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In a 2 L round-bottomed flask placed on a steam bath and fitted with a stirrer, a separatory funnel, and a reflux condenser is placed 540 g (1.9 moles) of commercial (88%) bromoformNote 1. There is then added 10 mL of a solution of sodium arsenite made by dissolving 230 g. (1.16 moles) of c. p. arsenious oxide and 440 g (11 moles) of sodium hydroxide in 1.4 L of water. The mixture is warmed gently to start the reaction, and then the remainder of the sodium arsenite solution is added during about one hour at such a rate that the solution refluxes gently. When the addition is complete, the flask is heated for four hours on the steam bath. The reaction mixture is distilled with steam, the lower layer of methylene bromide separated, and the water layer extracted once with 100 mL of etherNote 2. The methylene bromide is dried with 10 g of calcium chloride and distilled. The yield of slightly yellow liquid boiling at 97-100°C is 290-300 g. (88-90% of the theoretical amount).
Looking at "Phase Transfer Catalysis" by Starks, Liotta and Halpern (Chapman & Hall, 1994) I open book on page 349 to find three pages of info on the subject of "Bromide-Chloride Exchanges" with 21 references.
As far as phase-transfer catalysis goes I can find no specific mention of the reaction needed, but it looks like it would work with DCM, LiBr and Aliquat 336.
Summary of relevant part:
An equilibrium is established using a phase-transfer catalyst: RCl(l) + M+Br-(aq) ←→ RBr + M+Cl-
Cation dependence results from unequal (or non-proportional) solubilities of MCl and MBr for different kinds of M+ in the aqueous phase. The degree of anion hydration affects both the transferability of halide from aqueous to organic phase and the reactivity after transfer.
The equilibrium position of the PTC reaction of a primary alkyl chloride with M+Br- (M = Li, Na, K, Ca) to give the alkyl bromide was found to depend strongly on the nature of the cation M+, and the amount of water present1. RCl was converted to RBr in 80-94% yield by treatment with slight excess of LiBr containing 1% water and 5 mol% Aliquat 336 at 85-110°C. R-Br was converted to R-Cl by treatment with 25% aq. solutions of metal chlorides containing Aliquat 336 at 95°C. Equilibria data is available2. Bulky quaternary salts are the preferred catalysts3,4,5 although even Cetrimide will work6.