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What are the IUPAC systematic name for $\beta$-glucose and lactic acid (as in the monomers used in polymerization for PLA)?

For lactic acid, I found this:

2-hydroxypropanoic acid

Is this correct?

For $\beta$-glucose, I have not find anything conclusive so far. I did find on this site (Carbohydrates) that "carbohydrates have been given non-systematic names". Is this true, and does that imply that "$\beta$-glucose" is simply $\beta$-glucose?

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A compound that can be isolated from a natural source is commonly given a trivial name. These trivial names are usually related to the biological origin of the material. When the full structure is known, a systematic name may be generated in accordance with the IUPAC recommendations for nomenclature of organic chemistry; however, this name may be too cumbersome (for example for repeated use in the text of scientific publication).

According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), the preferred IUPAC name (PIN) for lactic acid is the systematic name 2-hydroxypropanoic acid. Nevertheless, the name lactic acid is retained for use in general nomenclature.

P-65.1.1.2.3 The names citric acid, lactic acid, glyceric acid, pyruvic acid, and tartaric acid, related to natural products, are also retained; no substitution is recommended, but the formation of salts and esters is allowed.

For carbohydrates such as glucose, the current IUPAC recommendations do not identify a PIN. The systematic name for the β-pyranose form of ᴅ-glucose is (2⁠R,3⁠R,4⁠S,5⁠S,6⁠R)-6-(hydroxymethyl)oxane-2,3,4,5-tetrol. Since this name may be too cumbersome, a semisystematic name may be formed based on the retained name ᴅ-glucose as follows.

P-102.3.4 Cyclic forms of monosaccharides

Most monosaccharides exist as cyclic hemiacetals or hemiketals. Two aspects of the internal cyclisation must be examined: first, the size of the ring, and secondly, the conformation of the newly created chirality center.

P-102.3.4.1 Ring size

Out of the various possible heterocyclic ring sizes resulting from hemiacetal or hemiketal formation, those with five and six members, including an oxygen atom, prevail and are discussed in this Section. Their names are based on those of the parent heterocycles furan and pyran, respectively. Names are formed by including the terms ‘furan’ and ‘pyran’ before the ending ‘ose’ in the name of a sugar. (…)

Thus, the retained name ᴅ-glucose is changed to ᴅ-glucopyranose to indicate the cyclic form having a six-membered ring.

P-102.3.4.2 Anomeric forms; the stereodescriptors ‘α’ and ‘β’

P-102.3.4.2.1 In the cyclic form, the configuration of the newly created chirality center ‘C-1’ must be expressed. This center is called the ‘anomeric center’. The two stereosiomers are called ‘anomers’; they are designated by the stereodescriptors ‘α’ and ‘β’ according to the configurational relationship between the anomeric center and the so called ‘reference center’.

P-102.3.4.2.2 Configurations ‘α’ and ‘β’ for monosaccharides

The anomeric reference center in a monosaccharide having a retained name is the configurational atom as defined in P-102.3.3. (…)
The anomeric stereodescriptor ‘α’ or ‘β’, followed by a hyphen, is placed immediately before the configurational stereodescriptor ‘ᴅ’ or ‘ʟ’ of the carbohydrate name.

Therefore, the resulting name for the β-pyranose form of ᴅ-glucose is β-ᴅ-glucopyranose.

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