21.2 Nuclear Equations

Nuclear equations let you track what happens when an atomic nucleus changes. By balancing mass numbers and atomic numbers on both sides of an equation, you can predict the products of radioactive decay, fission, and fusion.

Nuclear Reactions

Balanced nuclear equations

The key rule for nuclear equations: the sum of mass numbers (top numbers) and the sum of atomic numbers (bottom numbers) must be equal on both sides. This is how you check your work on any nuclear equation problem.

Alpha decay involves an unstable nucleus emitting an alpha particle (two protons and two neutrons). The parent atom loses 4 from its mass number and 2 from its atomic number, producing a new element.

• General equation: $^{A}_{Z}X \rightarrow ^{A-4}_{Z-2}Y + ^{4}_{2}\alpha$
  • $X$ is the parent nucleus, $Y$ is the daughter nucleus, and $\alpha$ is the alpha particle
• Example: $^{238}_{92}U \rightarrow ^{234}_{90}Th + ^{4}_{2}\alpha$
  • Check: mass numbers $238 = 234 + 4$, atomic numbers $92 = 90 + 2$. Both sides balance.

Beta decay occurs when a neutron converts into a proton inside the nucleus, emitting an electron (beta particle) and an antineutrino. The mass number stays the same, but the atomic number increases by 1.

• General equation: $^{A}_{Z}X \rightarrow ^{A}_{Z+1}Y + ^{0}_{-1}\beta^{-} + \bar{\nu}$
  • $\beta^{-}$ is the beta particle (electron), and $\bar{\nu}$ is the antineutrino
• Example: $^{14}_{6}C \rightarrow ^{14}_{7}N + ^{0}_{-1}\beta^{-} + \bar{\nu}$
  • Carbon-14 becomes Nitrogen-14. The mass number stays at 14, while the atomic number goes from 6 to 7.

Gamma emission happens when an excited nucleus releases energy as a high-energy photon (gamma ray). No particles are gained or lost, so the mass number and atomic number don't change. Only the energy state of the nucleus changes.

• General equation: $^{A}_{Z}X^{*} \rightarrow ^{A}_{Z}X + \gamma$
  • The asterisk ($*$) indicates the nucleus is in an excited state
• Example: $^{60}_{27}Co^{*} \rightarrow ^{60}_{27}Co + \gamma$

Particles in nuclear reactions

You'll see these particles show up repeatedly in nuclear equations. Knowing their notation is essential for balancing.

• Protons are positively charged particles in the nucleus, written as $^{1}_{1}p$. The number of protons defines the element (hydrogen has 1, helium has 2, lithium has 3).
• Neutrons are neutral particles in the nucleus, written as $^{1}_{0}n$. Different numbers of neutrons create different isotopes of the same element. For example, Carbon-12 has 6 neutrons, Carbon-13 has 7, and Carbon-14 has 8, but all three have 6 protons.
• Alpha particles consist of two protons and two neutrons, written as $^{4}_{2}\alpha$ (identical to a helium-4 nucleus). They're emitted during alpha decay. For example, $^{222}_{86}Rn$ undergoes alpha decay to become $^{218}_{84}Po$.
• Beta particles are high-speed electrons emitted from the nucleus, written as $^{0}_{-1}\beta^{-}$. They have essentially zero mass number and a charge of $-1$.

Antimatter and nuclear processes

Antimatter is made of antiparticles, which have the same mass as their corresponding particles but opposite charge. A positron, for instance, has the same mass as an electron but a positive charge. In nuclear equations, a positron is written as $^{0}_{+1}\beta^{+}$.

Annihilation occurs when a particle meets its antiparticle. They destroy each other and convert all of their mass into energy (released as gamma rays). The energy produced follows Einstein's equation $E = mc^2$, where $m$ is the combined mass of the particle and antiparticle.

• Example: an electron and a positron collide, producing two gamma ray photons.

Pair production is the reverse of annihilation. A high-energy gamma ray photon interacts near an atomic nucleus and converts into a particle-antiparticle pair (such as an electron and a positron). The photon must have at least enough energy to account for the rest mass of both particles: $E = 2mc^2$.

Nuclear Energy Processes and Decay

Nuclear fission splits a heavy nucleus into lighter nuclei, releasing energy and usually additional neutrons. Those released neutrons can trigger more fission events, creating a chain reaction. This is the process used in nuclear power plants and atomic weapons. A classic example is Uranium-235 absorbing a neutron and splitting into smaller nuclei like Barium and Krypton, plus additional neutrons.

Nuclear fusion combines light nuclei into heavier ones, releasing even more energy per unit mass than fission. Fusion powers the Sun, where hydrogen nuclei fuse to form helium under extreme temperature and pressure.

Half-life is the time it takes for half of a radioactive sample to decay. Each isotope has its own characteristic half-life. Carbon-14 has a half-life of about 5,730 years, which makes it useful for dating archaeological samples. Uranium-238 has a half-life of about 4.5 billion years, useful for dating geological formations.

To calculate how much of a sample remains after a certain number of half-lives:

1. Determine how many half-lives have passed (divide total time by the half-life).
2. Multiply the original amount by $\left(\frac{1}{2}\right)^n$, where $n$ is the number of half-lives.

Radioactive decay series describes how an unstable nucleus decays through a chain of intermediate isotopes until it reaches a stable one. The Uranium-238 decay series, for example, passes through thorium, radium, radon, and several other elements before finally arriving at stable Lead-206.

Binding energy is the energy required to completely separate a nucleus into individual protons and neutrons. A higher binding energy per nucleon means a more stable nucleus. This concept explains why both fission (splitting very heavy nuclei) and fusion (combining very light nuclei) release energy: in both cases, the products have higher binding energy per nucleon than the starting materials, and that difference is released as energy.