My physical chemistry professor in his electrochemistry class derived before us, the formula of the electrode potential for a metal- metal insoluble salt- anion half cell and after a few minutes of putting formulas here and there he finally proved before us that the electrode potential for a metal- metal ion electrode and metal- metal insoluble salt-anion electrode is/comes out to be the same, the formula appeals to me, however, I don't quite get the intuition and the reason behind the same. Is it because the anion plays a dormant/ insignificant role in the process, if so then why to take it in the first place?
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The excess of anion ( like $\ce{Cl-}$ ) determines via the solubility product ( like $K_\mathrm{sp}(\ce{AgCl}) = [\ce{Ag+}][\ce{Cl-}]$ ) the concentration of the metal cation ( like $\ce{Ag+}$ ).
So for the silver chloride reference electrode:
$$\begin{align} E(\ce{Ag/Ag+})&=E^{\circ}(\ce{Ag/Ag+}) + 0.059 \log {([\ce{Ag+}])} \\ &= E^{\circ}(\ce{Ag/Ag+}) + 0.059 \log {(K_\mathrm{sp}(\ce{AgCl}))} - 0.059 \log {([\ce{Cl-}])} \\ &= E^{\circ}(\ce{Ag/AgCl/Cl-}) - 0.059 \log {([\ce{Cl-}])} \end{align}$$
