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I’m teaching myself electrochemistry, and this question is from the general introduction of the book I’m using. According to Bard’s Electrochemical Methods book, in a given cell, the background limits ‘are the potentials where the cathodic and anodic currents start to flow at a working electrode when it is immersed in a solution containing only an electrolyte added to decrease the solution resistance (a supporting electrolyte).’ I don’t fully understand the significance of the italicized, but my main question comes from the following figure, which shows that the onset of H+ reduction and Br- oxidation (Where the background limits are) occur at different potentials. This is counter-intuitive for me, since, according to introductory electrochemistry from gen chem courses, one half of a reaction cannot occur without the other for a given applied potential. From Allen J. Bard’s Electrochemical Methods

How can this occur?

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the background limits ‘are the potentials where the cathodic and anodic currents start to flow at a working electrode when it is immersed in a solution containing only an electrolyte added to decrease the solution resistance (a supporting electrolyte).

Bard's book is notoriously complex. Some bright scientists are not good at communication. Richard Feynman was an exception.

All Bard is saying that the background limits define the potential window [see flat line in your figure] where electrolysis does not occur on the working electrode. The background limits are dependent on the electrolyte used, it is also dependent on the solvent and the electrode material as well.

Now keep in mind that modern potential sweep experiments there are three electrodes in a cell. There is a reference electrode, in your case, an Ag/AgBr electrode. Then you have a working electrode, Pt/H+ and then the writer is quiet about the third one, which is called the counter electrode. Most of the time it is understood that it is their. It could be another Pt electrode but not necessarily a hydrogen electrode i.e., nobody is bubbling hydrogen gas over the counter electrode.

Now the reference electrode has no role in the cell. There is no redox reaction taking place at the reference electrode because the circuitry is such that no or very little current passes through the reference electrode. Electrochemists don't care what is happening at the counter electrode. The potential difference is measured between the reference electrode and the working electrode. This is the only purpose of the reference electrode.

When you are sweeping (=continuously changing) the potential of the working electrode with respect to the reference electrode, it can become a cathode or an anode during scanning of the potential. You can scan in any direction. You are right, isolated reduction or oxidation cannot occur. If the working electrode was a cathode at a particular moment, the counter electrode is the anode and vice versa.

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You are correct that the potential of an isolated half-cell can't be measured. However the whole table of reduction potentials is measured relative to the standard hydrogen electrode which would be a third electrode in the system.

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