[sd79812]

I imagine the battery drives the graphite into a carbocation, then chloride is oxidized to donate the electron to the carbocation, but what pushes the lost electrons of chloride to be shared between two chlorides? That's not favored by gibbs free energy.

The battery creates the carbocation out of graphite? or does the battery pull out the electron from chloride or both?

If the battery pulled the electron out of chloride into the circuit, then there wouldn't be electrons to bond two chlorine atoms together unless the oxidized chloride can pull electrons out of the graphite.

[Hunter2]

There is no carbocation. Graphite is only conductor. The electrical current is the key. From Chloride electrons are attracted from the anode the positive pol. The chloride is oxidised to chlorine and on cathode side the negative pol sodium will be reduced from the donated electrons.

Na+ + e- => Na and Cl- => Cl + e-
Two Cl combine to Chlorine molecule Cl2