Is a Daniell Cell Rechargeable?

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SMD1990
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I will admit that I still have a bit of trouble when it comes to reading the table of standard electrode potentials. (Though, I do think my understanding is improving.)

However, unless I am mistaken, should not Daniell cells (zinc anode, copper cathode, sulfate solutions) not rechargeable?

I mean, during discharge, the zinc could be thought of as converting to zinc sulphate, and the copper sulfate to copper.

Now, to reverse the process...

Clearly, the problem is not the zinc. Zinc sulphate can be used to electroplate zinc onto a cathode.

Since copper appears above oxygen on the table of standard electrode potentials, should it not be what is oxidized at the anode?
 
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As I understand it, the Daniell cell when using copper (II) and zinc (II) sulfate solutions are reversible if you apply an EMF greater than the voltage of the cell (EMF > 1.1 V). However, if the Daniell cell is made as the original cell (Zn electrode in sulfuric acid & CuSO4 solution on a copper pot), the reaction is not reversible, because the reduction of sulfate in acidic solution

SO42-(aq) + 4 H+(aq) + 2 e- -----> SO2(g) + 2 H2O

has a more positive reduction potential (E0 = +0.20) than the reduction of Zn2+ (E0 = -0.76), and it will take precedence. So instead of depositing Zn, you will be seeing bubbles of SO2 coming out of the acidic Zn2+ solution.