Electrode potential

We can now visualize the electrode potential $E$^[1] on a $V_i$ diagram, along with its relatives — the 'redox potential' $E_h$, the 'equilibrium potential' $E_{\mathrm{eq}}$, and so on. They all share the form $E = V_{\mathrm{e}^-} - V^\circ_{\mathrm{e}^-}(\text{ref})$, and every subtlety comes down to which $V_{\mathrm{e}^-}$ and which reference we have in mind. We take the simple case first and add the complications afterward.

Electrode potential at equilibrium

At equilibrium, with no current flowing, an electrode and its solution settle into a tidy stack of levels. The electrode reaction equalizes the metal's electrons with the solution's redox couple, so the metal level and the reaction's implied level coincide,

$$V_{\mathrm{e}^-}(\text{electrode}) = V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}),$$

and the electrode potential is just the gap from that level down to the hydrogen reference level $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$:

$$E = V_{\mathrm{e}^-}(\text{electrode}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE}).$$

On the diagram this is a story of four levels — the metal's $V_{\mathrm{e}^-}$ and the reaction's $V_{\mathrm{e}^-}$ (sitting one on top of the other), the reaction's standard level $V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$, and the reference $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ — and $E$ is just one marked vertical distance among them.

What that reference level actually is, and how a real reference electrode pins it, we leave to the next topic; here we just take $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ as a level on the page.^[2]

The Nernst equation

Set the redundant metal level aside and three remain: the reaction's actual level $V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$, its standard level $V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$, and the reference $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$. The Nernst equation is just the statement that the first sits above the second by a concentration term,

$$E = E^\circ + \frac{RT}{zF}\ln\!\frac{a_{\mathrm{Ox}}}{a_{\mathrm{Red}}},$$

and the three quantities in it are the three gaps among those levels:

$$\begin{aligned} E &= V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE}), \\ E^\circ &= V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE}), \\ \tfrac{RT}{zF}\ln\tfrac{a_{\mathrm{Ox}}}{a_{\mathrm{Red}}} &= V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}) - V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}). \end{aligned}$$

The equation holds locally: every level is read off at the same spot in the solution.

Cracking open the subtleties

Everything so far assumed equilibrium and a single clean reference. Lift those and the complications appear, and each one is a question of which $V_{\mathrm{e}^-}$, or where the reference is.

Surface overpotential. Drive a current and the metal level peels away from the reaction level it was pinned to. That gap is the overpotential, in $V_i$ terms just a drop in $V_{\mathrm{e}^-}$ across the interface,

$$\eta = E - E_{\mathrm{eq}} = V_{\mathrm{e}^-}(\text{electrode}) - V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red}).$$

Even this has slippery edges, and the diagram helps by making them concrete: $V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$ depends on which half-reaction we say is running (for copper deposition, the one-electron $\mathrm{Cu}^+$ step or the overall two-electron one?) and which activities we assign it (a gas-evolving reaction may be wildly supersaturated).^[3]

A spatially varying $E_{\mathrm{eq}}$. The Nernst equation still gives the reaction's equilibrium level, but under load that level is no longer a single number: it varies through the solution, both because concentrations vary (concentration polarization) and because $V^\circ$ itself slopes (the ohmic drop). $E_{\mathrm{eq}}$ has become a field — drawn out, along with the overpotential step itself, just below.

Where is the reference? The deeper gotcha is then where the reference sits, more than what it is. An actual reference electrode samples $V^\circ$ at the point where its tip sits, and the reading then carries the ohmic drop between that point and the working electrode on top, so it depends on where you place it. The usual escape is to flatten the bulk — subtract the $iR$ drop, reach in with a Luggin capillary, or swamp the cell with supporting electrolyte — so the reference samples a position-independent bulk. That works in a roomy, stirred analytical cell; it fails in a battery polarized wall to wall, where there is no flat bulk at all and "the electrode potential" quietly loses its referent. This is exactly where reading $V_{\mathrm{e}^-}(x)$ and $V^\circ(x)$ off the diagram beats clinging to a single number $E$.^[4]

Mixed potentials. Finally, a real electrode may couple to several half-reactions at once, each with its own $E_{\mathrm{eq}}$; the electrode settles at a compromise mixed potential that matches none of them — one $V_{\mathrm{e}^-}$ line sitting among several dashed reaction levels.

Takeaways

At equilibrium the electrode potential is a clean four-level picture, and $E$, $E^\circ$, and the Nernst activity term are simply the vertical gaps among those levels. The famous confusions all arrive together once we leave equilibrium: the metal level splits from the reaction level (overpotential), the reaction level spreads out in space, and the reference stops having a single home. The $V_i$ diagram keeps every one of these as a line you can point at — and the realities of the reference electrode and the full cell come next.

NEXT TOPIC: Reference electrodes & cells