Write the formula and name of (a) a complex ion having \(\mathrm{Cr}^{3+}\) as the central ion and two \(\mathrm{NH}_{3}\) molecules and four \(\mathrm{Cl}^{-}\) ions as ligands (b) a complex ion of iron(III) having a coordination number of 6 and \(\mathrm{CN}^{-}\) as ligands (c) a coordination compound comprising two types of complex ions: one a complex of \(\mathrm{Cr}(\mathrm{III})\) with ethylenediamine (en), having a coordination number of 6 the other, a complex of \(\mathrm{Ni}(\mathrm{II})\) with \(\mathrm{CN}^{-}\), having a coordination number of 4

The complex ion formula represents the combination of a central metal ion with a set of surrounding ligands. In coordination chemistry, the entire assembly of the central ion and ligands is considered a single entity. Understanding the charge of the complex ion is crucial, as it determines the overall formula of the coordination compound.

For example, in the exercise solution, the complex ion \[Cr(NH_{3})_{2}Cl_{4}\] was formed by combining the central ion \(Cr^{3+}\) with 2 ammonia (\(NH_{3}\)) molecules and 4 chloride (\(Cl^{-}\)) ions. The numbering subscript corresponds to the quantity of each ligand around the central metal ion. Notably, the summation of the central ion's charge and the charges of the ligands gives the overall charge of the complex ion. In this case, the complex had an overall charge of -1, coordinating with the charge balance of the metal and ligands.

Ligands are atoms, ions, or molecules that can donate a pair of electrons to a metal ion to form a chemical bond in a complex ion. They are crucial for the structure and function of the complex. Ligands contribute to the stability and the geometrical arrangement of the complex ion.

There are various types of ligands, including monodentate, which bind through a single site (like \(NH_{3}\) or \(Cl^{-}\)), and polydentate, which attach through multiple binding sites simultaneously (like ethylenediamine). In the provided exercise, ammonia and chloride ions are examples of monodentate ligands, while ethylenediamine is a bidentate ligand, binding through two nitrogen atoms.

The coordination number of a metal ion in a complex represents the total number of ligand attachment points to the metal ion. It is a key feature in determining the geometry of a complex ion and can influence its color, magnetic properties, and reactivity.

In the solved problems, the coordination numbers varied, showing that \(Cr^{3+}\) has a coordination number of 6 when complexed with ethylenediamine, and \(Fe^{3+}\) has a coordination number of 6 with cyanide ions. However, \(Ni^{2+}\) forms a complex with a coordination number of 4 with CN^{-} ligands. The coordination number not only defines the number of ligands bound to the central metal but also indirectly informs us about the potential geometric arrangement of the complex ion, which might be octahedral, square planar, tetrahedral, and so on, depending on the number and spatial orientation of ligands.