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Consider an electrochemical cell with an inert Pt working electrode (WE) and some electrolyte that irreversibly decomposes upon oxidation. A minimum potential $E_{min}$ (vs. some reference) needs to be applied to the WE to allow oxidation to occur (but most likely a higher value is required to actually measure a current). I would call this potential the thermodynamic stability potential limit (SPL).

Does this minimum potential change when exchanging the WE material with another inert material, e.g. glassy carbon? I think it should not, as all that matters is the electrochemical potential/fermi level in the inert electrode.

Could some type of specific adsorption of electrolyte components change (lower) the thermodynamic SPL? To me this sounds like catalysis, which just influences kinetics, not thermodynamics.

What about adding some other substance? Could its reaction products cause degradation of the electrolyte components in some way?

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Technically you are correct. The thermodynamic potential of electrolyte decomposition should not be affected by the electrode composition.

which just influences kinetics, not thermodynamics.

The word "just" here is doing way more work than you're giving it credit for. There are situation where the kinetics of electrolyte decomposition are so slow that there is effectively no reaction. The stability window of electrolytes commonly used in Lithium-ion batteries, for example, is smaller than the potentials needed to charge the materials. However, once the reaction begins, initial decomposition forms an inter-phasic layer on the anode known as the solide-electrolyte-interphase (SEI). The kinetics of decomposition on this SEI are slow, which allows the battery to be operated at high potentials[1]. In fact, damage to the SEI has been linked to capacity loss in Li-Ion batteries over extended cycling[2].

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