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What would happen if I used a 5 speed chain?
On a more modern RD, it might rub or even jam in the pulley cage, because they are designed for narrower chain. 8-speed is typically 6.6mm pin length and will probably run OK in RDs designed for up to 11-speed (~5.6mm pin length). Before skinny flush riveted bushingless chains arrived in the 1980s, dérailleur bikes used standard 3/32" chain with pins of about 7.8mm, so pulley cages were made wide enough to accommodate it.
I'm mystified that anybody would take an old chain off and not replace it, if you shop around you can find basic 8-speed dérailleur chain for a fiver.
What would happen if I used a 5 speed chain? (don't really want to buy or replace the chain)
Would it just be a bit rough on the pull wheels? Or would it get totally stuck?