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What are the rules for choosing main branch?

Wikipedia states:

Identification of the parent hydrocarbon chain. This chain must obey the following rules, in order of precedence:

  1. It should have the maximum number of substituents of the suffix functional group. By suffix, it is meant that the parent functional group should have a suffix, unlike halogen substituents. If more than one functional group is present, the one with highest precedence should be used.
  2. It should have the maximum number of multiple bonds
  3. It should have the maximum number of single bonds.
  4. It should have the maximum length.

I don't understand how 3 and 4 differ. Which single bonds are we counting? Only carbon-carbon single bonds in the main chain, right?

Also, I heard there is a rule that says if you've two equal length chains, then choose the more substituted one. Why is it not given here?

And now now imagine you've two choices for a chain, like this:

enter image description here

The chemdoodle program gives the name as 2-(1-Hydroxybutyl)-3-methylhexanoic acid .

I simply do not understand why the methyl substituted chain is chosen instead of hydroxy substituted one and how that fits with IUPAC rules given in Wikipedia.

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For the given structure, the principal chain could be a 3-hydroxyhexanoic acid or a 3-methylhexanoic acid; i.e. the name of the structure could be 3-hydroxy-2-(pentan-2-yl)hexanoic acid or 2-(1-hydroxybutyl)-3-methylhexanoic acid.

2-(1-hydroxybutyl)-3-methylhexanoic acid

The IUPAC recommendations on this matter have been changed.

According to the old IUPAC recommendations (1979, retained in 1993), the principal chain was the 3-hydroxyhexanoic acid. Thus, the preferred name was 3-hydroxy-2-(pentan-2-yl)hexanoic acid.

However, the relevant rule in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:

P-45 SELECTION OF PREFERRED IUPAC NAMES

(…)

P-45.5 CRITERIA RELATED TO ALPHANUMERICAL ORDER OF NAMES

P-45.5.1 The preferred IUPAC name is the name that is earlier in alphanumerical order (…). Alphabetic letters are considered first in the order that they appear in the name; (…)

Therefore, the preferred name is 2-(1-hydroxybutyl)-3-methylhexanoic acid (not 3-hydroxy-2-(pentan-2-yl)hexanoic acid) since ‘hydroxybutyl…’ is earlier alphabetically than ‘hydroxy-pentan-yl…’.

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For point 3 carbon-carbon single bonds count. Heteroatoms would build up a functional group. However, I could not think of a scenario where point 4 is needed. It should be a scenario where two chains with an equal amount of multiple bonds and single bonds are present but different in length.

The rule, which you mention, in which the number of functional groups count is point 1.

According to what I have learned the name "2-(1-Hydroxybutyl)-3-methylhexanoic acid" is wrong. There are two equal chains, one has a methyl and one a hydroxyl group. According to IUPAC nomenclature of organic chemistry the group with the higher precedence is hydroxyl. Therefore, chemdoodle gave you the wrong name.
This is a case where this sentence applies which they put above: "Please give it a try and let us know if you encounter any issues."

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