\(\mathrm{Ag}_{2} \mathrm{~S}+\mathrm{NaCN} \longrightarrow \mathrm{A}\), \(\mathrm{A}+\mathrm{Zn} \longrightarrow \mathrm{B}+\mathrm{Ag}\) \(\mathrm{A}\) and \(\mathrm{B}\) are respectively (a) \(\mathrm{Na}_{3}\left[\mathrm{Ag}(\mathrm{CN})_{3}\right], \mathrm{Na}_{2}\left[\mathrm{Zn}(\mathrm{CN})_{4}\right]\) (b) \(\mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_{2}\right], \mathrm{Na}\left[\mathrm{Zn}(\mathrm{CN})_{5}\right]\) (c) \(\mathrm{Na}_{2}\left[\mathrm{Zn}(\mathrm{CN})_{4}\right], \mathrm{Na}_{2}\left[\mathrm{Ag}(\mathrm{CN})_{4}\right]\) (d) \(\mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_{2}\right], \mathrm{Na}_{2}\left[\mathrm{Zn}(\mathrm{CN})_{4}\right]\)

The interaction between silver sulfide (Ag2S) and sodium cyanide (NaCN) forms the basis of a complex chemical equation involving the formation of new compounds through a series of reactions. Silver sulfide, a compound formed between silver and sulfur, reacts with sodium cyanide, a cyanide salt of sodium, which is known for its ability to form complexes with metals.

During this reaction, the cyanide ions (CN-) act as ligands, which donate a pair of electrons to the silver ions to form a coordination complex known as the dicyanoargentate ion (Ag(CN)2-). The sodium ions (Na+) balance the overall charge as they accompany the negatively charged complex ion in solution, thus providing the counter ions necessary for electrical neutrality.

This process illustrates a critical aspect of chemical reactions: the ability of certain agents, in this case, cyanide ions, to induce the formation of soluble complexes with metals, often used in applications such as metal purification or extraction.

A displacement reaction is a type of chemical reaction where one element replaces another in a compound, often as a result of differences in reactivity. In the context of the given exercise, zinc (Zn), a more reactive metal compared to silver, is able to displace silver from its complex ion.

In this scenario, when zinc is introduced to the dicyanoargentate (Ag(CN)2-) complex, it essentially 'pushes out' the silver ions, taking their place and forming a new complex, tetracyanozincate (Zn(CN)4^2-). This displacement is made possible because zinc has a greater tendency to form a chemical bond with the cyanide ions.

Displacement reactions are not only important in the laboratory but also in industrial processes such as electroplating, metal refining, and even in the workings of batteries. Understanding the underlying principles of these reactions can help predict product formation and conditions for specific reaction processes.

Complex ions are entities composed of a central metal ion bonded to one or more ligands, which are molecules or ions that can donate a pair of electrons. The formation of complex ions, such as dicyanoargentate (Ag(CN)2-) and tetracyanozincate (Zn(CN)4^2-), demonstrates the variety of ways in which metals can bond with other species.

The properties of complexes, including stability, solubility, and color, can be heavily influenced by the nature of the ligands attached to the metal center. The theory behind complex ions is a fundamental part of coordination chemistry, which plays a vital role across many scientific fields, including biochemistry, medicinal chemistry, and materials science.

In practical applications, complex ion formation is essential for processes such as chelation therapy—a medical procedure where certain chemicals are used to remove heavy metals from the body—and in industrial catalysis, where complex ions often form the active sites of catalysts.