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Methanol to ethanoic acid.

Answer
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Hint :The process of conversion is when we use a number of steps or maybe a single step with specific reagents in order to convert one compound to the other. We also know that the ethanoic acid possesses two carbon atoms and cyanide is used as a reagent which helps in addition to carbon atoms.

Complete Step By Step Answer:
In order to answer this question we must first know what it is asking for. The question expects us to convert the methanol which has the formula $ C{H_3}OH $ to the ethanoic acid which can be represented by the formula $ C{H_3}COOH $ which is also known as the acetic acid. Now if we notice, the conversion should be carried out such that the number of carbon atoms would increase. So, we would have to use a reagent which would provide a carbon atom in the final product. We know that the cyanide is one such group which could provide a carbon atom, so we would use it in the process of conversion. Now, we will discuss the steps one by one.
Firstly, the conversion of the methanol to ethanoic acid can be done by the use of phosphorus pentachloride which is a reducing agent and it will reduce the hydroxide group which is present in the methanol and substitute it with chlorine. So, we will get methyl chloride as a product which can be shown as:
 $ C{H_3}OH\xrightarrow{{PC{l_5}}}C{H_3}Cl $
Now we will add ethanolic sodium cyanide to this product and the chloride would get substituted with the cyanide group which was present with the sodium in the reagent.
 $ C{H_3}Cl\xrightarrow{{Ethanolic\;NaCN}}C{H_3}CN $
Now, upon acidic hydrolysis, we would get ethanoic acid as the end product and it cab shown as:
 $ C{H_3}CN\xrightarrow[{{H_2}O}]{{{H^ + }}}C{H_3}COOH $
Hence, we can show the whole equation for this conversion as follows:
 $ C{H_3}OH\xrightarrow{{PC{l_5}}}C{H_3}Cl\xrightarrow{{Ethanolic\;NaCN}}C{H_3}CN\xrightarrow[{{H_2}O}]{{{H^ + }}}C{H_3}COOH $ .

Note :
The conversion of the methanol to ethanoic acid is done by using phosphorus pentachloride which reduces the alcoholic group to the chloride group.
And then we add sodium cyanide in order to increase the number of carbons in the basic change, which upon hydrolysis gets converted to a carboxylic group.