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In most of silver electroplating (or electrorefining) methods, it is mentioned that silver anode is oxidized in aqueous solution of silver nitrate, and that silver is deposited at the cathode (regardless of cathode material, let's assume it's also silver for the sake of simplicity). What I don't get is why would silver anode be oxidized before $\ce{OH-}$? The reduction potentials are ~0.8 V for silver and ~0.4 V for $\ce{OH-}$, at standard conditions.

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Hydroxide is bad for this process. You will have additional problems if hydroxide anion is present.

Silver cation reacts with hydroxide anion to form silver hydroxide, which spontaneously decomposes into silver oxide:

$$\begin{aligned}\ce{Ag+ + OH-}&\ce{ -> AgOH}\\ \ce{2AgOH}&\ce{ -> H2O +Ag2O}\end{aligned}$$

In addition to shutting down the oxidation of silver, hydroxide precipitates any silver cation that does dissolve.

For this reason, the electroplating process is probably run at a slightly acidic pH, and the competing oxidation reaction is not of hydroxide, but of water (reduction potential 1.229 V):

$$\ce{2H2O -> O2 + 4H+ + 4e-}\ E^\circ = +1.229\mathrm{\ V}$$

So long as you don't apply too much of an overpotential to the electroplating operating, silver will oxidize before water.

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