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I want to ask a question about alkenes and their reactivity with acids.

From high school studies, I recall that if you had an acid $\ce{H3O+}$ i.e. $H^+$ that it would react with an alkene as shown below:

enter image description here

However, when I was given a lecture today on Allylic alcohols in a $\ce{S_{N}1}$ reaction, the acid does not react with the alkene at all:

enter image description here

As you may understand, attempting to google or find an equivalent mechanism to explain this observation didn't yield a successful result.

The two mechanisms in my opinion appear to contradict each other, but I couldn't work out how the second mechanism for the Allylic Alcohol functioned whereby the alkene did not react with the acid. My assumption was that it was easier to protonate the alcohol group than react with the acid (with maybe a reasoning along the lines of pKa) but I couldn't find anything sufficient to justify my idea.

Why does the acid not react with the alkene group in the allylic alcohol shown above?

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My suspicion is that the resonance structures accessible to the allylic carbocation provide a lower energy state than the formation of the C--H single bond and the isolated carbocation that your expected pathway would require.

In order of reactivities, you are right to think that the alcohol is the controlling factor because of its pKa. Under acidic conditions it will behave more as a base than the alkene (It is much more basic than an alkene, or think of the Oxygen in an alcohol as being generally a stronger nucleophile than the alkene). This was one main consideration my O-Chem professor stressed was for us to be aware of how other functionalities in our molecules would behave under the reaction conditions of other steps (Eg, if going into a strongly acidic/basic environment after installing a reactive group). I am not an organic chemist by training, but I suspect that if you wanted to access the alkene reactivity, you would want some kind of alcohol protecting group.

This page has a brief discussion about three classes of nucleophile (and the next page describes what makes a good nucleophile): https://www.masterorganicchemistry.com/2011/03/04/the-three-classes-of-nucleophiles/

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Here the issue in question is about rate. Of course, in a real life scenario, both reactions probably take place, the question is which reaction is faster?

Well, looking at the reactants here, you are having a strong acid. The hydroxy group is way more basic/nucleophilic than the double bond simply due to electronic effects. Hence, the Sn1 substitution is expected to be faster and more kinetically favourable. Therefore, though alkene might be hydrated, the predominant reaction is expected to be the Sn1 reaction.

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