Reference electrodes & cells

In electrochemistry, a single electrode potential can never be measured on its own; you always need a second electrode potential to complete the picture. Together two electrodes form a cell. With the electrode potential machinery just discussed on the previous page, we have a clean visual of a cell to start our discussion.

A reference electrode is ostensibly a way to access (or infer) the standard reference levels (such as $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$) inside the cell. But is this actually possible? And, is the standard hydrogen electrode (SHE), the "zero level" of electrochemistry, actually a firm reference, or is the electrical ground ($V_{\mathrm{e}^-}=0$) of circuits more appropriate, or does the distinction really not matter in the end? What about the vacuum as a reference? A liquid junction potential complicates things further; what does that look like?

While I can't answer all the philosophical questions, what I can do is provide a rigorous visual picture where every quantity is perfectly represented.

Reference electrodes

In the language of half-reactions, a reference electrode is built on a couple chosen for fast, reproducible equilibration: the reaction pins the wire's $V_{\mathrm{e}^-}$ to the couple's implied level in the solution, a definite Nernst offset from the local $V^\circ_i$ ladder. Know the activities and you know exactly where the wire sits relative to that ladder. Two couples do most of this work in practice.

The silver/silver chloride electrode hangs off the chloride rung. Its reaction swaps an electron for a chloride ion, and equilibrium locks the wire to the solution, as we saw back in the Reactions topic:

$$V_{\mathrm{e}^-} = V_{\mathrm{Cl}^-} - \frac{\mu_{\mathrm{Ag}} - \mu_{\mathrm{AgCl}}}{F},$$

which the redox floating Nernst equation is equivalently:

$$\begin{aligned} V_{\mathrm{e}^-} &= V_{\mathrm{e}^-}(\mathrm{Ag/AgCl}) \\ &= V^\circ_{\mathrm{e}^-}(\mathrm{Ag/AgCl}) - \frac{RT}{F}\ln a_{\mathrm{Cl}^-} , \end{aligned}$$

with the solids' fixed chemical potentials absorbed into the couple's standard level, $V^\circ_{\mathrm{e}^-}(\mathrm{Ag/AgCl}) = V^\circ_{\mathrm{Cl}^-} - (\mu_{\mathrm{Ag}} - \mu_{\mathrm{AgCl}})/F$, a rung riding the ladder at a fixed distance below $V^\circ_{\mathrm{Cl}^-}$.

The hydrogen electrode hangs off the proton rung instead, interconverting hydrogen ions and hydrogen gas, $\mathrm{H}^+ + \mathrm{e}^- \rightleftharpoons \tfrac{1}{2}\mathrm{H_2}$:

$$\begin{aligned} V_{\mathrm{e}^-} &= V_{\mathrm{H}^+} - \frac{\mu_{\mathrm{H_2}}}{2F} \\ &= V^\circ_{\mathrm{e}^-}(\mathrm{SHE}) + \frac{RT}{F}\ln\frac{a_{\mathrm{H}^+}}{\sqrt{a_{\mathrm{H_2}}}}, \end{aligned}$$

with $V^\circ_{\mathrm{e}^-}(\mathrm{SHE}) = V^\circ_{\mathrm{H}^+} - \mu^\circ_{\mathrm{H_2}}/2F$. This is the electrode behind the previous topic's reference level: at standard activities the wire lands on $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ itself, the rung the whole $E$ scale hangs from. In practice that standard form is finicky to realize, its nominal $a_{\mathrm{H}^+}=1$ implying an awkward pH of 0 and its "1 bar" of $\mathrm{H_2}$ competing with water vapour, so the standard hydrogen electrode is more idealization than instrument.

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 $\mathrm{HCl}$. Each wire is pinned by its own couple, and the two standard rungs ride the same ladder at a fixed, tabulated spacing, so the cell voltage follows by subtracting the two readout forms. (For now we run everything ideal-dilute, each activity read as a plain concentration, $a_i = m_i/m^\circ$ and $a_{\mathrm{H_2}} = p_{\mathrm{H_2}}/p^\circ$; the real activities get their turn below.)

The measured cell voltage comes out as

$$\begin{aligned} \Delta V & = V_{\mathrm{e}^-}(\text{right}) - V_{\mathrm{e}^-}(\text{left}) \\ & = E^\circ_{\mathrm{cell}} + \frac{RT}{F}\ln\!\bigg(\frac{\sqrt{p_{\mathrm{H_2}}/p^\circ}}{(m_{\mathrm{H}^+}/m^\circ)\,(m_{\mathrm{Cl}^-}/m^\circ)}\bigg), \end{aligned}$$

with

$$E^\circ_{\mathrm{cell}} = V^\circ_{\mathrm{e}^-}(\mathrm{Ag/AgCl}) - V^\circ_{\mathrm{e}^-}(\mathrm{SHE}) = 0.222~\mathrm{V},$$

the familiar standard potential of the silver chloride electrode against the SHE, now visibly a gap between two standard rungs.

The traditional picture and its limitations

The usual practice in electrochemistry is to put the standard hydrogen electrode (or some other standard reference) at 0 V, making the picture solution-centered:

It's such a simpler picture than all the floating $V_{\mathrm{e}^-}$ and $V^\circ_{\mathrm{e}^-}$ levels, so why am I overcomplicating things? Why do I keep choosing one of the $V_{\mathrm{e}^-}$ levels as zero instead?

In short, zeroing $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ only makes sense when you're talking about a single idealized cell. When you go add complications, $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ loses its firm grounding. We already saw one issue for cells under bias (in the previous topic, the SHE had a gradient), but in the next two sections I want to show two of the complications that appear even in quiescent electrochemical cells: junctions (the SHE in every compartment is different) and nonideality (the SHE is only virtual anyway).

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; that 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 in the previous section.^[1] Whenever the $V^\circ_{\mathrm{e}^-}(\mathrm{rxn})$ levels vary in space (across a junction, a Donnan membrane, or a cell polarized wall to wall like the previous topic's battery), "the SHE" itself varies from place to place.

Unlike the junction-free cells above, a junction is a nonequilibrium object, idling at a steady interdiffusion: no species' $V_i$ runs flat across it. Even so, the $V^\circ_i$ rungs step rigidly together across it, by the LJP, the $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ rung along with them, so the reference reads the test solution through exactly the $\mathrm{LJP}$ term above, a step that drifts with the very solution being measured. The figure draws this for a typical construction, the silver chloride electrode sitting in its own $3\ \mathrm{mol/L}$ KCl filling solution behind a porous frit.

Activities: who is moving?

Take activities seriously and the story splits into a reassuring half and an unsettling half.

The reassuring half belongs to the junction-free cell. Its voltage touches the ions only through the charge-neutral product $a_{\mathrm{H}^+}a_{\mathrm{Cl}^-}$, so nonideality enters as real, measurable physics with no convention in sight: run our reference cell across a range of $\mathrm{HCl}$ concentrations (this is the classic Harned cell, and it is exactly the cell drawn in the figure above, there in its ideal-dilute version) and the measured voltage peels away from the ideal-dilute prediction, the gap being exactly $\tfrac{2RT}{F}\ln\gamma_\pm$. A mean activity coefficient, read straight off a voltmeter. The same cell, extrapolated to the dilute limit where the single-ion ambiguity dies away, is how the standard levels get pinned down in the first place;^[2] any tabulated $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$ is, in the end, a theoretical extrapolated level tied to the standard state of the aqueous proton.

The unsettling half is the split. The measured $\gamma_\pm$ is a property of the pair; divide it between $\mathrm{H}^+$ and $\mathrm{Cl}^-$ and no experiment can referee the division — the single-ion ambiguity of the nonideal appendix. A named convention (MacInnes, Bates–Guggenheim) is a rule that assigns a split smoothly across every composition, but the freedom underneath is pointwise: each solution's split is its own free choice. The figure's split slider exercises that freedom directly, exaggerated well past any published convention (the named ones stay within a few tens of millivolts of one another even at $4~\mathrm{mol/kg}$). Watch what moves when you drag it: not the wires, not the implied levels, not $\Delta V$ — only the rungs. Tradition puts this the other way around, saying the single-ion problem "contaminates" the electrode potentials; the diagram shows what that means. Each $E$ is a fine gap, but its lower edge, $V^\circ_{\mathrm{e}^-}(\mathrm{SHE})$, is the conventional object, so $E(\text{left})$ and $E(\text{right})$ shift together with the split while the cell voltage, wire-to-wire, is agreed on by every convention.

Now put two solutions in play, joined by a junction or simply drawn side by side on one plot, and each carries its own ladder with its own independent split. The wires' $\Delta V$ stays fixed while the decomposition $E(\text{right}) - E(\text{left}) + \mathrm{LJP}$ shuffles underneath as the two choices change.^[3] The figure strips the junction construction down to its skeleton to show it: a hydrogen electrode in dilute $\mathrm{HCl}$ on the left, our $\mathrm{KCl}$-jacketed silver chloride reference on the right, and one free split per compartment.

So, which is moving? With two solutions there is only one tenable reading: the wires' $V_{\mathrm{e}^-}$ are the invariants, drawn before any convention is chosen, and "the SHE" is a rung that slides per solution at the whim of bookkeeping. The zero selector lets you try each reading: pin a wire and you have chosen an inertial frame, arbitrary but agreeing with every other such choice about what is moving; pin a local SHE and you are in a rotating frame, where the sweep of the whole sky is an artifact of your anchor. The circuit wins; of the two candidate zeros in the introduction, it is the humble electrical ground that holds firm, and when a reference must be picked, a wire is where to pin it. (Often none must be picked: stay in the $V_i$ and $V_{\mathrm{e}^-}$ levels themselves and every measurable quantity is already a gap, no zero assigned.) The solution-centered view survives in a more modest role, as a local convention: perfectly serviceable inside any one solution, with every cross-solution comparison quietly routed through an equally conventional LJP. The machinery of these conventions, and how far apart they land in practice, is the nonideal appendix's business.

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). Even the number's pedigree is telling: the measurement chain beneath it had to exit the solution through a single-ion activity convention and a model of the water surface.^[4] 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 half-reaction kept reliably at equilibrium, pinning its wire at a known Nernst offset from a standard rung; 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 levels come in two kinds: wire levels, whose gaps a voltmeter delivers, and standard rungs, which sit wherever each solution's activity convention puts them. Anchoring everything to one rung works within a single solution; across solutions, only the circuit-centered quantities keep their meaning. The $V_i$ diagram shows all of them as the separate lines they always were.

NEXT TOPIC: Interface kinetics