Past exam of the chemistry course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / 2 / a

For a hydrogen-like ion, the Coulomb potential is stronger by the nuclear atomic number $Z$, and the energy levels scale as $Z^2$:Here $Z$ is the number of protons in the atomic nucleus; the ion has exactly one electron, so its net electric charge is $(Z-1)e$. The zero of energy is a separated stationary electron and atomic nucleus at infinite distance. This is the nonrelativistic hydrogenic approximation using the supplied $R_H$; a more precise isotope-specific result also corrects the reduced mass.

The lowest two energy levels have $n=1$ and $n=2$. For carbon, $Z=6$, even though the ion's net charge is $5+$:Increasing $n$ raises the energy toward zero; the most negative value is the ground-state energy.

An emitted photon has positive energyFor spontaneous emission into $n_f=2$, the conservation of energy equation isConsequentlyThe principal quantum number must be an integer, so the transition is $\boxed{n_i=3⟶n_f=2}$. Its calculated wavelength is approximately $18.22\mathrm{nm}$, agreeing with the rounded observation.