Explain nuclear stability and radioactive decay, write balanced alpha and beta decay equations, use half-life graphs, and compare fission and fusion

Learning goal: Explain why some nuclei are unstable in terms of the strong nuclear force and the electrical repulsion between protons; describe alpha, beta and gamma radiation by mass, charge and penetrating power; predict the product nucleus of an alpha or beta decay and write the balanced nuclear equation; define half-life and use a decay graph over whole numbers of half-lives, including for radioactive dating; and compare nuclear fission and nuclear fusion.

Chemical versus nuclear. A chemical reaction rearranges electrons and bonds and leaves every nucleus untouched, which is why atoms are conserved. A nuclear reaction changes the nucleus itself: the number of protons or neutrons changes, so one element can become another, and the energy involved is typically far larger than in any chemical reaction. The chemistry of the previous lesson still holds; nuclear change is a different process, happening at the center of the atom.

Why nuclei hold together and why some do not. Protons all carry positive charge and repel one another, and in a nucleus they are packed extremely close. The nucleus survives because of the strong nuclear force, an attraction between protons and neutrons that is extremely weak at ordinary distances but, over the very short distances inside a nucleus, is stronger than the electrical repulsion. Neutrons add attraction without adding repulsion, which is why heavier nuclei need more neutrons than protons. When the attractive nuclear forces and the repulsive electrical forces are not in balance, the nucleus is unstable. An unstable nucleus eventually emits radiation, fast-moving particles and energy, and becomes a new nucleus. This process is radioactive decay, and nuclei that undergo it are radioactive. Whether a given nucleus is stable depends on its particular proton and neutron counts: carbon-12 is stable, carbon-14 is not.

Three kinds of radiation. An alpha particle is two protons and two neutrons, the same as a helium-4 nucleus, with a charge of 2+ and the largest mass of the three; it is stopped by a sheet of paper or the outer layer of skin, so it is dangerous mainly if the source is swallowed or inhaled. A beta particle is a fast electron emitted from the nucleus when a neutron changes into a proton and an electron; it has charge 1− and very little mass, and it is stopped by a few millimeters of aluminum or thick plastic. Gamma radiation is high-energy radiant energy with no mass and no charge; it is the most penetrating and needs thick lead or concrete to absorb most of it. Each has a characteristic mass, charge and ability to alter and penetrate the material it strikes, which is why sources are shielded differently.

Predicting the product of a decay. Nuclear equations are balanced by keeping the total mass number and the total atomic number the same on both sides. In alpha decay, the nucleus loses 2 protons and 2 neutrons, so the mass number falls by 4 and the atomic number by 2, and the element changes. Uranium-238 (92 protons) emits an alpha particle: mass number 238 − 4 = 234, atomic number 92 − 2 = 90, which is thorium. Written with mass numbers on top and atomic numbers below: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. Check: 238 = 234 + 4 and 92 = 90 + 2. In beta decay, a neutron becomes a proton and an electron, and the electron leaves; the mass number is unchanged and the atomic number rises by 1. Carbon-14 (6 protons) undergoes beta decay: mass number stays 14, atomic number 6 + 1 = 7, which is nitrogen: ¹⁴₆C → ¹⁴₇N + ⁰₋₁e. Check: 14 = 14 + 0 and 6 = 7 + (−1). Gamma emission changes neither number; it releases energy from a nucleus left in an excited state, often after an alpha or beta decay.

Half-life. Decay is random for any single nucleus but predictable for a large sample. The half-life of a radioisotope is the time for half of the nuclei in a sample to decay, and it is characteristic of that isotope. For the decays met in this course it is not measurably changed by temperature, pressure or chemical combination, which is what lets it serve as a clock. After one half-life half remains, after two a quarter, after three an eighth, and so on. A graph of the amount remaining against time falls by half over every equal interval, and the half-life can be read from the graph as the time to go from any amount to half of it. Worked model: a hypothetical isotope has a half-life of 10 days and a sample starts with 400 units. After 10 days 200 remain, after 20 days 100, after 30 days 50, after 40 days 25. From the same graph, the time for the sample to fall from 400 to 50 is three half-lives, 30 days. This course uses only whole numbers of half-lives.

Radioactive dating and other uses. Because a half-life is fixed, the fraction of a radioisotope remaining tells how much time has passed since the sample stopped exchanging that isotope with its surroundings. Carbon-14, with a half-life of about 5,730 years, forms continuously in the upper atmosphere and is taken up by living things; when an organism dies the uptake stops and the carbon-14 decays. A piece of wood with one-eighth of the carbon-14 of living wood has been dead for three half-lives, about 17,190 years. The method suits once-living material up to a few tens of thousands of years old; older samples have too little carbon-14 left to measure, and rocks, which were never alive, are dated with isotopes that have much longer half-lives. Radioisotopes are also used in medicine, for example short-lived isotopes such as technetium-99m that are injected in tiny amounts so that a camera can image an organ and the activity fades within a day, and iodine-131, which the thyroid gland takes up, for treating some thyroid conditions. The same property that makes them useful, emitting radiation, is why their doses are tightly controlled. Radioactivity also drove the discovery of new elements: Marie and Pierre Curie identified polonium and radium in 1898 by tracking the radiation from uranium ore, and elements heavier than uranium have since been made artificially in nuclear reactors and particle accelerators.

Fission and fusion. Nuclear fission is the splitting of a large nucleus, such as uranium-235 struck by a neutron, into two smaller nuclei plus a few neutrons, releasing a large quantity of energy; the released neutrons can split more nuclei, and a controlled chain reaction in a power reactor heats water to drive turbines. Nuclear fusion is the joining of small nuclei into a larger one, such as hydrogen nuclei combining to form helium, also releasing large quantities of energy; it needs the enormous temperatures and pressures found in the sun and other stars, and it is the source of their energy, and fusion in stars made most of the carbon and oxygen and many of the other elements up to iron (most hydrogen and helium date from the early universe, and many elements heavier than iron formed in other processes in massive stars and stellar explosions). Fission fuel and the waste it leaves are radioactive: some products are intense but short-lived, a hazard for days or years, while others remain radioactive for thousands of years and must be isolated, which is the long-term problem of nuclear waste. The explanation of where the energy comes from in terms of mass and energy is left for physics.

Investigation: a half-life simulation with coins. No radioactive source is used. Materials: 100 identical coins (two-color counters work the same way; if dice are used, treat 1, 2 and 3 as "tails" so that each die still has a one-in-two chance of being removed), a tray with a lid, a tally sheet, graph paper. Steps: put all 100 coins in the tray heads up; shake, remove every coin that lands tails, and record the number remaining; repeat for six shakes, counting each shake as one "time interval" (if the coins run out sooner, record 0 for the remaining intervals); plot remaining coins against interval number. Record: a table of interval and coins remaining, and the graph. A result that supports the claim: the count falls by roughly half each interval (about 100, 50, 25, 12, 6, 3, 2), so the half-life of the model is one interval, and the curve has the same shape as a real decay graph; combining the whole class's data gives a smoother curve. Safety: none beyond keeping the coins in the tray. What the model cannot show: it does not show that real nuclei become a different element when they decay, it uses 100 objects rather than the trillions in a real sample so the halving is rough, and it cannot represent radiation type, energy or the different half-lives of different isotopes.

Worked example 1 — alpha decay. Radium-226 (atomic number 88) emits an alpha particle. Mass number: 226 − 4 = 222. Atomic number: 88 − 2 = 86, which is radon. Equation: ²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He. Check totals: 226 = 222 + 4 on top, 88 = 86 + 2 below. The identity changed from radium to radon because the proton count changed.

Worked example 2 — beta decay. Strontium-90 (atomic number 38) undergoes beta decay. A neutron becomes a proton and an emitted electron, so the mass number stays 90 and the atomic number becomes 39, which is yttrium: ⁹⁰₃₈Sr → ⁹⁰₃₉Y + ⁰₋₁e. Check: 90 = 90 + 0 and 38 = 39 + (−1). The beta decay taught here, the emission of an electron, always produces the element with the next higher atomic number (one more proton) without changing the mass number.

Worked example 3 — read a half-life from a graph. A decay graph shows 80 units at time 0, 40 units at 6 hours and 20 units at 12 hours. The amount halves every 6 hours, so the half-life is 6 hours. After 18 hours (three half-lives) 10 units remain, and after 24 hours (four half-lives) 5 units. The graph also answers the reverse question: 80 units falling to 10 units took three half-lives, 18 hours.

Worked example 4 — a carbon-14 date. Living wood has a known level of carbon-14. A wooden beam is found to contain one-quarter of that level. One-quarter is two half-lives, and each is about 5,730 years, so the tree was cut about 2 × 5,730 = 11,460 years ago. If the beam had one-sixteenth of the living level, that would be four half-lives, about 22,920 years, still within the method's range; the practical limit is roughly 50,000 years, about nine half-lives, beyond which too little carbon-14 remains to measure. The date is for when the wood stopped taking in carbon, not when it was used in a building.

Representation task: three radiations and two equations. Make a table for alpha, beta and gamma radiation with rows for composition, mass (largest, very small, none), charge, what stops it, and one use or hazard. Then write balanced nuclear equations for the alpha decay of polonium-210 (atomic number 84) and the beta decay of carbon-14 (atomic number 6), showing the mass-number and atomic-number checks beside each.

Independent performance: model, graph and interpret decay. Carry out the coin half-life simulation and produce a report. Success criteria: the data table lists the starting count and all six intervals; the graph has labeled axes and a smooth curve through the points; the half-life of the model is stated in intervals and justified from two different pairs of points; the class-combined curve is compared with your own and the difference explained by sample size; a paragraph applies the idea to carbon-14 dating with a correct worked example using a whole number of half-lives; and a final paragraph lists three things the coin model cannot show about real radioactive decay.