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I want to understand the mechanism of desilylation of trimethylsilyl under fluoride conditions in methanol, because I can't find any useful info that makes me understand the mechanism that takes place.

I did find this:

enter image description here

But my case is a little bit different because a terminal alkyne is bonded to the silyl group instead of oxygen:

enter image description here

Why would that bond be broken and not any other carbon-hydrogen bond?

Thanks in advance!

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The mechanism is the same with the generation of the alkynyl anion as the abstract of this paper ref 1 makes clear:

Tetrabutylammonium triphenyldifluorosilicate (TBAT) can be employed as a fluoride source to cleave silicon−carbon bonds thus generating in situ carbanions that coupled with a variety of electrophiles, including aldehydes and ketones, in moderate to high yields. Among the examples reported is the first instance of fluoride-induced intermolecular coupling between allyltrimethylsilane and imine derivatives. Also, of particular note is the TBAT-initiated coupling of primary alkyl halides with allyltrimethylsilane. TBAT is an easily handled crystalline solid that has several advantages over tetrabutylammonium fluoride (TBAF) as a fluoride source; it is anhydrous, nonhygroscopic, soluble in most commonly used organic solvents, and less basic than TBAF.

Bear in mind that the $\mathrm{p}K_\mathrm{a}$ of an alkynyl proton is approximately 25 (ref 2) so this is a far more stable anion than any possible alternative. Clearly with $\ce{MeOH}$ as the solvent it will be quenched immediately.

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