Nuclear Decay — AP Physics 2
1. Nuclear Decay Fundamentals and Notation ★★☆☆☆ ⏱ 3 min
Nuclear decay (also called radioactive decay) is the spontaneous breakdown of an unstable atomic nucleus, which emits ionizing radiation to reach a more stable lower-energy state. Nuclei become unstable due to an incorrect neutron-to-proton ratio, excessive total nucleon count, or after being left in an excited state from a prior nuclear reaction. While it is impossible to predict when any single nucleus will decay, large collections of unstable nuclei follow predictable statistical behavior.
2. Types of Decay and Conservation Laws ★★☆☆☆ ⏱ 4 min
All nuclear decay reactions must satisfy two fundamental conservation laws tested consistently on the AP Physics 2 exam: conservation of total nucleon number ($A$) and conservation of total electric charge ($Z$). The sum of $A$ values on the reactant side equals the sum of $A$ values on the product side, and the same equality holds for $Z$ values.
- **Alpha decay**: Occurs for very heavy nuclides ($A > 200$) that are too large to be stable. Emits an alpha particle ($^4_2 \alpha$, identical to a helium nucleus), resulting in transmutation to a new element.
- **Beta-minus decay**: The most common beta decay type, occurring when a nucleus has too many neutrons. A neutron decays into a proton, a high-energy electron ($^0_{-1} \beta^-$), and an antineutrino, resulting in transmutation.
- **Gamma decay**: Occurs when a nucleus is in an excited energy state. Emits a high-energy gamma photon ($^0_0 \gamma$) to release excess energy, with no change to $A$ or $Z$, so no transmutation occurs.
3. Exponential Decay, Half-Life, and Activity ★★★☆☆ ⏱ 5 min
Radioactive decay is a statistical process where the instantaneous rate of decay is proportional to the number of undecayed nuclei remaining $N(t)$. This gives the differential decay law $\frac{dN}{dt} = -\lambda N$, where $\lambda$ is the decay constant (units of inverse time, larger $\lambda$ means faster decay). Solving this gives the exponential decay law:
N(t) = N_0 e^{-\lambda t}
Where $N_0$ is the initial number of undecayed nuclei at $t=0$. Half-life ($T_{1/2}$) is defined as the time required for half of the original unstable nuclei to decay. Substituting $N = N_0/2$ at $t=T_{1/2}$ gives the core relation between half-life and decay constant:
T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}
Activity $R(t)$ is the measurable decay rate (number of decays per unit time), equal to $R = \left|\frac{dN}{dt}\right| = \lambda N(t)$. Activity also follows the exponential decay law $R(t) = R_0 e^{-\lambda t}$, where $R_0 = \lambda N_0$ is initial activity. For calculations, it is often easier to write decay directly in terms of half-life: after $n = t/T_{1/2}$ half-lives, $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$ and $R(t) = R_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$.
4. Mass-Energy Equivalence and Decay Q-Value ★★★☆☆ ⏱ 4 min
For a nuclear decay to be spontaneous, the total mass of the decay products must be less than the mass of the parent nuclide. The mass difference $\Delta m$ is converted to kinetic energy of the decay products, per $E = \Delta m c^2$. This energy is called the Q-value of the decay, defined as:
Q = (m_{\text{parent}} - \sum m_{\text{products}}) c^2
If $Q>0$, decay is spontaneous (exothermic), which is true for all naturally occurring nuclear decay. If $Q<0$, decay cannot occur spontaneously. A convenient shortcut for AP problems: use atomic masses (not nuclear masses), because the total number of electrons is the same on both sides of alpha and beta-minus decay, so electron masses cancel out. Recall that $1 \text{ u} \cdot c^2 = 931.5 \text{ MeV}$.
5. AP Style Concept Check ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: Students confuse the tiny mass of an electron with a nucleon, and mix up the sign of the beta particle’s charge.
Why: Students misremember which quantity goes in the numerator of the half-life/decay constant relation.
Why: Students mix up the definition of Q-value with mass defect for binding energy.
Why: Students misinterpret half-life as the total lifetime of the entire sample.
Why: Students assume all decay changes the nuclide identity, forgetting gamma is only energy emission.