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Up till high school, I've been taught that reactivity and acidity of a substance in a relation that if a substance is acidic and if it's acidity is higher (comparatively to some other substance) than its reactivity will also be higher. Examples were mostly given of $\ce{HCl}$ and $\ce{H2SO4}$ stating that since $\ce{H2SO4}$ provides more hydrogen ions when ionized hence its reactivity is much higher than $\ce{HCl}$.

However today I stumbled onto a contradiction of this theory. It was stated in a website that the acidity of phenol is much greater than an average alcohol due to a greater stability of conjugate base. Some links ahead it was also stated that the reaction of acyl chloride is less vigorous with phenol as compared to other alcohols due to reactivity of $\ce{-OH}$ group being stabilized by delocalization.

My question is, why does the theory "Acidity is proportional to reactivity" not applies for reaction of acyl chlorides with phenols and alcohols?

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The issue is that reactivity is not an inherent property of a compound. It can be reactive towards specific types of reactions and completely inert to others. Acyl chloride reacting with phenol doesn't rely on phenol behaving as an acid, the type of reaction for which you would describe it as more reactive.

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