Explain atomic energy levels and spectra and analyse nuclear decay and mass-energy in reactions.

What this dot point is asking

This dot point combines the quantised atom with the physics of the nucleus, both central revolutions in modern physics.

Atomic energy levels and spectra

In the Bohr model, electrons in an atom can only exist in certain allowed energy levels, not anywhere in between. An electron moves to a higher level by absorbing exactly the right amount of energy and falls to a lower level by emitting a photon. The photon's energy equals the difference between the two levels:

Because only specific energy differences are possible, atoms emit and absorb light at specific frequencies, producing line spectra. An emission spectrum is a set of bright coloured lines; an absorption spectrum is dark lines on a continuous background. These line spectra act as fingerprints that identify elements, which is how astronomers determine the composition of distant stars.

Radioactive decay

Unstable nuclei decay to become more stable, emitting radiation. The three common types are:

• Alpha ($\alpha$): emission of a helium nucleus ($^4_2\text{He}$); mass number falls by $4$, atomic number by $2$.
• Beta-minus ($\beta^-$): a neutron becomes a proton, emitting an electron and an antineutrino; atomic number rises by $1$, mass number unchanged.
• Gamma ($\gamma$): a high-energy photon released as the nucleus drops to a lower energy state; no change in mass or atomic number.

Decay is random for any single nucleus, but a large sample decays predictably. The half-life $t_{1/2}$ is the time for half the nuclei in a sample to decay:

Half-lives span an enormous range, from fractions of a second to billions of years, and this is the basis of radiometric dating. Carbon-14, with a half-life of about $5730$ years, lets archaeologists date once-living material, while uranium isotopes with half-lives of billions of years date rocks and the age of the Earth. The activity of a sample (decays per second) falls in step with the number of remaining nuclei, so it too halves every half-life.

Mass defect and binding energy

The mass of a nucleus is slightly less than the total mass of its separate protons and neutrons. This difference is the mass defect $\Delta m$. The missing mass is the energy that was released when the nucleons bound together, given by mass-energy equivalence:

This binding energy is what holds the nucleus together. The binding energy per nucleon peaks around iron ($^{56}\text{Fe}$), which is the most stable nucleus. In nuclear reactions the products have a larger total binding energy per nucleon than the reactants, so mass is converted to energy. Fusion of light nuclei (below iron) and fission of heavy nuclei (above iron) both move toward iron and release energy, which is the source of the energy in nuclear power and in stars.

When balancing nuclear equations, conserve both the mass number (top) and the atomic number (bottom) on each side. For energy questions, convert atomic mass units to kilograms if needed ($1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$, or use $931.5\ \text{MeV/u}$) before applying $E = \Delta m\,c^2$.