Watch this science video tutorial from Nurd Rage on how to make copper sulfate from copper and sulfuric acid in three ways. They show you how to make copper sulfate from copper and sulfuric acid using two chemical methods and one electrochemical method.
Copper does not directly react with sulfuric acid so some form of oxidant is needed. Hydrogen peroxide and nitric acid are excellent oxidants and the first two methods demonstrate this. The nitric acid method however produces toxic nitrogen dioxide gas so you'll need to perform this one outside or in a fumehood.
Because both chemical methods are rather expensive. A very cheap electrochemical method can be performed by running a current through two copper electrodes immersed in sulfuric acid. The copper is converted to copper ions at the positive electrode and hydrogen gas is formed at the negative electrode.
However if the electrodes are placed side by side then the copper ions will diffuse to the negative electrode and get converted back into copper. This is wasteful as you've used power to accomplish nothing. So the better way is to separate both by gravity with the positive electrode at the bottom. The denser solution of copper sulfate will remain near the bottom and thus increase efficiency.
As a side note, the electrochemical method is actually very expensive if you use batteries as your power source. The charge on a battery generates a comparatively small quantity of total copper sulfate compared to the same cost of chemicals for the chemical methods. The cheapest way is to use a DC power supply. The cost of municipal electricity is very small compared to the cost of batteries and chemicals.
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