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Why is the following compound named 3-phenyl-2-butanol and not 2-phenyl-3-butanol? Does the alcohol or the phenyl group have priority?

3-phenylbutan-2-ol

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You have already correctly identified the compound as a phenylbutanol. The only remaining problem is the numbering of locants. The most important simplified criteria for the numbering in such cases are:

  1. lower locants for the principal characteristic group that is expressed as suffix
  2. lower locants for multiple bonds (first to double bonds if there is a choice)
  3. lower locants for prefixes
  4. lower locants for substituents cited first as a prefix in the name

The corresponding actual wording in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:

P-14.4 NUMBERING

When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:

(…)

(c) principal characteristic groups and free valences (suffixes);

(…)

(e) saturation/unsaturation:

  (i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;

  (ii) low locants are given first to multiple bonds as a set and then to double bonds (…);

(f) detachable alphabetized prefixes, all considered together in a series of increasing numerical order;

(g) lowest locants for the substituent cited first as a prefix in the name;

(…)

Clearly, the $\ce{-OH}$ group is the principal characteristic group in this case since you have already used it as principal characteristic group to find the correct suffix ‘ol’.

Therefore, a low locant is assigned first to the $\ce{-OH}$ group according to Rule (c), which yields phenylbutan-2-ol.

After that, a low locant is assigned to the phenyl group according to Rule (f); however, the only possible numbering left is 3-phenylbutan-2-ol.

3-phenylbutan-2-ol

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