Half-reaction at the anode, Chemistry

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During the electrolysis of water, what are the two half-reactions taking place?

HALF-REACTION AT THE ANODE:

HALF-REACTION AT THE CATHODE:

What is the relative ratio between hydrogen and oxygen when the two half-reactions are balanced and combined? What was oxidized? What was reduced?

Why was no reaction noted in the first attempt at electrolysis? Why did that change when some vinegar was added?

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