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I am trying to understand electrolysis. My question is: when electrolysis is done with a NaCl solution I have learnt that sodium metal is not formed. Why is this?

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Let's look at these two redox couples: $$\ce{Na+ + e- <=> Na} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E^0(\mathrm{red})=-2.71 V $$ $$\ce{2H2O + 2e- <=> H2 +2OH- }\,\,\,\, E^0(\mathrm{red})=-0.83 V $$ Because it is much easier to reduce water than $\ce{Na+}$ ions, the only product formed at the cathode is hydrogen gas.

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The electrolysis of molten NaCl indeed gives elemental sodium and chlorine gas:

$\ce {2NaCl (l) -> 2Na (l) + Cl2 (g)}$

However, in solution (namely aqueous solution such that you are doing the electrolysis on a solution of NaCl in water), the highly reactive Na produced reacts instantly with water to give sodium hydroxide, hence why no solid Na is ever observed.

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