A method of making acetic anhydride or a mixture of acetic anhydride and acetic acid comprising: (a) catalytically reacting a feedstock containing methyl acetate and/or dimethyl ether with carbon monoxide in the presence of a homogeneous rhodium catalyst and methyl iodide in a reactor vessel which contains a substantially anhydrous liquid reaction mixture including acetic acid, acetic anhydride, methyl acetate and/or dimethyl ether, methyl iodide and the homogeneous catalyst, the reactor vessel being operated at a reactor pressure; (b) withdrawing reaction mixture from the reaction vessel and feeding the withdrawn reaction mixture along with additional carbon monoxide to a pre-flasher/post reactor vessel operated at a reduced pressure below the reactor vessel pressure; (c) venting light ends in the pre-flasher vessel and concurrently consuming methyl acetate and/or dimethyl ether in the pre-flasher/post reactor vessel to produce a pre-flash mixture which is enriched in acetic anhydride and diminished in methyl iodide and methyl acetate and/or dimethyl ether as compared with the reaction mixture; (d) withdrawing the pre-flash reaction mixture from the pre-flasher/post reactor vessel and feeding the pre-flash mixture to a flash vessel; and (e) flashing a crude product stream from the mixture in a flash vessel operated at a pressure substantially below the pressure of the pre-flasher/post reactor vessel.