Which \(\mathrm{M}^{3+}\) ions (where \(\mathrm{M}\) is a metal) are predicted to have the following ground-state electron configurations: (a) \([\mathrm{Ar}] 3 \mathrm{~d}^{6}\); (b) \([\mathrm{Ar}] 3 \mathrm{~d}^{5}\); (c) \([\mathrm{Kr}] 4 \mathrm{~d}^{5}\); (d) \([\mathrm{Kr}] 4 \mathrm{~d}^{3}\) ?

Electron configuration is like a map that outlines the locations of electrons around an atom's nucleus. These configurations follow a set of rules known as the Aufbau Principle, the Pauli Exclusion Principle, and Hund's Rule. The Aufbau Principle suggests that electrons fill the lowest energy orbitals first before moving on to higher levels. The Pauli Exclusion Principle states that no two electrons can have the same set of four quantum numbers, which means an orbital can hold a maximum of two electrons with opposite spins. Hund's Rule tells us that electrons will fill degenerate orbitals—one electron per orbital with parallel spins—before pairing up.

Electron configurations are expressed in a notation that includes the energy level (principal quantum number), the type of orbital (s, p, d, f), and the number of electrons in those orbitals, written as superscripts. For example, the notation \(1s^2 2s^2 2p^6\) implies that the first energy level 's' orbital has 2 electrons, the second level 's' orbital has 2 electrons, and the second level 'p' orbitals have a total of 6 electrons. This approach helps students to predict and understand the chemical behavior of elements.

Transition metals are found in the central block of the periodic table, specifically in the d-block ranging from groups 3 through 12. These metals are unique because their d-orbitals are being filled with electrons. The distinctive properties of transition metals, such as varied oxidation states and the ability to form colored compounds, can be attributed to these d-orbitals.

In the case of the exercise, we are looking at transition metal ions with a +3 charge, which means three electrons have been removed from a neutral atom. When determining the ground-state electron configuration of these ions, we take into account the stability of half-filled and fully-filled d-orbitals, which often renders them exceptionally stable configurations. The offered electron configurations in the exercise help us understand that these metal ions were derived by removing electrons from the d-orbitals of their neutral states—such a process is classic to transition metals due to their versatile electron configurations.

The periodic table is a systematic array of elements, ordered by increasing atomic number, which is the number of protons in an element's nucleus. It is not only a vital tool for chemists but also for anyone studying the sciences, as it helps predict the types of chemical reactions that an element can participate in, based on its position.

The table is organized into rows called periods and columns known as groups or families. Elements in the same group typically have similar properties and the same valence electron configuration, which determines their behavior in chemical reactions. For students working through exercises involving electron configurations and the properties of metal ions, the periodic table provides critical information. For example, by referring to the periodic table, one can verify whether removing three electrons to form a +3 cation is possible for a given element, just like in our original exercise where a transition metal's electron configuration helps us locate its position and identity on the periodic table.