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$\hspace{6.4cm}$image of ethyl acetate

I would like to know how to name the alkyl groups connected to the above molecule. For example, would the far left alkyl group be considered a methyl ester group? Would the alkyl group bound to the oxygen be a methylene ester group?

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You are going about naming them incorrectly, try breaking it into two parts: the conjugate base of a carboxylic acid and an alkyl group.

$\hspace{5.9cm}$ester parts

Combining the two, we get:

$\hspace{6.9cm}$ester

Both the carboxyl and alkyl carbon chains can be numbered, with the carboxyl chain starting at the carboxyl carbon, and the alkyl chain starting at the carbon that places its bond to the ester oxygen at the lowest position. See below:

$\hspace{5.5cm}$numbering example

To refer to a specific carbon, simply say "at position $x$ on the carboxyl/alkyl chain."

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