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In many electrochemistry scholarly articles/studies they talk about certain specific reactions occurring at certain voltage potentials, while in other articles they talk about current density as the determiner for which reactions happen.

Why is this? From what I know about electrical engineering you can either set the voltage, and the current will adjust accordingly, or you can set the current and the voltage will adjust to meet the current. You cannot set both of them at the same time. That would defy the laws of physics.

So what do many of these articles mean when they talk about, for instance:

Reported reactions of this type often occur at potentials smaller than -3 V

How does this relate to the current density that is set by a power supply for instance?

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Reaction rate in electrochemistry is limited by diffusion of reactants from solution towards electrode. Once all reactants in near proximity for given type of reactions are exhausted, and if voltage is still applied, other reactions (if voltage is strong enough) may occur. For example, consider positive electrode in $\ce{KCl}$ solution (leaving aside paired negative electrode). It is true, that $\ce{Cl^-}$ anion is oxidized easier then water. However, once all chloride near the electrode is oxidised, the next possible substrate to be oxidised is $\ce{OH^-}$ that is always present is water via reaction $\ce{2H_2O <=> H_3O^+ + OH^-}$. Thus, depending on the current density on the electrode you will have different major products.

In short: resistance of the electrochemical cell is a complex function of temperature, voltage, current and solution composition.

The current density in electrochemistry may be adjusted not only by adjusting voltage applied, but also by adjusting surface area of the electrodes used. Also, it is critical to maintain conditions for effective transportation of reactants towards electrodes.

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