Describe the Bohr model of the atom and radioactive decay, apply the half-life formula, and connect at least one major idea across two or more units from the year.
The Bohr model of the atom -- an early, still-useful approximation -- pictures electrons orbiting the nucleus only in specific, allowed energy levels, each with a fixed energy E_n = -13.6 eV / n^2 for hydrogen (n=1, 2, 3... is the level number, with n=1 the lowest, most tightly bound "ground state," and the energy negative because the electron is bound to the atom). An electron can jump between levels only by absorbing or emitting a photon whose energy exactly matches the difference between the two levels -- which is why atoms absorb and emit light only at specific, characteristic wavelengths (a spectrum) rather than a smooth rainbow, and it is how astronomers identify which elements are present in distant stars.
A nucleus can also be unstable, undergoing radioactive decay to become more stable. Alpha decay ejects a helium nucleus (2 protons, 2 neutrons); beta decay converts a neutron into a proton, ejecting an electron; gamma decay releases a high-energy photon with no change to the nucleus's protons or neutrons at all. Every radioactive isotope has a characteristic half-life, the time for half of a given sample to decay: N = N0 * (1/2)^(t / half-life) -- a purely statistical process with no way to predict when any single atom will decay, only the overall population trend.
Mastery looks like: They compute the photon energy for a hydrogen electron transition and apply the half-life formula to find the remaining amount of a radioactive sample, and can connect at least one major idea across two or more units from the year.
Common stumbles: Forgetting to correctly subtract the more-negative lower energy level when finding an emitted photon's energy; assuming a sample completely "runs out" after a couple of half-lives rather than continuing to halve indefinitely.