Generally, the calculation of bandgaps from electrochemical measurements are carried out in an organic solvent in the presence of ferrocene as a reference. How do I calculate the bandgaps by electrochemical measurements in an aqueous medium?
You can do electrochemistry in aqueous solution if you want. Common supporting electrolytes are 0.1-0.5 M KCl or NaCl. However, you will be limited by the "solvent window" (the potentials at which water is reduced to hydrogen and water is oxidized to hydroxyl or oxygen). You can reference using ferricyanide or functionalized ferrocenes that are water soluble (such as ferrocenemethanol). You can also use a Ag/AgCl reference electrode, which has a known value.
The solvent window limitation will mean that you will only be able to measure the band gap if is it relatively small and within the solvent window.