How many electrons are in the valence \(d\) orbitals in these transition-metal ions? (a) \(\mathrm{Co}^{3+},(\mathbf{b}) \mathrm{Cu}^{+}\), (d) \(\mathrm{Os}^{3+}\). (c) \(\mathrm{Cd}^{2+}\)

Transition metal ions play a vital role in chemistry, in part due to the unique properties of their valence d orbitals. These ions are formed when atoms of transition metals lose electrons, often from their outermost shells, resulting in a positively charged ion. The term 'transition' denotes their position on the periodic table, sandwiched between the s-block and p-block elements. They are known for their color, catalytic properties, and the ability to form a variety of complex compounds.

The valence d orbitals are particularly important because they can house up to ten electrons and are largely responsible for the magnetic and chemical behaviors of these metals and their ions. During ion formation, the energy required to remove electrons usually means that the s orbital electrons are lost before the d orbital electrons, despite the counterintuitive order on the periodic table. This behavior of transition metal ions is crucial in fields such as materials science, biochemistry, and industrial chemistry, where they are employed in everything from electronic structures to enzyme functions.

The electron configuration of an atom or ion tells us how electrons are distributed among various orbitals. For transition metals, this configuration becomes more complex due to the involvement of d orbitals, which are of higher energy than s and p orbitals in the same shell. The unique arrangement of electrons in these d orbitals gives transition metals their distinctive properties, including variable oxidation states.

When transition metals form ions, they tend to lose the s orbital electrons first, followed by the d orbital electrons, if necessary. Understanding and predicting the electron configuration of transition metal ions is central to explaining their reactivity, the types of bonds they can form, and their overall chemistry. For instance, the loss of electrons to form \( \(\mathrm{Co}^{3+}\), \(\mathrm{Cu}^{+}\), \(\mathrm{Cd}^{2+}\), and \(\mathrm{Os}^{3+}\) \) ions alters the electron configuration and directly affects the number of electrons in their valence d orbitals, which can be determined using the specifically defined steps in the step-by-step solution.

The periodic table is an essential tool for chemists and a systematic way to organize the elements according to their atomic structure and properties. Its layout is such that it reflects the periodicity of element properties, which is instrumental in predicting the behavior of atoms during chemical reactions. For transition metals, the periodic table provides information on their position between Groups 3 to 12.

Through the periodic table, we can determine an element's atomic number, which is equal to its number of protons and, in an electrically neutral atom, its number of electrons. This assists us in figuring out the ground-state electron configurations of elements and their ions, as well as provides clues about their possible interactions with other elements. For example, knowing the position of Cobalt, Copper, Cadmium, and Osmium on the periodic table, and understanding their propensity to form specific ions, we can deduce their valence electron configurations and make predictions about their physical and chemical behavior.