Final answer:
During the electrolysis of brine, the half equation for the reaction at the cathode is 2H2O(l) + 2e- → H2(g) + 2OH-(aq), which illustrates the production of hydrogen gas and hydroxide ions.
Step-by-step explanation:
The half equation for the reaction that occurs at the cathode during the electrolysis of brine involves the reduction of water rather than sodium ions. This is because the reduction of water has a higher reduction potential compared to that of sodium ions. The half equation for the reduction of water to produce hydrogen gas at the cathode is as follows:
2H2O(l) + 2e- → H2(g) + 2OH-(aq)
This equation indicates that two water molecules accept two electrons to form one molecule of hydrogen gas and two hydroxide ions. During the process, hydrogen gas is produced at the cathode, which can be observed as bubbles forming at the cathode electrode.