This is a basic Claisen condensation reaction in which the  functions to take off the acidic hydrogen (between the ketone and the ester) and then the same molecule can attack a carbonyl carbon to yield .

The attack at the ester carbonyl leads to the leaving of  group.

Treatment of ethyl acetate with methoxide will yield the enolate ion (II). However, side trans-esterification reactions will also result in formation of product III.

It is extremely useful in organic synthesis to utilize methods by which less reactive carboxylic acids (and their derivatives) may be converted to more reactive species, like acid chlorides. Such reactions are not always intuitive as they proceed by transforming low-energy species to more energetic states. Reagents such as thionyl chloride  and phosphorus tribromide , which you have likely seen previously in examination of common alcohol reactions, readily convert carboxylic acids to acid chlorides. The reaction is energetically favorable due to the formation of sulfur dioxide, a very stable gas.

A carboxylic acid can be synthesized directly into an acid anhydride by introducing the carboxylic acid to an acid chloride.

This question is asking us to determine what the product will be when a carboxylic acid is reacted with thionyl chloride, . In this reaction, the carboxylic acid is converted into an acyl chloride, with additional production of  and  as side products. Generally, this reaction is very useful for converting carboxylic acids into a more reactive form, so that further reactions can take place.

The Grignard reagent also acts as a base, and will remove any protons that have a low-enough pKa. This means that the alpha hydrogen will be removed, and the resulting carboxylate ion is stabilized through resonance and is unlikely to react with the protonated Grignard reagent.

This reaction is an acid-catalyzed esterification of a carboxylic acid. In the reaction the carbonyl is attacked by the alcohol oxygen, and the carbonyl is reformed by kicking off what was the hydroxyl group in the original carboxylic acid. What is left is an ester (of the form RCOOR', with R' containing the same number of carbons as was in the original alcohol).

The initially chiral carbon has an  hybridization. Once treated with ,an oxidizing agent, the secondary alcohol in the compound is oxidized to a ketone. The central carbon is no longer a chiral center (a carbon with a double bond cannot be chiral), and the double bond (pi-bond) formed between carbon and oxygen gives the molecule an  hybridization.

 hybridization is formed with triple bonds, and an  hybridization does not exist. 

In acidic solution, a hydrogen will be added to the hydroxy group of the given compound. This intermediate is a good leaving group. As it leaves, a hydrogen will be abstracted and a double bond will form. The bond is across carbons two and three becasue those two are more highly substituted than carbon 1, which is primary. The naming begins at the double bond, so the name is 4-bromo-2-hexene.

Chromic acid (formed in-situ via ), the correct answer, is a powerful oxidizing agent that converts primary alcohols to carboxylic acids. PCC is also an oxidizing agent, but oxidizes primary alcohols one "rung" up the ladder to aldehydes. ,  and  are reducing agents, which react in the opposite direction of the given transformation.