The Number We Agree to Trust: pH and the Architecture of Convention

"At the end of my last essay, I left off with a question: If a voltmeter never really measures $\phi$ , if it only reports a difference in electrochemical potential, then what happens when we turn to a quantity that feels even more familiar than voltage—pH?"

A pH meter gives a number. Every student learns to read it before learning what it means. Seven is neutral, below is acidic, above is basic. The number feels as solid as a length or a mass. It is calibrated against buffer solutions and reported without hesitation in nearly every experimental paper in chemistry.

But pH is not what it appears to be. Under the same Gibbs–Guggenheim reasoning that undid my confidence in $\phi$ , pH carries its own hidden fragility. I think working through it is the clearest way to see how a field builds a trustworthy number out of a quantity that is, strictly speaking, undefined.

The Quantity Nobody Can Isolate

pH is defined as the negative logarithm of hydrogen ion activity:

\text{pH} = -\log_{10} a(\text{H}^+)

It looks like an ordinary definition. But activity, for a single ionic species, is not something thermodynamics actually hands you. In 1923, Gilbert Lewis and Merle Randall raised the idea of an individual ion activity almost in passing, noting it would be of theoretical interest if such a quantity could be determined. Six years later, Guggenheim, building on Gibbs' philosophy, showed why it could not.

The activity of an isolated ion depends on the electrical state of the phase it sits in, and that electrical state is not something you can separate from the ion's chemical surrounding. You cannot move a hydrogen ion across a phase boundary and account for solvation, bonding, and electrostatics as three independent contributions. The electrical part and the chemical part arrive together or not at all.

This is the same trap described in my previous essay regarding $\phi$ :

\tilde{\mu}_i = \mu_i + z_i F \phi

is a modeling choice, not a physical separation. Apply that same logic to a single ion's activity coefficient, and it dissolves the same way. Individual ion activity is not a hard-to-measure quantity. It is not a quantity at all, in the strict thermodynamic sense. IUPAC's own 2002 recommendation on pH measurement says as much, describing the standard definition as "notional."

Thus the number printed on a pH meter is reporting something that, by the field's own admission, cannot be pinned down by any thermodynamically valid method.

Choosing to Trust, Not to Discover

The Bates–Guggenheim convention does not discover the "true" activity of a hydrogen ion. It cannot, because that quantity does not exist in the sense we usually mean by "true." What it does is fix enough of the system, by agreement, that a number can be produced reproducibly, by anyone, anywhere, using the same procedure.

Key InsightThis is exactly the move I was circling at the end of the last essay without naming it. $\Delta \tilde{\mu}_{e^-}$ is operational. It is not a claim about what $\phi$ truly is; it is a statement about what a specific measurement, under a specific protocol, reliably returns. pH is the same idea taken one step further. It is not a claim about the true state of a proton. It is a number that a specific cell, under a specific convention, reliably returns.

The field didn't solve the problem of individual ion activity. It set the problem aside and built something more useful: a convention rigorous enough that everyone measuring pH is, in effect, measuring the same thing, even though none of them are measuring the thing the equation appears to name.

A Pattern That Keeps Repeating

I have started to notice that this move shows up again and again in the history of electrochemistry, and I find it useful to think of it as two separate lineages, each asking a different kind of question about the same physical systems:

Lineage I

The Operationalist Lineage

Cares about measurability first. Its central question is not "what is happening," but "what can we actually pin down, and by what agreed convention." This is the line that runs from Nernst's electromotive force framework, through Gibbs's observation that certain potentials are not independently accessible, through Guggenheim's refutation of individual ion activities, through the Bates–Guggenheim convention that rescues pH as a usable number, and into more recent work on measurement architectures for batteries and electrolytes.

Lineage II

The Mechanistic Lineage

Asks a different question entirely: what is the microscopic mechanism? This is the line that runs from Butler's early charge-transfer picture, through Marcus and Hush's electron-transfer theory, through Gerischer, then Chidsey's application to electrode kinetics, into proton-coupled electron transfer (PCET), and now into coupled ion-electron transfer (CIET) theory for battery intercalation.

Both lineages use the same words: activity, potential, kinetics, overpotential. But they are not answering the same question, and I think conflating them is a quiet source of confusion in how electrochemists talk to each other across subfields.

Why This Matters for Batteries

The question of whether Butler–Volmer (BV), modified-BV or coupled ion-electron transfer (CIET) theory correctly describes Li-intercalation kinetics belongs to the mechanistic lineage. It asks what is physically happening at the interface when a lithium ion and an electron cross together.

But my earlier essay on the Measurement Challenge showed something uncomfortable: every popular method for extracting battery kinetics runs into the same wall that individual ion activity ran into. The quantity we want is entangled with a shifting thermodynamic reference (which is called chemical capacitance) that we cannot cleanly subtract away. We are, in a sense, trying to answer a mechanistic-lineage question before we have done the operationalist-lineage work of deciding what, exactly, can be pinned down and agreed upon.

This is not unique to batteries. For instance, in the catalytic field, the Surendranath group has run into a version of the similar problem in electrocatalysis, and their response is instructive: they borrow tools built for measurability (wireless potentiometry, membrane-separated double cells) to ask mechanistic questions about proton-coupled electron transfer. The tool comes from one lineage. The question comes from the other. That pairing, not a new mechanism, is often what breaks a stalemate.

"If measurability, not mechanism, has always been the harder problem, if pH only became a usable number once the field stopped asking 'what is the true proton activity' and started asking 'what can we consistently define and reproduce,' then the CIET-versus-Butler-Volmer question in batteries may be arriving one step too early. Before we can decide which kinetic law holds at high overpotential, we may first need our own Bates–Guggenheim moment: not a better mechanism, but a convention rigorous enough to make the measurement itself trustworthy."

What that convention should look like, for a lithium-intercalation electrode, is the question I want to spend the next several years on.

References & Key Literature

Lewis, G. N. & Randall, M. Thermodynamics and the Free Energy of Chemical Substances. McGraw-Hill (1923).
Guggenheim, E. A. "The conceptions of electrical potential difference between two phases and the individual activities of ions." J. Phys. Chem. 33, 842–849 (1929).
Buck, R. P. et al. "Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002)." Pure Appl. Chem. 74, 2169–2200 (2002).
Pethica, B. A. "Are electrostatic potentials between regions of different chemical composition measurable? The Gibbs–Guggenheim principle reconsidered, extended and its consequences revisited." Phys. Chem. Chem. Phys. 9, 6253–6262 (2007).