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Is it possible to fully esterify phenol with oxalic acid in order to get diphenyl oxalate? What catalyst should I use and is the synthesis possible to perform at home with hobby equipment?

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I don't know for sure, but it could work provided some conditions be fulfilled

  1. use of oxalyl chloride
  2. use low temperature, in order to try to suppress fragmentation of the intermediate (semi-ester) into the simple monoacyl chloride (by CO loss): in such cases $\ce{(COCl)2}$ behaves much like phosgene ($\ce{COCl2}$.
  3. use the phenol as a dissociated salt (sodium, potassium) in polar solvents such as sulfolane or acetonitrile, this is optional and possibly risky, due to enhanced competitive "C-acylation" if ortho/para positions are free.
  4. use Piridine-based solvents as reactivity enhancers (they convert acyl chloride to acyl immonium salts). Optionally, even stronger catalysts such as DMAP (p-dimethylamino pyridine) could work

Anyway, to sum up, the major risks stem from:

  1. "C-acylation" (which is generally kinetically slower but irreversible and thermodynamically favored over "O-acylation"), if some suitable position is avail.
  2. formation of diaryl carbonates (on extrusion of CO). I guess this last drawback coul be at least reduced by working at low temperature.
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Diphenyl oxalate can be obtained by transesterification of dimethyl oxalate with phenol.

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