Sodium metal is produced at the cathode during the electrolysis of molten NaCl with inert electrodes, where reduction takes place. Concurrently, chlorine gas is liberated at the anode, the site of oxidation. The electrode half-reactions are delineated as:
At the cathode (reduction):
\[
Na^+ + e^- \rightarrow Na \quad \text{(Sodium is deposited)}
\]
At the anode (oxidation):
\[
2Cl^- \rightarrow Cl_2 + 2e^- \quad \text{(Chlorine gas is evolved)}
\]
Therefore, the observed phenomena are the deposition of sodium at the cathode and the evolution of chlorine gas at the anode.