\(2.21\) What is the ground-state clectron configuration expected for each of the folluwing ions: (a) \(B i^{34}\); (b) \(\mathrm{Sn}^{4+} ;\) (c) \(\mathrm{P}^{3-} ;\) (d) \(\mathrm{Br}^{-} ;\)(e) \(\mathrm{Ni}^{2+}\) ?

The Aufbau principle is a fundamental concept in understanding how electrons occupy available energy levels within an atom. It states that electrons fill the lowest energy orbitals first before moving on to higher energy orbitals.

This principle helps us predict the electron configurations of atoms and ions, with electrons filling sublevels in a specific order: 1s, 2s, 2p, 3s, and so on. It's based on the idea that lower energy states are more stable and thus, more preferable for electrons. When an atom transforms into an ion, the Aufbau principle still applies, but the total number of electrons changes based on the charge of the ion.

The Pauli exclusion principle is another key concept that governs the arrangement of electrons in an atom or ion. It asserts that no two electrons in an atom can have the same set of four quantum numbers, which implies that each orbital can hold a maximum of two electrons, and they must have opposite spins.

Under this principle, when we write electron configurations, we ensure that electrons are paired correctly in orbitals. For example, in a Sn^4+ ion, the electrons are paired up until the 4d sublevel, adhering to the Pauli exclusion principle, and accommodating the loss of four electrons due to the ion's positive charge.

Hund's rule addresses the distribution of electrons across orbitals of the same sublevel. It states that electrons will occupy empty orbitals singly before doubling up in any one orbital, and all singly occupied orbitals will have electrons with the same spin.

For instance, in the case of the P^3- ion, after following the Aufbau principle to reach the 3p sublevel, Hund's rule dictates filling each of the three 3p orbitals with one electron before any pairing occurs. When the phosphorus atom gains three electrons to become a P^3- ion, these additional electrons fill the remaining 3p orbitals, thereby obeying Hund's rule and completing the sublevel.

Noble gas notation is a shorthand representation used to simplify the expression of electron configurations, especially for atoms with many electrons. It involves starting with the electron configuration of the nearest noble gas element with fewer electrons and then showing the remaining electrons in their respective orbitals.

For example, for Br^-, which has one more electron than the neutral bromine atom, the noble gas notation begins with [Ar], the noble gas prior to bromine, and then adds the configuration for the extra electron: 4p^6. This notation is helpful in quickly identifying the valence electron configuration, which shows the electrons that are important in chemical bonding.