In this hydrocarbon there can be two main chains: one with two double bonds and one with a double and a triple bond
Both main chains contain seven carbon atoms. Which will be considered correct and why?
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According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), as already mentioned in the question, the first relevant criterion to be considered in choosing a principal chain is the length of the chain.
When there is a choice for the principal chain, the following criteria are applied, in the order listed, until a decision is reached:
P-44.4.1 If the criteria of P-44.1 through P-44.3, where applicable, do not effect a choice of a senior parent structure, the following criteria are applied successively until there are no alternatives remaining. (…)
The senior ring, ring system, or principal chain:
(a) has the greater number of multiple bonds (P-22.214.171.124);
(b) has the greater number of double bonds (P-126.96.36.199);
(h) has the lower locant for an attached group expressed as a suffix (P-188.8.131.52);
Since both possible chains in this case have two multiple bonds, Rule (a) is not enough to make a choice.
According to Rule (b), the principal chain is the hepta-1,6-diene because it has the greater number of double bonds.
Therefore, the preferred IUPAC name (PIN) is 4-(prop-2-yn-1-yl)hepta-1,6-diene.