31.4 Nuclear Decay and Conservation Laws

Nuclear Decay and Conservation Laws

Nuclear decay is the spontaneous process by which unstable atomic nuclei emit radiation to reach a more stable state. The conservation laws that govern these reactions let you predict exactly what comes out of a decay and how much energy gets released. These two ideas together form the foundation for understanding everything from radiometric dating to nuclear power.

Nuclear Decay

Process of nuclear decay

An unstable nucleus has an unfavorable ratio of protons to neutrons, so it spontaneously emits radiation to move toward a more stable configuration. This emitted radiation comes in three main forms: alpha particles, beta particles, and gamma rays.

The half-life of a radioactive isotope is the time it takes for half of a sample's unstable nuclei to decay. Each isotope has its own characteristic half-life, ranging from fractions of a second to billions of years.

Nuclear decay has several practical applications:

• Radiometric dating uses known half-lives to determine the age of rocks, fossils, and archaeological artifacts
• Nuclear medicine relies on radioactive tracers for diagnostic imaging (like PET scans) and targeted cancer treatment
• Nuclear power plants harness the energy released during decay and fission to generate electricity
• Health risks arise because the emitted radiation is ionizing, meaning it carries enough energy to damage DNA and living tissue

Conservation laws in nuclear reactions

Three conservation laws constrain every nuclear reaction. If you can remember these three rules, you can check whether any proposed nuclear equation is valid.

Conservation of mass-energy: The total mass-energy of a closed system stays constant. Mass and energy are interchangeable through Einstein's relation $E = mc^2$. When a nucleus decays, a small amount of mass "disappears" and shows up as kinetic energy of the products and emitted radiation.

Conservation of charge: The total electric charge before the reaction must equal the total charge after. In a nuclear equation, this means the sum of atomic numbers ($Z$) on the left side equals the sum on the right side.

Conservation of nucleon number: The total number of nucleons (protons + neutrons) stays constant. The sum of mass numbers ($A$) of the reactants must equal the sum of mass numbers of the products.

Radioactive Decay

Parent vs. daughter nuclei

The parent nucleus is the original unstable nucleus before decay. The daughter nucleus is what remains after the decay. The daughter may itself be stable or unstable. If it's unstable, it will decay further, sometimes producing a chain called a radioactive decay series.

Two key examples:

• Alpha decay: Uranium-238 (parent) emits an alpha particle ($^4_2\text{He}$) and becomes Thorium-234 (daughter). The mass number drops by 4 and the atomic number drops by 2.
• Beta decay: Carbon-14 (parent) emits a beta particle (an electron) and an antineutrino, converting a neutron into a proton. The daughter is Nitrogen-14. The mass number stays the same, but the atomic number increases by 1.

Notice how conservation laws apply in each case. In alpha decay, $238 = 234 + 4$ (nucleon number) and $92 = 90 + 2$ (charge). In beta decay, $14 = 14$ and $6 = 7 + (-1)$.

Energy release in decay reactions

The energy released in a decay comes from the mass difference between the parent and the products. This difference, multiplied by $c^2$, gives the $Q$-value of the reaction.

Alpha decay:

$Q_\alpha = (m_p - m_d - m_\alpha) c^2$

• $m_p$ = mass of parent nucleus
• $m_d$ = mass of daughter nucleus
• $m_\alpha$ = mass of the alpha particle

A positive $Q$-value means the decay releases energy and can occur spontaneously.

Beta decay:

$Q_\beta = (m_p - m_d - m_e) c^2$

• $m_e$ = mass of the emitted electron (beta particle)

In beta decay, the released energy is shared between the beta particle and an antineutrino, which is why beta particles emerge with a continuous spectrum of energies rather than a single value.

Gamma emission:

$E_\gamma = hf$

• $h$ = Planck's constant ($6.626 \times 10^{-34} \text{ J·s}$)
• $f$ = frequency of the gamma photon

Gamma rays are emitted when a daughter nucleus is left in an excited state after alpha or beta decay. The nucleus drops to a lower energy level and releases the excess energy as a high-frequency photon. No change in $A$ or $Z$ occurs during gamma emission.

Nuclear Reactions and Energy

Nuclear binding energy

Nuclear binding energy is the energy required to completely disassemble a nucleus into separate protons and neutrons. A higher binding energy per nucleon means a more stable nucleus.

Iron-56 has the highest binding energy per nucleon of any element, which is why nuclei lighter than iron tend to release energy through fusion, while nuclei heavier than iron tend to release energy through fission.

Nuclear fission and fusion

Nuclear fission splits a heavy nucleus (like Uranium-235) into two lighter nuclei, releasing energy because the products have higher binding energy per nucleon than the original heavy nucleus. This is the process used in nuclear power plants and atomic weapons.

Nuclear fusion combines light nuclei (like hydrogen isotopes) into a heavier nucleus, also releasing energy for the same reason. Fusion powers the Sun and other stars, where hydrogen nuclei fuse into helium under extreme temperature and pressure.

Both processes convert a small amount of mass into a large amount of energy, as described by $E = mc^2$.