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How to know that reaction that occur at the anode electrode is $\ce{2H2O -> 4H+ + O2}$?

Why the reaction is not the oxidation of $\ce{SO4^{2-}}$?

This problem does not give the E0 of $\ce{SO4^{2-}}$ too.

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How to know that reaction that occur at the anode electrode ...

There are two kinds of anions: oxidizable (ones that contain central element in lower oxidation states), like $\ce{ClO3-}$, $\ce{Cl-}$, $\ce{NO2-}$, and unoxidaziable, where central element is in hightest oxidation state ($\ce{ClO4-}$, $\ce{NO3-}$. Fluoride obviously is unoxidizable as well. As a rule of thumb, it is usually assumed that unoxidizable anions do not react in electrolysis. A rationalization for it is that in such cases an oxygen should be oxidized, and oxygen in water reacts faster. For the sake of basic chemistry course it is enough.

However, it is not entirely true: electrolysis of cold solution of ammonium sulfate in sulfuric acid results in formation of ammonium persulfate $\ce{(NH4)2S2O8}$, where anion has structure $\ce{ (O3SOOSO3)^{2-}}$ Similiarly, electrolysis of very cold solutions of carbonates produces various percarbonates. A synthetic method exists, that is based on electrolysis of carboxylic acid. The anions on oxidation form $\ce{RCOO\cdot}$, that further loose $\ce{CO2}$ and resulting radicals pairs, resulting in net reaction $\ce{2RCOO- \rightarrow R2 + 2CO2 + 2 e-}$ However, in general, formation of peroxocompounds requires low temperatures, usually below zero.

The exact mechanism of the process highly depends on the nature of the electrodes, but likely includes formation and transformation of surface particles on the electrode, making reasoning difficult and electrode-specific. Since commonly anodes used have oxide coating, it can be more-or-less safely assumed, that the process includes dissociative adsorption of water with subsequent oxidation of adsorbed (hydro)oxide atoms.

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