The oxidation number of \(\mathrm{Fe}\) in \(\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}, \mathrm{Cr}\) in \(\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{NO}_{2}\right)_{3}\right]\) and \(\mathrm{Ni}\) in \(\left[\mathrm{Ni}(\mathrm{CO})_{4}\right]\) are respectively (a) \(0,+3,+2\) (b) \(+3,+3,0\) (c) \(+3,0,+3\) (d) \(+2,+3,0\)

Coordination chemistry is a branch of inorganic chemistry that deals with the study of compounds formed when an array of ligands, which are molecules or ions that can donate a pair of electrons, align themselves around a central metal atom or ion. Such assemblies are termed coordination compounds or complexes. The central metal atom or ion, also known as the coordination center, accepts these electron pairs from the ligands, forming coordinate covalent bonds.

Understanding the nature of these complexes is crucial because they play significant roles in various processes, including biological systems and industrial applications. For example, the hemoglobin in your blood is a coordination compound that helps transport oxygen throughout the body. An intriguing aspect of coordination compounds is their ability to exhibit different colors based on the metal and the ligands involved, which is why they are frequently used in dyes and pigments.

It is also essential to understand the coordination number, which is the number of ligand attachment points to the metal center. The geometry of the complex (whether it's square planar, tetrahedral, octahedral, etc.) depends heavily on this coordination number. In the context of our exercise, for instance, \(\mathrm{Fe}(\mathrm{CN})_{6}\)^{4-}\ has an octahedral geometry because of its six cyanide ligands coordinating the Fe center.

Chemical bonding refers to the attraction between atoms or molecules that allows the formation of chemical compounds. There are mainly three types of bonds: ionic, covalent, and metallic. In coordination compounds, the bond formed between the ligands and the central metal is known as a coordinate covalent bond or dative bond.

In these bonds, both of the electrons in the bond come from the same atom, typically the ligand, which contrasts with a normal covalent bond where each atom supplies one electron. This distinction is particularly crucial when predicting properties like reactivity or even magnetic characteristics of the coordination compound. Moreover, in coordination complexes, the type of bonding between the ligands and the metal center can significantly influence the stability and shape of the molecule. Understanding these bonding types can help in deciphering the structure and reactivity of coordination compounds, which relates to the exercise we are discussing.

Oxidation number calculations are pivotal in the study of chemical reactions, particularly redox reactions, where electron transfer is involved. An oxidation number denotes the charge an atom would have if the compound was composed of ions. It helps chemists understand electron distribution within a compound, which is fundamental in predicting reactivity and product formation.

In coordination chemistry, identifying the oxidation states of metal ions is essential for naming complexes and understanding their reactivity and properties. When calculating oxidation numbers in coordination complexes, we first identify the known oxidation states of the ligands. For example, in the complex \(\mathrm{Fe}(\mathrm{CN})_{6}\)^{4-}\, we know that the cyanide ion (CN) commonly has a charge of -1. Given the charge of the entire complex, we can deduce the oxidation number of the central metal by balancing the total charge of the complex with those of the ligands. Simple arithmetic is used, where the sum of the oxidation numbers multiplied by their frequency plus the metal's oxidation state equals the complex's overall charge. Through such calculations, we arrived at the solution for the oxidation states of \(\mathrm{Fe}\), \(\mathrm{Cr}\), and \(\mathrm{Ni}\) in the given exercise.