Reference electrodes & cells

A single electrode's potential can never be measured on its own; you always need a second electrode to close the circuit. That second electrode is the reference, and the pair of them is a cell. This topic is about the reference electrode, the full cell, the junction that usually sits between the two half-cells, and finally the tempting idea of referencing everything to the vacuum.

Reference electrodes

Two electrodes do most of the reference work in practice.

The silver/silver chloride electrode couples chloride to electrons, as we saw back in the equilibrium topic:

$$V_{\mathrm{Cl}^-} - V_{\mathrm{e}^-} = \frac{1}{F}(\mu_{\mathrm{Ag}} - \mu_{\mathrm{AgCl}}).$$

The hydrogen electrode interconverts hydrogen ions and hydrogen gas, $\mathrm{H}^+ + \mathrm{e}^- \rightleftharpoons \tfrac{1}{2}\mathrm{H_2}$, giving

$$V_{\mathrm{e}^-} = V_{\mathrm{H}^+} - \frac{\mu_{\mathrm{H_2}}}{2F},$$

with $\mu_{\mathrm{H_2}} = \mu^\circ_{\mathrm{H_2}} + RT\ln a_{\mathrm{H_2}}$ depending on the gas pressure. Its standard form is the reference the whole $E$ scale is built on.

A reference cell

Now stick the two together: a hydrogen electrode on the left, a silver chloride electrode on the right, both dipping into the same dissolved $\mathrm{HCl}$. The left electrode couples $V_{\mathrm{e}^-}(\text{left})$ to $V_{\mathrm{H}^+}$, the right couples $V_{\mathrm{Cl}^-}$ to $V_{\mathrm{e}^-}(\text{right})$, and the middle is bridged by the solution's fixed ladder gap $V^\circ_{\mathrm{Cl}^-} - V^\circ_{\mathrm{H}^+} = 1.3601~\mathrm{V}$ (from the standard-state data).

Walking $V_{\mathrm{e}^-}$ across, the measured cell voltage comes out as

$$V_{\mathrm{e}^-}(\text{right}) - V_{\mathrm{e}^-}(\text{left}) = E^\circ_{\mathrm{cell}} + \frac{RT}{F}\ln\!\bigg(\frac{\sqrt{a_{\mathrm{H_2}}}}{a_{\mathrm{H}^+}a_{\mathrm{Cl}^-}}\bigg),$$

with

$$E^\circ_{\mathrm{cell}} = \frac{\mu^\circ_{\mathrm{H_2}}}{2F} + (V^\circ_{\mathrm{Cl}^-} - V^\circ_{\mathrm{H}^+}) - \frac{1}{F}(\mu_{\mathrm{Ag}} - \mu_{\mathrm{AgCl}}) = 0.222~\mathrm{V},$$

the familiar standard potential of the silver chloride electrode against the SHE.

The single-ion activities $a_{\mathrm{H}^+}$ and $a_{\mathrm{Cl}^-}$ are individually ambiguous (just like the placement of the $V^\circ_i$ ladder), but the ambiguity cancels in the charge-neutral product $a_{\mathrm{H}^+}a_{\mathrm{Cl}^-}$, so the measured voltage is unambiguous, as it must be.

Two readings of this $\Delta V$ coexist happily: an engineer sees the electrodes' $V_{\mathrm{e}^-}$ as reservoirs and the reaction as a generic electromotive force pump; a chemist sees a reversible free-energy change, $\Delta G = -zF\,\Delta V$, per formula unit ($z$ electrons passed).

Two ways to read a cell

That derivation took a peculiar route, and it is worth seeing why. We walked the chain

$$V_{\mathrm{e}^-}(\text{left}) \to V_{\mathrm{H}^+} \to V^\circ_{\mathrm{H}^+} \to V^\circ_{\mathrm{Cl}^-} \to V_{\mathrm{Cl}^-} \to V_{\mathrm{e}^-}(\text{right}),$$

stepping from one real species voltage to the next. Call this the circuit-centered reading: track the actual $V_i$ straight across, dipping into single-ion activities only at the two $V_i \to V^\circ_i$ excursions (whose sum, the mean activity, is unambiguous).

Traditional electrochemistry takes the other route, which is solution-centered (really potential-centered). It starts in the middle, at the solution's $\phi$ — a stand-in for the $V^\circ_i$ ladder — and works outward to each electrode:

$$\begin{aligned} \phi(\text{soln}) &\to V^\circ_{\mathrm{H}^+} \to V_{\mathrm{H}^+} \to V_{\mathrm{e}^-}(\text{left}), \\ \phi(\text{soln}) &\to V^\circ_{\mathrm{Cl}^-} \to V_{\mathrm{Cl}^-} \to V_{\mathrm{e}^-}(\text{right}). \end{aligned}$$

The cell voltage is the difference of the two, and $\phi$ cancels. Its appeal is that each electrode is described against a common reference; its cost is that each half now leans on a single-ion activity, and on a $\phi$ that no ion can measure. The $V_i$ picture keeps the same split into two electrodes but anchors it to the real, ladder-based $V^\circ_i$ rather than to $\phi$.

This is the distinction Boettcher et al. draw between the electrode potential (the electrode's own electronic level, our $V_{\mathrm{e}^-}(\text{electrode})$) and the solution potential (the solution's level, our $V^\circ_{\mathrm{e}^-}(\mathrm{rxn})$): two different "potentials" that the bare word runs together.

The liquid junction potential

Real reference electrodes are usually kept in their own clean compartment and wired to the test solution through a porous frit or salt bridge — which means a junction, and a junction means a step. For a cell whose two half-cells are different solutions, the measured voltage splits as

$$\begin{aligned} \Delta V &= V_{\mathrm{e}^-}(\text{right}) - V_{\mathrm{e}^-}(\text{left}) \\ &= E(\text{right}) - E(\text{left}) + \mathrm{LJP}, \end{aligned}$$

where the liquid junction potential is the step in the reference level across the junction, $\mathrm{LJP} = V^\circ_{\mathrm{e}^-}(\mathrm{H}^+/\mathrm{H_2}, \text{right}) - V^\circ_{\mathrm{e}^-}(\mathrm{H}^+/\mathrm{H_2}, \text{left})$; that reference level is the local standard hydrogen level $V^\circ_{\mathrm{e}^-}(\mathrm{H}^+/\mathrm{H_2})$, made explicit below.^[1] The point worth dwelling on is that whenever the $V^\circ_{\mathrm{e}^-}(\mathrm{rxn})$ levels vary in space — across a junction, a Donnan membrane, or under load — "the SHE" itself varies from place to place. There is a reason a perfectly defined reference is a fiction.

What a "standard electrode" really is

The chains above show what the standard reference levels are: a "standard electrode" is the hypothetical electrode that would sit at a given reaction's standard level $V^\circ_{\mathrm{e}^-}(\mathrm{rxn})$ — the floating standard-redox levels we tabulated in the half-reactions topic. Setting the reactant activities to one in our two electrodes gives

$$\begin{aligned} V^\circ_{\mathrm{e}^-}(\mathrm{SHE}) &= V^\circ_{\mathrm{H}^+} - \frac{\mu^\circ_{\mathrm{H_2}}}{2F}, \\ V^\circ_{\mathrm{e}^-}(\mathrm{Ag/AgCl}) &= V^\circ_{\mathrm{Cl}^-} - \frac{1}{F}(\mu_{\mathrm{Ag}} - \mu_{\mathrm{AgCl}}), \end{aligned}$$

and their difference is again the $0.222~\mathrm{V}$ from above. Re-drawn with only the electronic levels, the cell is just two $V_{\mathrm{e}^-}$ values sitting against two standard levels:

In practice the SHE is finicky to pin down: its nominal $a_{\mathrm{H}^+}=1$ implies an awkward pH of 0, its "1 bar" of $\mathrm{H_2}$ competes with water vapour, and like every standard level it must be reached by extrapolation from dilute cells (the junction-free Harned cell being the classic).^[2] Any "$V_{\mathrm{e}^-}(\mathrm{SHE})$" is, in the end, a theoretical extrapolated level tied to the standard state of the aqueous proton, $V^\circ_{\mathrm{H}^+}$.

The "absolute" electrode potential

Could we sidestep all this by referencing to the vacuum instead — an "absolute" electrode potential? On a $V_i$ diagram the vacuum is just one more level, $\phi_{\mathrm{vac}} = V_{\mathrm{e}^-} - \Phi/e$, sitting $\Phi/e$ below the metal's electrons on this voltage axis (equivalently a work function $\Phi$ above them in electron energy — the step we drew for capacitors). The widely-quoted "absolute" value of about $4.44~\mathrm{V}$ for the SHE is best read as an electrode's work function: a genuine surface property that drifts with preparation and contamination, not a cleaner fundamental reference, and the in-material $\phi$ it leans on is not well defined to begin with (the subject of $\phi$ under the microscope). The vacuum offers no escape — it is one more floating level, handy for lining up work functions, not a universal zero. Where the $4.44~\mathrm{V}$ comes from, and what vacuum levels are honestly good for, is covered in Vacuum levels.

Takeaways

A reference electrode is a device for pinning one $V^\circ_{\mathrm{e}^-}(\mathrm{rxn})$ level so a working electrode's $V_{\mathrm{e}^-}$ can be read against it; a cell is two such electrodes; and a junction between them adds a liquid-junction step. The whole zoo of "potentials" — electrode potential, solution potential, cell voltage, liquid junction potential, the absolute reference — are particular gaps among the $V_{\mathrm{e}^-}$ and $V^\circ_{\mathrm{e}^-}(\mathrm{rxn})$ levels, and the $V_i$ diagram simply shows them as the separate lines they always were.

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