Nuclear fission and fusion are opposite ways nuclei change, but both follow the CED constraints: conservation of nucleon number, conservation of energy/momentum, and mass–energy equivalence (E = mc^2). - Fission: a heavy nucleus (like U-235 or Pu-239) splits into two or more smaller nuclei + neutrons. Some nuclear binding energy is released as kinetic energy and photons because the products have a higher binding energy per nucleon (mass defect → energy). Fission can be spontaneous or neutron-initiated; if emitted neutrons induce more fissions you can get a chain reaction (critical mass matters). (CED: 15.7.A.6–8) - Fusion: two or more light nuclei combine to form a heavier nucleus. Fusion releases energy when the resulting nucleus has greater binding energy per nucleon than the reactants (again mass → energy). Fusion requires extreme temperatures/pressures to overcome the Coulomb barrier (strong force acts at nuclear scales). (CED: 15.7.A.6, 15.7.A.1, 15.7.A.4–5) Both release energy via kinetic energy of products and photons; both are constrained by conservation laws and describable using mass–energy equivalence. For quick review, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI). For unit context and extra practice, check Unit 15 (https://library.fiveable.me/ap-physics-2-revised/unit-15) and the practice problem bank (https://library.fiveable.me/practice/ap-physics-2-revised).

Radioactive decay is a quantum, spontaneous change of an unstable nucleus into a different nucleus (often emitting an alpha particle, beta particle, or gamma photon). Whether a nucleus is unstable depends on binding energy, the strong force, and conservation laws (nucleon number, energy, momentum). We can’t predict when a single nucleus will decay because decay is inherently probabilistic—quantum mechanics gives only a decay constant λ (or half-life t1/2 with λ = ln2 / t1/2) that sets the probability per unit time. For large samples the randomness averages out and you get exponential decay: N = N0 e^(−λt), so half-life and decay rates are predictable for ensembles even though individual events are not (CED 15.7.B.1 and 15.7.B.2). If you want more AP-aligned review of these ideas, check the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-15). For lots of practice problems, see (https://library.fiveable.me/practice/ap-physics-2-revised).

The strong nuclear force is the fundamental interaction that holds protons and neutrons (nucleons) together in the nucleus. It’s what provides the nuclear binding energy that overcomes the electrostatic repulsion between positively charged protons (CED 15.7.A.1, binding energy/BED ideas). Why only short range? There are two useful ways to think about it: - At the quark level the force is carried by gluons (quantum chromodynamics). Gluons cause “color confinement,” so the color force doesn’t spread out the way electric force does—it effectively binds quarks tightly and doesn’t produce a long-range pull between separate nuclei. - The residual strong force between whole nucleons is effectively mediated by massive particles (mesons). Because those carriers have mass, the force falls off very rapidly with distance (exponentially), so the strong force is only significant at nuclear distances (~1–3 fm, 10^-15 m). Net result: inside the nucleus the strong force dominates and gives binding energy; beyond a few femtometers it drops away and electromagnetic repulsion wins, which matters for fission, fusion, and decay (see Topic 15.7 study guide for more: https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI). For extra practice on related AP problems, check the unit practice set (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of half-life as a statistical rule for a large group, not a timetable for each atom. Radioactive decay is random: any given nucleus has a certain probability per second (the decay constant λ) of decaying, but you can’t predict when an individual one will go. For a large sample the number follows exponential decay: N = N0 e^{−λt}, and λ = ln 2 / t1/2. After one half-life N = N0/2; after two half-lives N = N0/4 (not zero) because each remaining nucleus still only has the same probability to decay in the next interval. So “half the atoms decay every half-life” means the expected fraction decays, not that the sample is exhausted in a fixed number of steps. This probabilistic, exponential behavior is exactly what the CED calls radioactive decay and half-life (15.7.B.1–B.2). For a quick refresher, check the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

E = mc^2 means mass and energy are the same thing in different forms: a tiny amount of mass m can be converted into a huge amount of energy E because c^2 (speed of light squared) is huge. In nuclear physics this shows up as mass–energy equivalence (CED 15.7.A.4). In fission or fusion, the total mass of the products is a little different from the mass of the starting nuclei. That “missing” mass becomes released energy (or vice versa)—usually as kinetic energy of particles or as photons (CED 15.7.A.5). Fusion in the Sun releases energy because combining light nuclei makes a more tightly bound nucleus with less mass (binding energy released, CED 15.7.A.1, 15.7.A.6). Fission of heavy nuclei can do the same if products have higher total binding energy per nucleon (CED 15.7.A.7–8). For AP-style review, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Because nuclear binding energy per nucleon isn’t the same for all nuclei, combining very light nuclei (like H isotopes) into a heavier nucleus can make the result more tightly bound. “More tightly bound” means the final nucleus has less total mass-energy than the two separate nuclei—that missing mass is the mass defect and gets released as energy via E = mc^2. The strong nuclear force is what provides that binding at short ranges (CED 15.7.A.1, 15.7.A.4). So fusion releases energy when the product has a higher binding energy per nucleon than the reactants; the excess shows up as kinetic energy of products and photons (15.7.A.5). Fission also releases energy for heavy nuclei because splitting them moves products toward the peak binding-energy-per-nucleon curve. For more review on these ideas (binding energy, mass–energy equivalence, conservation laws), see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and the Unit 15 overview (https://library.fiveable.me/ap-physics-2-revised/unit-15). For extra practice, check the AP Physics 2 practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

The decay constant λ is the probability per unit time that any single nucleus will decay. It shows up in the exponential decay law: N = N0 e^(−λt), where N0 is the initial number of nuclei and N is the number left after time t. You can rearrange to solve for anything you need: λ = −(1/t) ln(N/N0) or t = (1/λ) ln(N0/N). The half-life t1/2 is related: λ = ln 2 / t1/2. The activity (decay rate) is A = −dN/dt = λN (units: s^−1). Quick example: if t1/2 = 5 days, λ = ln2/5 ≈ 0.1386 day^−1. After 10 days N = N0 e^(−0.1386·10) = N0 e^(−1.386) ≈ 0.25 N0 (two half-lives). These are exactly the CED equations for Topic 15.7.B (use λ to predict remaining nuclei or age a sample). For more review and practice problems, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and the unit practice set (https://library.fiveable.me/practice/ap-physics-2-revised).

Scientists date fossils and rocks by measuring radioactive decay of unstable isotopes. A sample contains a parent isotope that decays into a daughter isotope at a predictable exponential rate: N = N0 e^(−λt), where λ = ln2 / t1/2 (CED 15.7.B). You measure how much parent (or daughter) is left now, estimate the original amount (N0) or use a parent/daughter ratio, and solve for t: t = (1/λ) ln(N0/N). Different isotopes fit different timescales (e.g., C-14 for ~10^4–10^5 years, uranium–lead for millions–billions), because half-lives vary hugely (CED 15.7.B.3). On the AP exam you should be ready to use the exponential decay law and the λ ↔ t1/2 relation in calculations (CED 15.7.B.1–2). For a focused review, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Whether a nucleus fissions spontaneously or needs input comes down to binding energy and the reaction Q-value. If splitting a nucleus into fragments yields products whose total mass is less than the original (so Δm < 0), the mass-energy difference appears as released energy (Q > 0) and fission can be energetically favorable—it may occur spontaneously or after a small perturbation. If the fragments have higher total mass (Q < 0), you must supply energy to make fission happen. Two practical points for AP-style thinking (use CED vocabulary): the strong force and binding energy per nucleon determine stability; heavy nuclei with relatively low binding energy per nucleon (like some isotopes of uranium and plutonium) can have exothermic fission once a neutron is absorbed. Whether that neutron capture triggers fission also depends on nuclear structure (fissile vs. just fissionable), neutron energy, and conservation laws (nucleon number, energy/momentum). For more review, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and the Unit 15 overview (https://library.fiveable.me/ap-physics-2-revised/unit-15). For extra practice problems, check Fiveable’s practice set (https://library.fiveable.me/practice/ap-physics-2-revised).

Short answer: all the energy released in a nuclear reaction shows up as other measurable energy—mostly as kinetic energy of the reaction products and as photons (gamma rays) —and it obeys conservation of energy (including mass–energy equivalence, E = mc^2). Explanation: when a nucleus fissions, fuses, or decays, the total rest mass can change; the “missing” mass becomes energy (15.7.A.4). That energy appears as high kinetic energy of fragments and emitted particles (neutrons, alpha/beta particles), as gamma photons, and sometimes as neutrino energy. In reactors or samples that stay nearby, that particle energy quickly becomes thermal energy (heating the surroundings). Momentum is also conserved, so products move fast. AP-relevant facts: memorize E = mc^2, that energy shows up as kinetic energy or photons (15.7.A.4–A.5), and apply conservation laws for problems. For review and practice, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and try problems at Fiveable Practice (https://library.fiveable.me/practice/ap-physics-2-revised).

Use N = N0 e^{-λt} exactly the way you’d use any exponential decay—isolate the unknown, take natural logs, and plug in numbers. Step-by-step: 1. Identify what you know: N0 (initial nuclei), N (remaining nuclei), and either λ (decay constant) or t (time). If you’re given half-life t1/2, get λ = ln 2 / t1/2 (CED 15.7.B.1.iii). 2. Start with N = N0 e^{-λt}. Divide both sides by N0: N/N0 = e^{-λt}. 3. Take natural log: ln(N/N0) = -λt. This is the derived equation in the CED. 4. Solve for the unknown: - If solving for t: t = -[1/λ] ln(N/N0). - If solving for λ: λ = -[1/t] ln(N/N0). - If solving for N: N = N0 e^{-λt} (just compute). 5. Keep units consistent (t in seconds or years to match λ). Check signs: ln(N/N0) is negative when N < N0, so t and λ come out positive. Quick example: N0 = 1000, N = 250, find t if λ = 0.693/day (half-life 1 day). ln(250/1000)=ln(0.25)=-1.386 → t = -(-1.386)/0.693 = 2 days. This form and the ln-derivation show up on AP free-response and multiple-choice (practice in CED Topic 15.7). For more review and practice problems, check the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and the unit practice bank (https://library.fiveable.me/practice/ap-physics-2-revised).

Fusion happens a lot in stars because gravity gives enormous pressure and temperature so nuclei can get close enough for the strong force to bind them. On Earth it’s hard because positively charged nuclei repel each other (Coulomb barrier). To overcome that repulsion you need very high kinetic energy—temperatures of millions of kelvins—and/or very high densities and long confinement times so collisions happen frequently. Achieving all three (temperature, density, confinement) so that net energy comes out is tough: you either need magnetic confinement (tokamaks) or inertial confinement (lasers), both of which struggle with instabilities, material damage from high-energy neutrons, and engineering losses. The Lawson criterion summarizes the required product of density × confinement time × temperature for net gain. For AP, focus on the roles of the strong force vs. electrostatic repulsion, mass–energy equivalence (E = mc^2) in energy release, and why extreme conditions (not present on Earth naturally) are needed (see the Topic 15.7 study guide for a concise review: https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI). For extra practice, try problems at (https://library.fiveable.me/practice/ap-physics-2-revised).

Alpha: nucleus emits a 4He nucleus (2 protons + 2 neutrons). Mass number A drops by 4 and atomic number Z drops by 2. Alpha particles are heavy and carry a lot of kinetic energy (typically a few MeV), so they deposit energy locally. Beta: a neutron converts to a proton + electron (β−) or a proton converts to a neutron + positron (β+). Z changes by ±1 while A stays the same. The emitted beta particle plus an antineutrino/neutrino share the decay energy; typical beta energies are up to a few MeV but are often spread over a spectrum. Gamma: an excited nucleus emits a high-energy photon (γ). A and Z don’t change; gamma carries away discrete photon energies (keV–MeV) to get the nucleus to a lower energy state. Which releases the most energy? It depends on the isotope and specific decay, but typically alpha decays release the largest kinetic energy per decay (a few MeV) because of big changes in binding energy. Gamma rays can be very energetic too, but they’re usually “leftover” de-excitation energy and often smaller than the alpha Q-value for heavy nuclei. Remember conservation of energy/mass (E = mc^2) and conservation of nucleon number when predicting products (see the Topic 15.7 study guide for AP-aligned details: https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI). For extra practice, hit the AP Physics 2 problems (https://library.fiveable.me/practice/ap-physics-2-revised).

You were supposed to see that radioactive decay follows an exponential law and how to work with it on graphs and problems. Key points from the CED: N(t) = N0 e^{−λt}, λ = ln 2 / t1/2, and ln(N/N0) = −λt. That means: - The number of undecayed nuclei drops by half every t1/2 (half-life), but when a single nucleus decays is unpredictable—decay is probabilistic (15.7.B.1.i–iii). - On a semi-log plot (ln N vs. t) the data are linear with slope −λ. Use that slope to find λ and then t1/2 (useful for AP free-response and MCQs; see sample MCQ #4 in the CED). - Different isotopes have hugely different half-lives—memorize the relationships, not specific numbers. If you want a quick refresher, check the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI) and practice problems for extra decay-curve exercises (https://library.fiveable.me/practice/ap-physics-2-revised).

Conservation of nucleon number means the total number of nucleons (protons + neutrons, i.e., the mass number A) is the same before and after any nuclear reaction. It’s a constraint on possible reactions (CED 15.7.A.2). For example, in a typical fission: U-235 + n → Ba-141 + Kr-92 + 3 n Check A: 235 + 1 = 141 + 92 + 3·1 = 236, so nucleon number is conserved. In beta decay (C-14 → N-14 + e- + ν̄) the mass number stays 14 even though a neutron becomes a proton (charge changes but nucleon count does not). Why it matters: it eliminates impossible reactions and helps you balance nuclear equations alongside conservation of charge, energy (including E = mc^2), and momentum (CED 15.7.A.3–A.5). On the AP exam you’ll use nucleon-number conservation to check or construct nuclear equations and to reason about decay chains and fission/fusion outcomes. For a focused review, see the Topic 15.7 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-7/7-fission-fusion-and-nuclear-decay/study-guide/nVULWHgdGjkiGhbI). For extra practice problems, try the AP Physics 2 practice set (https://library.fiveable.me/practice/ap-physics-2-revised).