Wittig reaction
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The Wittig reaction is a beautifully direct way to turn a carbonyl into an alkene: it stitches a C=O and a special phosphorus reagent together and pops out a carbon-carbon double bond exactly where the C=O used to be. Where most carbonyl additions leave behind a single C-O bond, the Wittig swaps the oxygen out entirely for carbon.
The reagent is a phosphorus ylide (pronounced ILL-id), R2C=PPh3, a carbon that carries a lone pair and a negative charge stabilized by an adjacent positively charged phosphorus. You make it by deprotonating a phosphonium salt (itself made from a haloalkane and triphenylphosphine) with a strong base. The ylide's nucleophilic carbon attacks the carbonyl carbon, forming a four-membered ring intermediate (an oxaphosphetane) that then collapses: the very strong phosphorus-oxygen bond drives the ring apart, expelling triphenylphosphine oxide (Ph3P=O) and forming the new C=C of the alkene. The whole reaction is essentially irreversible because that P=O bond is so stable.
The Wittig is prized because it places the new double bond at a known, predictable position, the old carbonyl carbon, with no ambiguity about which way an elimination might have gone. Unstabilized ylides tend to give the Z (cis) alkene, while stabilized ylides (bearing an electron-withdrawing group) tend to give the E (trans) alkene, so you even get a handle on geometry. It is a workhorse for synthesizing alkenes, including in the industrial synthesis of vitamin A.
Cyclohexanone plus the ylide Ph3P=CH2 gives methylenecyclohexane, an alkene with the new C=C exactly where the ketone carbonyl was, plus Ph3P=O as byproduct.
The Wittig swaps C=O for C=C at the same carbon, with the phosphorus carrying off the oxygen.
The driving force is the formation of the very strong P=O bond in triphenylphosphine oxide, which is also the price: you generate a stoichiometric amount of Ph3P=O waste that must be separated from the product.