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We might have a half cell consisting of the redox couple $\ce{Ag+}/\ce{Ag}$, which for example's sake might be fixed to be at the cathode (i.e. undergoing reduction).

The reaction at the cathode is then

$$\ce{Ag+ + e- -> Ag}$$

The standard electrode potential of the $\ce{Ag+}/\ce{Ag}$ couple is $+0.80\ \mathrm V$ with respect to SHE. I've seen the above reaction written as such:

$$\ce{Ag+ + e- <=> Ag} \quad E = +0.80\ \mathrm V$$

I'm confused as to why we write this electrode potential next to the reaction, considering that the electrode potential of a redox couple is reasonably disjoint to the concept of the chemical reaction. Although, at the end of the day, the chemical reaction results in the interfacial potential difference which causes the electrode potential, $E$ is a property of the electrode, not the chemical reaction.

Is it perhaps just because we want to write the equation of the reaction occurring at a certain electrode and the electrode potential of the electrode on the same line?

I'm aware that for associating potentials with written reactions, there exist two different conventions: European, where all reactions are labelled with the reduction potentials of the couple, and American, where oxidation reactions are labelled with the oxidation potentials of the redox couple and likewise reductions with reduction potentials. However, my question is why we need to attach the electrode potential to the chemical reaction in the first place!

That is, ultimately, aren't redox potentials associated with redox couples/electrodes, not reactions?

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Your question is a valid question, and ignore downvotes. They don't mean anything.

Your understanding is very good and that you realized that the electrode potential is a property of the electrode and it really does not care how the reaction is written. However, a equation is $needed$ to keep track of the electrons lost or gained in the Nernst equation. So as a tradition and a matter of convenience, electrode potentials are quoted along with a balanced half-cell.

Another very important thing to remember (even some university teachers and PhDs lack this understanding), that the sign of the electrode potential in invariant. It does not care how the reaction is written. It is the electrostatic sign of the electrode. This part takes care of the European and American conventions. American convention is due to Gibbs and Latimer and no longer used. Most international electrochemists have agreed to stick to the European convention i.e., write all half reactions as reduction.

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    $\begingroup$ Thank you for your answer, this clears up my doubts. I also agree that the American convention for electrode potential is slightly nonsensical, since it makes little sense to label oxidation reactions with oxidation potentials - as you say, why not just stick with the electrostatic sign...! That is, even if the reaction proceeds as an oxidation at a particular electrode, we may as well just write the oxidation equation (as usual) and label it with a standard electrode potential, representing the electrostatic potential of that electrode - not its negative! $\endgroup$
    – 13509
    Commented Mar 15, 2020 at 22:46
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    $\begingroup$ Having been taught the the American convention it does not seem nonsensical. It is simply based on the fact that if the sum of the the oxidation potential and the reduction potential is positive, then the reaction is spontaneous. It is easy to tell if the reaction is an oxidation or reaction based on which side the electrons are shown in the half-cell reaction. I'll point ot that even in the American convention that all the tables that I have seen are compiled for the reduction potentials. // Having said all of that there is logic in teaching/learning one convention. $\endgroup$
    – MaxW
    Commented Mar 15, 2020 at 23:11
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    $\begingroup$ The main difference in American and European convention is that Gibbs and Latimer treat the electrode potential as a thermodynamic quantity where one can flip flop signs, but the European convention stuck to the sign invariance because they took the plus minus as the sign of electrostatics. It is a very subtle difference but it took ages to settle. $\endgroup$
    – ACR
    Commented Mar 15, 2020 at 23:15
  • $\begingroup$ @JamesWirth, If you don't mind, what grade are you in? I am surprised that modern students are aware of the American vs. European conventions. Are they still teaching them? Not that I am very old but I cannot recall seeing this in modern textbooks. I learned about the because of my own interest in chemical history. $\endgroup$
    – ACR
    Commented Mar 15, 2020 at 23:17
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    $\begingroup$ I'm a senior but my textbook teaches the American convention. It caused quite a lot of confusion because, from learning basic circuit theory, I didn't understand why they were negating electric potentials. So I did some research and found out about the two conventions, which helped a little in terms of rationalising the intentions of the author. Of course, the two methods are algebraically equivalent, but IMO one is more intuitive than the other. $\endgroup$
    – 13509
    Commented Mar 16, 2020 at 6:38

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