What is the half-reaction that occurs at the cathode during the electrolysis of molten potassium bromide?

The complete ionization of KBr into its constituents is:

KBr (s) - - - > K + (aq) + Br - (aq)

During electrolysis, oxidation takes place at the anode electrode. This means that an ion is stripped off its electron hence becoming more positive:

2 Br - (aq) - - - > Br2 (g) + 2e-

We can see that Bromine gas Br2 is evolved at the anode.

Meanwhile at the cathode, the reduction reaction occurs. Which means that the electron from the anode electrode is used to make an ion more negative:

2K + (aq) + 2e - - - - > 2K (s)

Hence, through reduction, solid potassium is deposited on the plate.

Half reactions:

Anode: 2 Br - (aq) - - - > Br2 (g) + 2e-

Cathode: 2K + (aq) + 2e - - - - > 2K (s)