When purifying copper by electrolysis, does the copper sulphate solution need to be molten?



Answer:
No, the ions free in solution can conduct electricity needed for electrolysis.

Something like Al2O3 needs to be melted as it is insoluble.
Copper sulfate decomposes before it melts. The ions are freed in solution instead.
Well molten is not the word I believe you're looking for, its diluted...and yes it must be diluted (not all just the amount you want to convert back into copper) in a liquid that's soluble, like water. Thus making the liquid and diluted compound a solution.
P.S. also you can convert copper oxide (the black or white powder that is left when copper burns in air) back into copper by heating it and passing ammonia gas over it. Be careful ammonia gas is some nasty smelling stuff. Anyway I hope that was of some help to you.
Right now, Aqueous copper sulfated solution is the predominate Copper Electrorefining processes. However, molten salt electrolytes are being development. This method would almost halve the energy requirement to refine copper as can use Cu(I) verses Cu(II). In an electrorefining process, the anode is the impure metal and the impurities must be lost during the passage of the metal from the anode to the cathode during electrolysis.

The anode electrode reaction is
M → Mx+ + xe
And the cathode electrode reaction is
Mx+ + xe- → M

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