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$\ce{FeBr3}$ catalyzes bromination and $\ce{AlCl3}$ chlorination. I'd like to know whether bromination and chlorination can be done in sequence without changing the catalyst? So - will using the $\ce{FeBr3}$ for chlorination and $\ce{AlCl3}$ for bromination work? Is it less efficient or slower or is there another reason?

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The reason you use the respective halides for halogenation is that the halogen and Lewis acid can inter-react to form a chemical equilibrium if you don't.

$$\ce{FeBr3 + Cl2 <=> Cl+ + [FeBr3Cl]- <=> Br+ + [FeBr2Cl2]- }\\ \ce{AlCl3 + Br2 <=> Br+ + [AlBrCl3]- <=> Cl+ + [AlBr2Cl2]- }$$

Given these equilibrium reactions you can see that now you have two electrophile species which means your halogenation is not as controlled. It will work and halogenation will still occur, but now and you will have bromination and chlorination occurring in competition with each other when you only wanted one or the other.

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