\(\mathrm{R}-\mathrm{CH}_{2}-\mathrm{CH}_{2} \mathrm{OH}\) can be converted into \(\mathrm{RCH}_{2} \mathrm{CH}_{2}\) COOH. The correct sequence of reagent is: (a) \(\mathrm{PBr}_{3}, \mathrm{KCN}, \mathrm{H}^{+}\) (b) \(\mathrm{HCN}, \mathrm{PBr}_{3}, \mathrm{H}^{+}\) (c) \(\mathrm{KCN}, \mathrm{H}^{+}\) (d) \(\mathrm{PBr}_{3}, \mathrm{KCN}, \mathrm{H}_{2}\)

When we talk about organic chemistry, one of the transformations that often comes up is the conversion of an alcohol to a carboxylic acid. This process is fundamental in synthesizing various compounds, including pharmaceuticals and polymers. The transformation begins by substituting the hydroxyl group (-OH) of an alcohol with a more reactive halogen, typically a bromine, using phosphorus tribromide (PBr₃). After the conversion to a halide, a cyano group (-CN) is introduced through a nucleophilic substitution reaction. Lastly, this cyano group is transformed into a carboxylic acid (-COOH) group by hydrolysis. Understanding each step and the reagents involved is crucial in mastering organic synthesis.

It's essential to note that in the initial step of this conversion, PBr₃ is used to convert the -OH group to a bromide (-Br), thereby forming a more reactive intermediate that is susceptible to further substitution. Without this initial step, the subsequent reactions would not proceed efficiently. The meticulous sequence of these reactions and the use of appropriate reagents is what allows for the precise conversion from an alcohol to a carboxylic acid.

Nucleophilic substitution reactions are a cornerstone of organic chemistry, vital in altering the structure of a molecule. They involve the replacement of an atom or group (usually halogen) by a nucleophile. A nucleophile is a chemical species that donates an electron pair to form a chemical bond in relation to a reaction. In the context of the presented exercise, once the alcohol is converted to a bromoalkane, a nucleophilic substitution reaction is used to replace the bromine atom with a cyano group. Potassium cyanide (KCN) acts as the nucleophile in this reaction.

The bromoalkane is an ideal substrate for this reaction due to its reactivity, and the effectiveness with which the cyano group can replace the bromine atom is pivotal in the synthetic pathway towards creating a carboxylic acid. This step is not just a simple swap, but a carefully coordinated dance of electrons, where the bromine leaves with its pair of electrons and the nucleophilic cyanide ion comes in, bringing along its electrons to form a new carbon-nitrogen bond.

The final step required for converting an alcohol to a carboxylic acid is the hydrolysis of a nitrile. Nitrile hydrolysis is an acid-catalyzed reaction where the cyano group of a nitrile transforms into a carboxylic acid. In the presence of an acid catalyst (H⁺), water reacts with the carbon-nitrogen triple bond of the cyano group (C≡N), leading to the addition of an -OH group and the conversion to a carboxylic acid (COOH). This reaction is significant due to its ability to extend the carbon skeleton during the conversion process, an important aspect in the synthesis of a variety of organic compounds.

For students tackling chemistry problems, understanding the role of the acid catalyst and the mechanism of addition of water across the triple bond in nitriles can provide a deeper comprehension of organic reaction pathways. It is the precise manipulation of such reactions that enables chemists to construct complex molecules with potential applications in different fields of chemistry.