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What happens when strongly basic water is electrolyzed? I know that the electrolysis of water in strongly acidic solution involves hydrogen protons; will these protons still be involved in strongly basic solution?

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In a basic solution $\ce{H2O}$ will act as the $\ce{H}$-source for the hydrogen evolution reaction. The reaction will look something like:

\begin{equation} \ce{2H2O + 2e- -> 2OH- + H2} \end{equation}

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