Using your roadmaps as a guide, show how to convert \((E)\)-8-hexene into propyl propionate. You must use \((E)\)-3-hexene as the source of all carbon atoms in the target molecule. Show all reagents and all molecules synthesized along the way.

When examining the synthesis of propyl propionate from \(E\)-3-hexene, we must first understand this starting molecule. \(E\)-3-hexene is an alkene, meaning it has a double bond between two of its carbon atoms. The \(E\) notation indicates that the two largest groups attached to the carbons in the double bond are on opposite sides (from the German word 'Entgegen,' meaning opposite). In organic synthesis, alkenes like \(E\)-3-hexene are valuable as they can be transformed into a variety of other compounds through reactions such as hydrohalogenation.

In the exercise provided, \(E\)-3-hexene is the backbone for the target molecule, and the first step in the synthesis involves adding a halogen to the molecule to create an alkyl halide. The carbon chain of \(E\)-3-hexene dictates the basic structure of the final product, ensuring that all carbon atoms in propyl propionate come from this source.

The Grignard reagent is a cornerstone of organic synthesis and plays a pivotal role in the creation of propyl propionate from \(E\)-3-hexene. Named after French chemist Victor Grignard, this powerful reagent is formed when an alkyl, allyl, or aryl halide reacts with magnesium in dry ether. The general formula for a Grignard reagent is \(RMgX\), with 'R' representing an organic group and 'X' indicating a halogen atom.

In the process described in the solution, we see the conversion of 3-hexyl bromide to 3-hexyl magnesium bromide. This Grignard reagent acts as a nucleophile, capable of attacking electrophiles such as carbonyl compounds. The Grignard addition to an ester, as outlined in the steps, leads to the formation of a tertiary alcohol, which is a crucial intermediate towards the synthesis of the desired ester, propyl propionate.

The process of converting alcohols into esters is an art in organic chemistry, and Fischer esterification is a classic example of this transformation. This reaction, named after the German chemist Hermann Emil Fischer, involves an acid-catalyzed reaction between an alcohol and a carboxylic acid. The reaction proceeds by heating the two reactants with a strong acid, typically sulfuric acid, which acts as a dehydrating agent, facilitating the removal of water and thus driving the reaction towards ester formation.

In our synthesis puzzle, we convert the tertiary alcohol obtained from the Grignard addition step into propyl propionate using Fischer esterification. The alcohol reacts with propionic acid under acidic conditions, yielding the ester and water as a byproduct. This step is critical as it leads to the formation of the desired ester functional group present in propyl propionate.

The hydrohalogenation reaction is a method to add halogen acids like HBr or HCl across the double bond of an alkene to form an alkyl halide. It involves the electrophilic addition, where the alkene acts as a nucleophile and the halogen acid as the electrophile. The result is the addition of a halogen atom on the more substituted carbon -- a pattern known as Markovnikov's rule.

In the given solution, hydrohalogenation is the initial step to converting \(E\)-3-hexene into 3-hexyl bromide. This new compound becomes the foundation for the formation of the Grignard reagent, which is critical to progressing towards the final product of propyl propionate. Understanding this initial transformation is essential, as it is the first move in the multi-step synthesis of the target molecule.