Exercises: Quantum numbers and electron configurations

Consider the principal quantum level n = 3 for the electrons in the electron shell of an atom.

1. a) State, with reasoning, all possible values of the secondary or azimuthal (l) and magnetic ($m_l$) quantum numbers associated with this principal level, specifying the type of subshell or orbital represented by each value of l.
2. b) Calculate the total maximum number of electrons that level n = 3 can hold by applying the Pauli exclusion principle to each of its subshells.

Determine, giving reasons, whether each of the following combinations of four quantum numbers $(n,l,m_l,m_s)$ is physically allowed or forbidden for describing the state of a single electron in an atom:

1. a) (3, 2, -2, -1/2)
2. b) (2, 2, 0, +1/2)
3. c) (4, 1, +2, -1/2)
4. d) (1, 0, 0, +1/2)

If a combination is not allowed, state explicitly which rule or restriction on the quantum numbers is violated.

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 ionises 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 behaviour.

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

1. a) Write the ground-state electron configuration predicted by the standard Moeller 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.

Consider the chemical species $\mathrm{S}^{2-}$ (Z = 16), $\mathrm{K}^+$ (Z = 19) and $\mathrm{Ti}^{2+}$ (Z = 22).

1. a) Write the complete electron configuration of each of the three species in its ground state and state, giving reasons, which of them are isoelectronic with each other.
2. b) Deduce the magnetic character (paramagnetic or diamagnetic) of each ion, justifying the number of unpaired electrons in each case, and propose a valid set of quantum numbers $(n,l,m_l,m_s)$ for one of the valence or outermost electrons of the species $\mathrm{Ti}^{2+}$.

Copper (Z = 29) usually forms two stable ions: the cuprous ion $\mathrm{Cu}^+$ and the cupric ion $\mathrm{Cu}^{2+}$.

1. a) Deduce the electron configuration of both ions in their ground state, starting from the actual configuration of the neutral copper atom and bearing in mind the order in which electrons are lost when transition metals ionise.
2. b) Justify in detail which of the two ions is diamagnetic and which is paramagnetic, based on the pairing of their electrons. Then state how many electrons with spin quantum number $m_s=+1/2$ the cation $\mathrm{Cu}^+$ contains in total.