The by-product of electrolysis of brine is sodium hydroxide. The
reaction that is quoted is 2H2O + 2e - H2 + 2OH- .....which occur
at cathode. The proble is that at cathode cations are discharged
such as here H+ ions are discharged. No idea how water being a
neutral specie migrates to cathode ?? Please explain.
Within the H2O molecule, because O atoms are more electronegative
than H atoms, hence the H atoms, having a partial positive charge,
are electrostatically attracted to, and accept electrons from, the
cathode. For every H2O molecule, 1 H atom accepts 1 electron,
generating an OH- ion and a H atom, which readily combines with
another H atom from the reduction of the next H2O molecule, thus
generating H2 gas (with positive, thermodynamically favorable
entropy change).
If you prefer, an alternative acceptable way to look at it, is that
H+ ions (or H3O+ ions) from the auto-dissociation of H2O molecules,
are electrostatically attracted to, and accept electrons from, the
cathode, to be reduced to H atoms, which readily combines with the
next H atom to generate H2 gas (again with positive,
thermodynamically favorable entropy change), thereby causing the
position of equilibrium of the auto-dissociation of water to shift
to the RHS as predicted by Le Chatelier's principle, thus
generating more OH- ions as a byproduct.
Both ways of looking at this are correct, use whichever you prefer,
whichever helps you understand better, and Cambridge certainly
won't ask for such explanatory details at A levels in any case.