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.

Phosphorus is an essential non-metallic 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) Analyse 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 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.

A neutral atom $X$ is in its ground state. Its outermost shell has $n=4$ and configuration $4s^2\,4p^2$; the inner shells are complete up to $[\mathrm{Ar}]\,3d^{10}$.

1. a) Deduce the atomic number $Z$, write its complete configuration and identify the group and period of the element.
2. b) Assign a set of four valid quantum numbers to each of its two unpaired electrons, justifying the application of Hund's rule. Is the isolated atom paramagnetic or diamagnetic?