When purifying copper by electrolysis, why does the copper deposit on the cathode rather than sinking ...?

...to the bottom

Answer:
Because the surface of the cathode is where the copper ions actually pick up the electrons.
The copper atoms all of a sudden become insoluble, and stick to the fresh atoms already on the surface using metallic bonding.
In the Copper electrorefining process, the anode is the impure metal and the Cathode becomes the "pure" metal. The impurities are "lost" in solution or fall to the bottom of the cell during the passage of the copper metal from the anode to the cathode during electrolysis. The pure copper "sticks" to the cathode as a result of the flow of electrons. At the Anode the copper is forced to give up its electrons by the reaction: Cu → Cu++ + 2e- and at the Cathode, the copper gets back electrons by the reaction: Cu++ + 2e- → Cu.

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