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In general, is it true that if I connect an aqueous oxidant to an aqueous reductant via a wire then a current flows along the wire? In particular, does current flow in the following scenario?

Imagine putting one end of a wire in hydrogen peroxide (a common oxidizer) and the other end inside water containing formic acid or pure carbon (reductants).

I understand that standard Energizer alkaline batteries are composed of separated $\ce{Zn}$ and $\ce{MnO2}$. Connecting the terminals allows electrons to flow from the $\ce{Zn}$ to the $\ce{MnO2}$, oxidizing/reducing them to $\ce{ZnO}$ and $\ce{MnOOH}$.

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is it true that if I connect an aqueous oxidant to an aqueous reductant via a wire, will a current flow along the wire? In particular, does current flow in the following scenario?

No true. Nothing will happen as the circuit is incomplete. The oxidant or the reductant species have to see each other "face to face". Guess what is the most powerful reducing and oxidizing agent in the world? It is the cathode or the anode respectively. Consider an electrolytic cell which is calm and unstirred. The anode and the cathode are dipping in the solution. The moment you turn on the current, only the species which is next to the electrode gets oxidized or reduced. Nothing happens to the bulk right away, so physical promixity is important in electrochemistry.

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