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The five-membered heterocycles pyrrole, furan, and thiophene undergo Friedel-Crafts acylation, but I have not seen them undergoing a simple Friedel-Crafts alkylation reaction with a Lewis acid such as aluminium chloride. Why is that so?

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Alkylation of the five-membered rings is perfectly possible - nobody says it cannot happen. But there are two problems:

  1. Alkylation further activates the ring, making it more reactive - this is exactly the same problem that makes alkylation an unattractive reaction in benzene rings
  2. The Lewis acid can, and will, lead to polymerisation of electron-rich rings such as the five-membered heterocycles.

With acylation, the first point is moot, and if you choose an appropriate catalyst you can avoid polymerisation. So acylation can be used in certain cases (I'm fairly sure that simply chucking in an acyl chloride with $\ce{AlCl3}$ will not be a clean reaction at all though).

From Joule & Mills, Heterocyclic Chemistry 5th ed., p 300 (chapter on pyrrole):

Mono-C-alkylation of pyrroles cannot be achieved by direct reaction with simple alkyl halides, either alone or with a Lewis-acid catalyst, for example pyrrole does not react with methyl iodide below 100 °C; above about 150 °C, a series of reactions occurs leading to a complex mixture made up mostly of polymeric material together with some poly-methylated pyrroles. The more reactive allyl bromide reacts with pyrrole at room temperature, but mixtures of mono- to tetra-allyl-pyrroles together with oligomers and polymers are obtained.

p 327 (chapter on thiophene):

Alkylation occurs readily, but is rarely of preparative use, an exception being the efficient 2,5-bis-t-butylation of thiophene. (J. Org. Chem., 1984, 49, 4161. DOI:10.1021/jo00196a012)

Presumably the steric bulk of the t-butyl group prevents further reactions.

p 350 (chapter on furan):

Traditional Friedel–Crafts alkylation is not generally practicable in the furan series, partly because of catalyst-induced polymerisation and partly because of polyalkylation.

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