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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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Yes, it is possible. The Rosenmun-von Braun reaction is the classic method of doing this. But it involves CuCN which is quite toxic.

I'd suggest that you investigate some more modern methods. These use palladium based catalysts. Many also use Zn(CN)2 which is still toxic but less so than CuCN. These two are good for starters:

  • Iqbal, Z., Lyubimtsev, A., & Hanack, M. (2008). Synthesis of Phthalonitriles Using a Palladium Catalyst. Synlett, 2008(15), 2287–2290. https://doi.org/10.1055/s-2008-1078269
  • Martynov, A. G., Birin, K. P., Gorbunova, Y. G., & Tsivadzea, A. Y. (2013). Modern synthetic approaches to phthalonitriles with special emphasis on transition-metal catalyzed cyanation reactions. Macroheterocycles, 6(1), 23–32. https://doi.org/10.6060/mhc130221m

This however would be a very silly and expensive way of making benzoic acid. Oxidation of toluene using permanganate would be much easier and quicker.

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