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Do carboxylic acids (or any compound with a carbonyl group) count as a branch chained organic compound due to the C=O? Or does it need a carbon - carbon branch?

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In organic chemistry, the terminology of branched vs. unbranched is generally used to refer to the structure of the hydrocarbon skeleton of a molecule. Hence, heteroatomic functional groups are usually not relevant. The structure in your question is therefore unbranched. For an authoritative reference, see the relevant sections of the IUPAC blue book.

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