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According to my textbook, vinyl has higher priority than isopropyl, because an atom which is connected by a double bond can be treated as two identical atoms (one of them is just an imaginary atom).

How about vinyl and secondary butyl? Should the imaginary atom be treated as that nothing else is connected to it? Does secondary butyl have higher priority?

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The assignment of CIP priorities has some intricate details when it comes to multiple bonds.

First, the general procedure (Wikipedia):

  1. If there is a tie, we must consider the atoms at distance 2 from the stereocenter — as a list is made for each group of the atoms bonded to the one directly attached to the stereocenter. Each list is arranged in order of decreasing atomic number. Then the lists are compared atom by atom; at the earliest difference, the group containing the atom of higher atomic number receives higher priority.

  2. If there is still a tie, each atom in each of the two lists is replaced with a sub-list of the other atoms bonded to it (at distance 3 from the stereocenter), the sub-lists are arranged in decreasing order of atomic number, and the entire structure is again compared atom by atom. This process is repeated, each time with atoms one bond farther from the stereocenter, until the tie is broken.

Now, concerning double bonds in particular:

If an atom A is double-bonded to an atom B, A is treated as being singly bonded to two atoms: B and a ghost atom that has the same atomic number as B but is not attached to anything except A. In turn, when B is replaced with a list of attached atoms, A itself is excluded in accordance with the general principle of not doubling back along a bond that has just been followed, but a ghost atom for A is included so that the double bond is properly represented from both ends.

I have constructed a table of these atom lists for the vinyl, isopropyl and sec-butyl rests: $$\small \begin{array}{llll} \hline \text{Dist} & \text{Vinyl} & \text{Isopropyl} & sec\text{-Butyl} \\ \hline 1 & \text{C} & \text{C} & \text{C} \\ 2 & \text{C} & \text{C} & \text{C} \\ & \text{(C)} & \text{C} & \text{C} \\ & \text{H} & \text{H} & \text{H} \\ 3 & \text{(C)} & \text{H} & \text{C} \\ & \text{H} & \text{H} & \text{H} \\ & \text{H} & \text{H} & \text{H} \\ & & \text{H} & \text{H} \\ & & \vdots & \vdots \\ \hline \end{array} $$

The sec-butyl rest has thus the highest priority of the three, followed by vinyl and then isopropyl.

The ghost atom at distance three is the ghost carbon at distance one, as seen from the ghost carbon that was doubly bonded to it. If the distance 4 was to be considered, no more ghost atoms would appear, as the ghost atom at position 1 does not have any more ghost atoms bonded to it (atom 1 however is bonded to the ghost atom at position 2).

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