\(2.9\) Predict the ground-state clectron configuration of the following metal ions: (a) \(\mathrm{Cr}^{2+}\); (b) \(\mathrm{V}^{3+}\); (c) \(\mathrm{Zr}^{2+}\); (d) \(\mathrm{Pd}^{2}\)

The ground-state electron configuration is the arrangement of electrons around the nucleus of an atom in its most stable, lowest energy state. Each element has a unique electron configuration that can be determined through the use of the periodic table. The electrons are filled into atomic orbitals in a sequence according to the increasing energy levels, following specific rules: the Aufbau principle, the Pauli exclusion principle, and Hund's rule.

When writing electron configurations, we list the occupied orbitals and the number of electrons in each, using a notation such as '1s² 2s² 2p⁶'. In the exercise, we analyze transition metal ions, which are known for their complex configurations due to the similar energy levels of their 'd' and 's' orbitals.

To predict the electron configuration of metal ions, you must consider the removal (for cations) or addition (for anions) of electrons from their neutral ground state. In the case of cations, you remove electrons starting from the orbitals with the highest 'n' quantum number, which typically begins with the outermost 's' electrons and then the 'd' electrons, as seen in the removal of electrons from chromium (Cr) and vanadium (V) in the exercise.

Atomic orbitals are regions of space around an atom's nucleus where there is a high probability of locating an electron. Each orbital can contain a maximum of two electrons with opposite spins. Orbitals are organized into levels and sublevels (s, p, d, f) based on their shape and energy.

The 's' orbital is spherical, the 'p' orbitals are dumbbell-shaped, and 'd' orbitals have more complex shapes. For transition metals, the 'd' orbitals play a significant role in their chemistry. When filling these orbitals, the electrons are placed to maximize unpaired spins due to Hund's rule, leading to configurations such as those seen in chromium [Ar] 3d⁵ 4s¹ where half-filled 'd' orbitals provide additional stability.

Transition metal ions display a wide variety of electron configurations because of the close energy levels of 's' and 'd' subshells in these elements. In the context of transition metal ions like Cr²⁺, V³⁺, and Pd²⁺, their ground-state configurations are based on the removal of electrons due to the ion's positive charge.

Typically, the electrons from the 's' orbital are removed first before the 'd' orbital, except in some cases where a full or half-full 'd' subshell imparts additional stability, as seen in chromium and palladium. For example, palladium (Pd) in its neutral form has a full 'd' subshell [Kr] 4d¹⁰, which becomes [Kr] 4d⁸ after forming Pd²⁺. These nuances are important when predicting the behavior and reactions of transition metal ions in various chemical contexts.