Poly-$$\beta$$-hydroxybutyrate-co-$$\beta$$-hydroxyvalerate (PHBV) is a copolymer of ____.

First, we remind ourselves of what the word “copolymer” means. A copolymer is a polymer that is formed when two different monomer molecules link together in the same long chain. Thus, to identify the monomers of PHBV we must recall the two specific β-hydroxy carboxylic acids that join alternately to build up this polymer.

The name given in the question is “poly-β-hydroxybutyrate-co-β-hydroxyvalerate.” Let us break this down:

• “β-hydroxybutyrate” is the conjugate base of β-hydroxybutyric acid.
• “β-hydroxyvalerate” is the conjugate base of β-hydroxyvaleric acid.

We must therefore translate these common names into proper IUPAC names so that we can match them with the options.

Step 1 - Identifying β-hydroxybutyric acid.
The parent chain “butyric” tells us there are four carbon atoms (butanoic acid). The “β-hydroxy” prefix tells us that the hydroxyl group lies on the β-carbon, which is the third carbon from the carboxyl carbon when we start numbering at the -COOH group. Hence, its IUPAC name is $$3\text{-hydroxybutanoic\ acid}.$$ The structure is $$CH_3-CH(OH)-CH_2-COOH.$$

Step 2 - Identifying β-hydroxyvaleric acid.
“Valeric” corresponds to a five-carbon straight chain (pentanoic acid). Once again “β-hydroxy” places the -OH on the β-carbon, i.e. the third carbon from -COOH. Therefore its IUPAC name is $$3\text{-hydroxypentanoic\ acid},$$ with structure $$CH_3-CH_2-CH(OH)-CH_2-COOH.$$

Now, let us compare these IUPAC names with the options given:

A. 3-hydroxybutanoic acid and 4-hydroxypentanoic acid ⟹ wrong second position
B. 3-hydroxybutanoic acid and 2-hydroxypentanoic acid ⟹ wrong second position
C. 3-hydroxybutanoic acid and 3-hydroxypentanoic acid ⟹ both positions correctly at 3 (β-position)
D. 2-hydroxybutanoic acid and 3-hydroxypentanoic acid ⟹ wrong first position

Only Option C contains both monomers in which the hydroxyl group is indeed on the β-carbon (carbon-3) of each respective acid, matching the common names β-hydroxybutyric acid and β-hydroxyvaleric acid that build PHBV.

Hence, the correct answer is Option C.