Exercises: Quantum numbers and electron configurations

Phosphorus is an essential nonmetallic element with atomic number Z = 15.

1. a) Write the electron configuration of the phosphorus atom in its ground state according to the Aufbau principle or Moeller diagram.
2. b) State how many unpaired electrons it has in its valence shell according to Hund's rule of maximum multiplicity, and deduce whether the neutral ground-state atom is paramagnetic or diamagnetic.

Consider the neutral carbon atom (Z = 6), whose complete electron configuration begins with the sublevel $1s^2\,2s^2$ and ends with two electrons in the 2p sublevel. The following diagram shows three possible arrangements (A, B, and C) for the two electrons in the three degenerate 2p orbitals:

1. a) Identify which of the three options corresponds to the ground state of carbon, justifying your choice using the quantum rules for filling orbitals.
2. b) Analyze in detail which quantum principle or rule is violated in each of the two remaining options.

Iron is a transition element with atomic number Z = 26.

1. a) Write the electron configuration of the neutral iron atom in its ground state and justify why, when it ionizes to form the cation $\mathrm{Fe}^{2+}$, it loses the 4s electrons before those of the 3d sublevel.
2. b) Write the electron configuration of the cation $\mathrm{Fe}^{3+}$ and determine, giving reasons, which of the two species, $\mathrm{Fe}^{2+}$ or $\mathrm{Fe}^{3+}$, will have the greater number of unpaired electrons and, consequently, stronger paramagnetic behavior.

Chromium (Z = 24) and copper (Z = 29) show recognized anomalies with respect to the usual mnemonic rule of the Madelung diagram.

1. a) Write the ground-state electron configuration predicted by the standard Madelung filling order for $\mathrm{Cr}$ and $\mathrm{Cu}$, and compare each with the experimentally verified actual configuration.
2. b) Justify the quantum and energetic cause of these anomalous configurations, based on the stability associated with half-filled or completely filled sublevels and on the small energy difference between the 4s and 3d orbitals.