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https://youtu.be/iHBz77E625o

In this video there are pieces of matboard that are soaked in water saturated with NaCl. The pennies themselves are made of zinc and copper. With the electrical energy from the zinc and copper would the chloride let go of its electron and the sodium gain back its electron producing a Na Cl battery within a Zn Cu battery?

Would the water in the process separate into hydrogen and oxygen through electrolysis? If so since hydrogen is very flammable would a combustion reaction occur in the air giving you water again?

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The electrolyte in that clip is a solution of $\ce{NaCl}$ and acetic acid (vinegar). The zinc anode is oxidized, resulting in $\ce{Zn^2+}$ ions going into the electrolyte. The electrons went through the red LED, lighting it, and then on to the copper cathode. From there, they reduced some $\ce{H+}$ to hydrogen gas. So two $\ce{H+}$ ions were reduced to one $\ce{H2}$ molecule (which left: it's a gas) and the $\ce{Zn^2+}$ ion simply replaced the two $\ce{H+}$ ions.

Acetic acid is a weak electrolyte, so a solution of it (vinegar) is not a great conductor of electricity. The $\ce{NaCl}$ just makes the electrolyte a better electrical conductor: movable ions are necessary for a steady flow of electrical current in this battery. Without the $\ce{NaCl},$ most of the generated voltage would have been wasted on the electrolyte's resistance.

No sodium is produced (by reducing sodium ions) because the $\ce{H+}$ ions get the electrons. And no chlorine gas is produced (by oxidizing chloride ions) because the $\ce{Zn}$ anode is more readily oxidized.

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