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I've considered synthesizing benzoic acid from bromobenzene using CN-, and subsequently treating the nitrile with acid and heat. But, is this possible?

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Economically, your approach is probably a bad idea.

But in general, the conversion of aryl halides to aryl nitriles does work.

I have prepared large amounts of 2,3,5,6-tetramethylterephthalonitrile from 1,2,4,5-tetramethylbenzene in a two-step procedure

  1. iodination
  2. reaction of the diiodide with $\ce{CuCN}$

back in the days for my PhD thesis. For the cyanation, I followed the procedure described in Synthesis, 1974, 53-55, but used N, N-dimethylformamide (DMF) as a solvent.

The halogen-cyanide exchange is know as the Rosenmund-von Braun reaction and you might find further information in a review on Cyanation of aromatic halides (Chem. Rev., 1987, 87, 779-794).

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