During the electrolysis of brine (sodium chloride solution), hydrogen gas (\(H_{2}\)) is produced at the cathode (negative electrode) because water molecules are more easily reduced than sodium ions (\(Na^{+}\)), forming hydrogen gas and hydroxide ions (\(OH^{-}\)). This process also generates sodium hydroxide (caustic soda) and chlorine gas at the anode, making it a crucial industrial process.