Which one of the following forms, with an excess of \(\mathrm{CN}^{-}\), a complex having coordination number two? (a) \(\mathrm{Ni}^{2+}\) (b) \(\mathrm{Cu}^{+}\) (c) \(\mathrm{Ag}^{+}\) (d) \(\mathrm{Fe}^{2+}\)

The process where central metal atoms or ions bind with molecules or anions referred to as ligands is known as complex formation. In this process, the ligands donate pairs of electrons to the metal center, creating a more stable arrangement called a coordination compound or complex. Complex formation is a crucial aspect of inorganic chemistry, particularly in the synthesis of new materials and in understanding biological systems.

When exploring complex formation, we consider the coordination number, which represents the number of ligands directly attached to the central metal ion. Taking the exercise where \(\mathrm{CN}^{-}\) acts as a ligand, it will offer a lone pair of electrons to metal ions such as \(\mathrm{Ag}^{+}\), resulting in the formation of a coordination complex with a specific geometry based upon the metal's preference and ligands' properties.

Transition metals are elements found in the d block of the periodic table known for their ability to form a variety of different compounds, with various oxidation states. The complex chemistry of transition metals is dictated by their electron configurations, particularly the d electrons that can participate in bonding.

In this context, assessing options like \(\mathrm{Ni}^{2+}\), \(\mathrm{Cu}^{+}\), \(\mathrm{Ag}^{+}\), and \(\mathrm{Fe}^{2+}\), their ability to form complexes varies. For instance, \(\mathrm{Cu}^{+}\) and \(\mathrm{Ag}^{+}\) often exhibit a 3d10 electronic configuration, which is associated with less complex formation with high coordination numbers. Instead, these ions frequently form complexes with lower coordination numbers, correlating their electronic configuration and the specific size and electronic needs of the ion.

Ligand-field theory is a sophisticated model that explains how ligands affect the energy levels of d orbitals in a transition metal ion. It is an adaptation of crystal field theory and molecular orbital theory that takes into account covalent, as well as ionic, aspects of coordination.

The theory aids in predicting the magnetic and spectral properties of metal complexes. When ligands such as \(\mathrm{CN}^{-}\) approach the metal ion, they interact with the d orbitals and split them into sets of orbitals with varied energy levels. This splitting influences the arrangement of electrons among these orbitals and impacts the observable properties of the complexes. Understanding how ligands like \(\mathrm{CN}^{-}\) influence the d orbitals helps to explain why \(\mathrm{Ag}^{+}\), a transition metal ion, can preferentially form a complex with a coordination number of two, as seen in the provided exercise where the \(\mathrm{CN}^{-}\) ligands create a stable complex featuring this specific coordination number.