one (powers the Sun). Fission: large unstable nucleus splits into smaller ones (nuclear reactors/bombs). Both release energy.
- Radioactive decay conserves total nucleons and total charge: parent nucleus → daughter nucleus + emitted particle.
- Five decay types differ in what's emitted and how Z/A change — alpha, beta-minus, beta-plus, gamma, electron capture (see table).
- Half-life math: n = n₀e^(−λt), with T½ = ln2/λ ≈ 0.693/λ — decay rate is proportional to the amount of undecayed material remaining.
Five Types of Radioactive Decay
| Decay type (particle emitted) | Change in Z / Change in A |
|---|---|
| Alpha (α) — helium-4 nucleus | Z −2, A −4 |
| Beta-minus (β⁻) — electron | Z +1, A unchanged |
| Beta-plus (β⁺) — positron | Z −1, A unchanged |
| Gamma (γ) — high-energy photon | Z unchanged, A unchanged (energy only) |
| Electron capture — absorbs an inner electron | Z −1, A unchanged |
Key Terms
- Atomic number (Z)
- Number of protons in the nucleus.
- Mass number (A)
- Total number of protons + neutrons (nucleons).
- Parent nucleus / daughter nucleus
- The original isotope before decay, and the isotope it transforms into after decay.
- Beta-minus decay
- A neutron converts to a proton, emitting an electron — Z increases by 1.
- Beta-plus decay (positron emission)
- A proton converts to a neutron, emitting a positron — Z decreases by 1.
- Electron capture
- Nucleus absorbs an inner orbital electron, which combines with a proton to form a neutron — Z decreases by 1, essentially the reverse of β⁻ decay.
Half-Life and Exponential Decay
Δn/Δt = −λn, n = n₀e^(−λt), T½ = ln2/λ ≈ 0.693/λ
- n = Number of undecayed nuclei remaining at time t
- n₀ = Initial number of undecayed nuclei
- λ = Decay constant — unique to each isotope
- T½ = Half-life — time for half the sample to decay, independent of initial amount
- Worked example: n₀ = 800 g, T½ = 6 h, t = 18 h → 3 half-lives → n = 800 × (1/2)³ = 100 g.
- Same answer via decay constant: λ = 0.693/6 ≈ 0.1155 h⁻¹ → n = 800e^(−0.1155×18) ≈ 100 g.
Common MCAT Trap
- Gamma decay changes neither Z nor A — it's pure energy release, don't treat it like particle emission changes the isotope's identity.
- Beta-plus decay and electron capture both decrease Z by 1 with A unchanged, but are different processes (positron emission vs. electron absorption) — don't conflate the mechanism even though the Z/A change looks identical.
- Half-life is independent of the starting amount — a bigger sample doesn't decay 'faster' in absolute terms, it still takes T½ for half of whatever amount is present to decay.
Quick Recall
²³⁸₉₂U undergoes alpha decay. What is the daughter nucleus?
A 400 g sample has a half-life of 10 days. How much remains after 30 days?
Which decay type releases a positron, and how does it change Z?